Physics Book - User contributions [en]

Field of a Charged Rod

2024-04-15T07:11:14Z

Sthomas386:

'''Sajan Thomas, Spring 2024'''

== The Main Idea ==

Previously, we've learned about the electric field of a point particle. While this skill does build the fundamentals of E&M physics, the real world is not simply made up of isolated particles. In fact, the world is consisted of many geometric shapes our one equation for the electric field of a particle cannot account for. One such example of these geometric shapes is a '''rod'''.

For the purposes of this course's scope, we assume that the charge is uniformly distributed along the rod. This is an important assumption to make since if it weren't the case, a lot of the calculations would change. This in mind, let's start with what we know: the electric field of a point particle. By itself, we can't represent a rod. But, if we stack a lot of charged particles together closely in a line, what do we get? a rod. That is the basis of how we calculate the electric field of a charged rod.

That's the basis of it but for now, we'll keep that for later when we're actually deriving how to find it. First, let's discuss the factors that can contribute to the electric field of the charged rod. Firstly, as with any electric field, we have to know the charge of the source-- in this case, the rod. Secondly, we have to know the length of the rod. Thirdly, we also should know the observation location to find the radius from the rod.

The process of finding the electric field due to charge distributed over an object has four steps:

1. Consider the rod, as previously discussed, as a collection of many, many, infinitesimally small charged particles. Make a diagram and draw the electric field <math>\Delta \vec{E}</math> contributed by one of the pieces.

2. By drawing the vectors for the <math>\Delta \vec{E}</math> contributed to one piece, and repeating it along the rod, you should realize by symmetry that several of the vector components will cancel out.

3. Add up the contributions of all pieces, either numerically or symbolically. While there is a formula we can just apply simply later, this step requires integration as you are adding all of the <math>\Delta \vec{E}</math> contributions of many, many, charged particles.

4. Check that the result is physically correct.

=== A Mathematical Model ===
The process of calculating a uniformly charged rod's electric field is tedious, but breaking the process down into several steps makes the task at hand easier. Consider a uniformly charged thin rod of length <math>L</math> and positive charge <math>Q</math> centered on and lying along the x-axis. The rod is being observed from above at a point on the y-axis.

[[Image:SajanThomas1.png|800px|center|thumb|Figure 2: Problem 2]]

'''Step 1: Divide the Distribution into Pieces; Draw <math>\Delta \vec{E}</math>'''

Imagine dividing the rod into a series of very thin slices, each with the same charge <math>\Delta Q</math>. This charge <math>\Delta Q</math> is a small part of the overall charge. Picture it as a point charge. Each slice contributes its own electric field, <math>\Delta E</math>. Summing all these individual slices of <math>E</math> gives you the total electric field of the rod. This process approaches taking an integral, as each thickness approaches 0 and the the number of slices approaches infinity. Note that in this example, the variable that is changing for each slice is its x-coordinate.

[[Image:SajanThomas2.png|800px|center|thumb|Figure 2: Problem 2]]

'''Step 2: Write an Expression for the Electric Field Due to One Piece'''

The second step is to write a mathematical expression for the field <math>\Delta E</math> contributed by a single slice of the rod. We use the formula of the electric field for a point charge because we are imagining each slice as a point charge. First, determine <math>r</math>, the vector pointing from the source to the observation location. For our example, this is <math> r = obs - source = <0,y,0> - < x,0,0> = <-x,y,0></math>. Now use this to calculate the magnitude and direction of <math>r</math>. So <math>|\vec{r}| = \sqrt{(-x)^2 + y^2} = \sqrt{x^2 + y^2}</math> and <math>\hat{r} = \frac{\vec{r}}{\hat{r}} = \frac{< -x,y,0>}{\sqrt{x^2 + y^2}} </math>. <math> \hat{r}</math> is the vector portion of the expression for the field. The scalar portion is <math> \frac{1}{4\pi\epsilon_0} \cdot \frac{\Delta Q}{|\vec{r}|^2}</math>. Thus the expression for one slice of the rod is:
<math> \Delta \vec{E} = \frac{1}{4\pi\epsilon_0} \cdot \frac{\Delta Q}{(x^2+y^2)^{3/2}} \cdot < -x,y,0> </math>.

'''Determining <math>\Delta Q</math> and the integration variable'''

In the first step, we determined that the changing variable for this rod was its x-coordinate. This means the integration variable is <math> dx</math>. We need to put this integration variable into our expression for the electric field. More specifically, we need to express <math>\Delta Q</math> in terms of the integration variable. Recall that the rod is uniformly charged, so the charge on any single slice of it is: <math>
\Delta Q = (\frac{\Delta x}{L})\cdot Q</math>. This quantity can also be expressed in terms of the charge density.

'''Expression for <math> \Delta \vec{E}</math>

Substitute the expression for the integration variable into the formula for the electric field of one slice. Separating the equation into separate x and y components, we get <math> \Delta \vec{E_x} = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{L} \cdot \frac{-x}{(x^2+y^2)^{3/2}} \cdot dx </math> and <math> \Delta \vec{E_y} = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{L} \cdot \frac{y}{(x^2+y^2)^{3/2}} \cdot dx </math>. Note that we have replaced <math> \Delta x </math> with <math> dx</math> in preparation for integration.

'''Step 3: Add Up the Contributions of All the Pieces'''

The third step is to sum all of our slices. We can go about this in two ways. One way is with numerical summation, or separating the object into a finite number of small pieces, calculating the individual contributing electric fields, and then summing them. Another, more precise method is to integrate. Most of the work of finding the field of a uniformly charged object is setting up this integral. If you have reached the correct expression to integrate, the rest is simple math. The bounds for integration are the coordinates of the start and stop of the rod. In this example the bounds are from <math>-L/2</math> to <math>+L/2</math>. So the expression is <math> \int\limits_{-L/2}^{L/2}\ \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{L} \cdot \frac{y}{(x^2+y^2)^{3/2}} \cdot dx. </math> Solving this gives the final expression <math> E_y = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{x} \cdot \frac{1}{(\sqrt{x^2+ (L/2)^2})} </math>. Note that the field parallel to the x axis is zero. This can be observed due to the symmetry of the problem.
This equation can be written more generally as <math> E = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{r} \cdot \frac{1}{(\sqrt{r^2+ (L/2)^2})} </math> where r represents the distance from the rod to the observation location.

'''Step 4:Checking the Result'''

Finally, the fourth step is to check the result. The units should be the same as the units of the expression for the electric field for a single point particle <math> E = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{r^2} </math>.
Our answer has the right units, since <math> \frac{1}{(\sqrt{x^2+ (L/2)^2})} </math>. has the same units of <math>\frac{Q}{r^2}</math>

Is the direction qualitatively correct? We have the electric field pointing straight away from the midpoint of the rod, which is correct, given the symmetry of the situation. The vertical component of the electric field should indeed be zero.

=== Computational Models ===

'''Finding the Electric Field from a Rod with Code'''
While computation done by hand does have its merits, and certainly was the
methodology used when these ideas were conceived, there are much more efficient,
powerful, and most importantly, pretty ways to go about finding and showing the
electric field. This is of course referring to computers, specifically computers
running glowscript code in the case of Physics 2212. With that idea in mind,
here are some demonstrations of said methodology:

This is some code that you can run which shows the electric field vector
at a given distance from the rod along its length. The rod is shown as a
series of green balls to help emphasize that when using the numerical
integrations mentioned on this page, you are measuring the field produced
by discrete parts of the rod being analyzed. At each point of analysis,
eight field arrows are shown so as to visualize the electric field.

Notice the edge-effects of the electric field of the rod. For reasons
discussed above, if we used the long rod approximation (L>>d), these
effects would be negligible.

[http://www.glowscript.org/#/user/yoderlukas/folder/Public/program/ElectricFieldAlongRodLength '''Click Here to Run the Code''']

For a more precise model, the two links below lead to code that generates a
forty element line of charge with given magnitude, and length, and then
iterates vectors representing the electric field in the space all around said
line of charge. Note that the vectors are small, but for the positively charged
rod, they lead radially outward, and for the negative, radially inward. This is
due to the fact that the rod(s) are treated as lines of positive or negative
charge, and the electric field behaves as such.

Try zooming in and out! You can really see the symmetry of the field far out,
and the edge effects when zoomed in.

[https://www.glowscript.org/#/user/michaelwise/folder/Public/program/LineofCharge-Positive '''Positive Charge''']

[https://www.glowscript.org/#/user/michaelwise/folder/Public/program/LineofCharge-Negative '''Negative Charge''']

==Examples==

Although this is not a very difficult topic, some reasonably difficult conceptual questions can be asked about it.

===Simple===

[[Image:LukasYoderNo.jpg|800px|center|thumb|Figure1: Problem 1]]

===Middling===

[[Image:LukasYoderMaybe.jpg|800px|center|thumb|Figure 2: Problem 2]]

===Difficult===

[[Image:LukasYoderYes.jpg|800px|center|thumb|Figure 3: Problem 3]]

== Connectedness ==
The electrical field of a charged rod has many real world applications. Even within other areas of physics, you can extend what we know about a charged rod to find out the electric field of other objects. For instance, a ring is also just a rod that is bent in a circle. Further, a disk is a collection of many concentric rings. This method of thinking can lead us to some very interesting derivations for physics.

Even if we are to only think about real life, rods exist everywhere. Since everything has an electrical field, any rod or even simply cylindrical shape or anything made of rods is an application. A very useful example that also comes up in physics is the concept of wires. Wires are, in essence, thin rods in which electrons can flow through. This categorizes them as a charged rod. Wires are not only an important subject in physics, but also real life as the device you are reading this on probably needed a wire for charge! What happens inside of the wire is important, but we also what happens outside of it is equally important. If we consider static electricity, a misunderstanding or a failure to account for static could be detrimental at a large scale as static can very easily cause malfunctioning to anything that uses wires to operate.

I, the most recent editor of this page, see the use of the electric field of charged rods as a CS major constantly for a similar reason: wires. Computer Science and any form of software really only exists with accompanying hardware. Almost always, the connection between software and hardware is wires and the electrical pulses that serve as the data that travels along the wire. Especially as someone looking into robotics, it is important to understand the outward effect that wires have because static can be very tricky if you don't know how to handle it!

== History ==

Physicists and scientists make use of electric fields and charged objects all the time. Many times, we may need to know which objects are contributing how much charge in certain areas. Charged objects may attract or repel (depending on the signs of their charge), so we often need to know how objects will interact with each other based on their charges. The phenomenon of this interaction, or electric force between charged particles, was finally confirmed and stated as a law in 1785 by French physicist Charles-Augustin de Coulomb, hence "Coulomb's Law."

== See also ==

The equation for the electric field of a charged rod was derived from the equation for the electric field of a charged particle. See the article "[[Electric Field]]" for more information.

=== Further Reading ===

The page on electric fields: [[Electric Field]]

=== External Links ===
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elelin.html

http://online.cctt.org/physicslab/content/phyapc/lessonnotes/Efields/EchargedRods.asp

https://pages.uncc.edu/phys2102/online-lectures/chapter-02-electric-field/2-4-electric-field-of-charge-distributions/example-1-electric-field-of-a-charged-rod-along-its-axis/

http://dev.physicslab.org/Document.aspx?doctype=3&filename=Electrostatics_ContinuousChargedRod.xml

== References ==

https://www.glowscript.org/#/

https://rhettallain_gmail_com.trinket.io/intro-to-electric-and-magnetic-fields#/electric-fields/multiple-charges

https://www.youtube.com/watch?v=BBWd0zUe0mI

(For the above reference, the textbook's method is followed in that the charge distribution was left undefined, and assumed to be constant)

Chabay and Sherwood: Matter and Interactions, Fourth Edition, Chapter 15

[[Category: Electric Field]]

Field of a Charged Rod

2024-04-15T04:28:23Z

Sthomas386:

'''Sajan Thomas, Spring 2024'''

== The Main Idea ==

Previously, we've learned about the electric field of a point particle. While this skill does build the fundamentals of E&M physics, the real world is not simply made up of isolated particles. In fact, the world is consisted of many geometric shapes our one equation for the electric field of a particle cannot account for. One such example of these geometric shapes is a '''rod'''.

For the purposes of this course's scope, we assume that the charge is uniformly distributed along the rod. This is an important assumption to make since if it weren't the case, a lot of the calculations would change. This in mind, let's start with what we know: the electric field of a point particle. By itself, we can't represent a rod. But, if we stack a lot of charged particles together closely in a line, what do we get? a rod. That is the basis of how we calculate the electric field of a charged rod.

That's the basis of it but for now, we'll keep that for later when we're actually deriving how to find it. First, let's discuss the factors that can contribute to the electric field of the charged rod. Firstly, as with any electric field, we have to know the charge of the source-- in this case, the rod. Secondly, we have to know the length of the rod. Thirdly, we also should know the observation location to find the radius from the rod.

The process of finding the electric field due to charge distributed over an object has four steps:

1. Consider the rod, as previously discussed, as a collection of many, many, infinitesimally small charged particles. Make a diagram and draw the electric field <math>\Delta \vec{E}</math> contributed by one of the pieces.

2. By drawing the vectors for the <math>\Delta \vec{E}</math> contributed to one piece, and repeating it along the rod, you should realize by symmetry that several of the vector components will cancel out.

3. Add up the contributions of all pieces, either numerically or symbolically. While there is a formula we can just apply simply later, this step requires integration as you are adding all of the <math>\Delta \vec{E}</math> contributions of many, many, charged particles.

4. Check that the result is physically correct.

=== A Mathematical Model ===
The process of calculating a uniformly charged rod's electric field is tedious, but breaking the process down into several steps makes the task at hand easier. Consider a uniformly charged thin rod of length <math>L</math> and positive charge <math>Q</math> centered on and lying along the x-axis. The rod is being observed from above at a point on the y-axis.

[[Image:SajanThomas1.png|800px|center|thumb|Figure 2: Problem 2]]

'''Step 1: Divide the Distribution into Pieces; Draw <math>\Delta \vec{E}</math>'''

Imagine dividing the rod into a series of very thin slices, each with the same charge <math>\Delta Q</math>. This charge <math>\Delta Q</math> is a small part of the overall charge. Picture it as a point charge. Each slice contributes its own electric field, <math>\Delta E</math>. Summing all these individual slices of <math>E</math> gives you the total electric field of the rod. This process approaches taking an integral, as each thickness approaches 0 and the the number of slices approaches infinity. Note that in this example, the variable that is changing for each slice is its x-coordinate.

[[Image:SajanThomas2.png|800px|center|thumb|Figure 2: Problem 2]]

'''Step 2: Write an Expression for the Electric Field Due to One Piece'''

The second step is to write a mathematical expression for the field <math>\Delta E</math> contributed by a single slice of the rod. We use the formula of the electric field for a point charge because we are imagining each slice as a point charge. First, determine <math>r</math>, the vector pointing from the source to the observation location. For our example, this is <math> r = obs - source = <0,y,0> - < x,0,0> = <-x,y,0></math>. Now use this to calculate the magnitude and direction of <math>r</math>. So <math>|\vec{r}| = \sqrt{(-x)^2 + y^2} = \sqrt{x^2 + y^2}</math> and <math>\hat{r} = \frac{\vec{r}}{\hat{r}} = \frac{< -x,y,0>}{\sqrt{x^2 + y^2}} </math>. <math> \hat{r}</math> is the vector portion of the expression for the field. The scalar portion is <math> \frac{1}{4\pi\epsilon_0} \cdot \frac{\Delta Q}{|\vec{r}|^2}</math>. Thus the expression for one slice of the rod is:
<math> \Delta \vec{E} = \frac{1}{4\pi\epsilon_0} \cdot \frac{\Delta Q}{(x^2+y^2)^{3/2}} \cdot < -x,y,0> </math>.

'''Determining <math>\Delta Q</math> and the integration variable'''

In the first step, we determined that the changing variable for this rod was its x-coordinate. This means the integration variable is <math> dx</math>. We need to put this integration variable into our expression for the electric field. More specifically, we need to express <math>\Delta Q</math> in terms of the integration variable. Recall that the rod is uniformly charged, so the charge on any single slice of it is: <math>
\Delta Q = (\frac{\Delta x}{L})\cdot Q</math>. This quantity can also be expressed in terms of the charge density.

'''Expression for <math> \Delta \vec{E}</math>

Substitute the expression for the integration variable into the formula for the electric field of one slice. Separating the equation into separate x and y components, we get <math> \Delta \vec{E_x} = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{L} \cdot \frac{-x}{(x^2+y^2)^{3/2}} \cdot dx </math> and <math> \Delta \vec{E_y} = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{L} \cdot \frac{y}{(x^2+y^2)^{3/2}} \cdot dx </math>. Note that we have replaced <math> \Delta x </math> with <math> dx</math> in preparation for integration.

'''Step 3: Add Up the Contributions of All the Pieces'''

The third step is to sum all of our slices. We can go about this in two ways. One way is with numerical summation, or separating the object into a finite number of small pieces, calculating the individual contributing electric fields, and then summing them. Another, more precise method is to integrate. Most of the work of finding the field of a uniformly charged object is setting up this integral. If you have reached the correct expression to integrate, the rest is simple math. The bounds for integration are the coordinates of the start and stop of the rod. In this example the bounds are from <math>-L/2</math> to <math>+L/2</math>. So the expression is <math> \int\limits_{-L/2}^{L/2}\ \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{L} \cdot \frac{y}{(x^2+y^2)^{3/2}} \cdot dx. </math> Solving this gives the final expression <math> E_y = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{x} \cdot \frac{1}{(\sqrt{x^2+ (L/2)^2})} </math>. Note that the field parallel to the x axis is zero. This can be observed due to the symmetry of the problem.
This equation can be written more generally as <math> E = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{r} \cdot \frac{1}{(\sqrt{r^2+ (L/2)^2})} </math> where r represents the distance from the rod to the observation location.

'''Step 4:Checking the Result'''

Finally, the fourth step is to check the result. The units should be the same as the units of the expression for the electric field for a single point particle <math> E = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{r^2} </math>.
Our answer has the right units, since <math> \frac{1}{(\sqrt{x^2+ (L/2)^2})} </math>. has the same units of <math>\frac{Q}{r^2}</math>

Is the direction qualitatively correct? We have the electric field pointing straight away from the midpoint of the rod, which is correct, given the symmetry of the situation. The vertical component of the electric field should indeed be zero.

=== Computational Models ===

'''Finding the Electric Field from a Rod with Code'''
While computation done by hand does have its merits, and certainly was the
methodology used when these ideas were conceived, there are much more efficient,
powerful, and most importantly, pretty ways to go about finding and showing the
electric field. This is of course referring to computers, specifically computers
running glowscript code in the case of Physics 2212. With that idea in mind,
here are some demonstrations of said methodology:

This is some code that you can run which shows the electric field vector
at a given distance from the rod along its length. The rod is shown as a
series of green balls to help emphasize that when using the numerical
integrations mentioned on this page, you are measuring the field produced
by discrete parts of the rod being analyzed. At each point of analysis,
eight field arrows are shown so as to visualize the electric field.

Notice the edge-effects of the electric field of the rod. For reasons
discussed above, if we used the long rod approximation (L>>d), these
effects would be negligible.

[http://www.glowscript.org/#/user/yoderlukas/folder/Public/program/ElectricFieldAlongRodLength '''Click Here to Run the Code''']

For a more precise model, the two links below lead to code that generates a
forty element line of charge with given magnitude, and length, and then
iterates vectors representing the electric field in the space all around said
line of charge. Note that the vectors are small, but for the positively charged
rod, they lead radially outward, and for the negative, radially inward. This is
due to the fact that the rod(s) are treated as lines of positive or negative
charge, and the electric field behaves as such.

Try zooming in and out! You can really see the symmetry of the field far out,
and the edge effects when zoomed in.

[https://www.glowscript.org/#/user/michaelwise/folder/Public/program/LineofCharge-Positive '''Positive Charge''']

[https://www.glowscript.org/#/user/michaelwise/folder/Public/program/LineofCharge-Negative '''Negative Charge''']

==Examples==

Although this is not a very difficult topic, some reasonably difficult conceptual questions can be asked about it.

===Simple===

[[Image:LukasYoderNo.jpg|800px|center|thumb|Figure1: Problem 1]]

===Middling===

[[Image:LukasYoderMaybe.jpg|800px|center|thumb|Figure 2: Problem 2]]

===Difficult===

[[Image:LukasYoderYes.jpg|800px|center|thumb|Figure 3: Problem 3]]

== Connectedness ==
The following two DIY experiments are shown to better understand and visualize the physical consequences of electric fields.

'''Charged Rod and Aluminum Can'''

In our first example we set up an experiment using two charged rods placed to the left and right of an aluminum can, distanced by a length d. If one of the rods is positively charged and the other is negatively charged, what will the can do? Because the positively charged rod induces a negative charge on the left side of the can, creating an attractive force between the rod and the can, and the negatively charged rod induces an equal positive charge on the right side of the can, which creates an attractive force between the can and that rod, the net force on the can is zero. Thus the can will stay still. The setup is depicted in the image below.

[[File:plusq.png|300px|center|thumb|Figure 4: apparatus diagram with +q]]

Next, lets consider a setup but with both rods having equal positive charge, as shown in the image below. What will the can do in this situation?

[[File:minusq.png|300px|center|thumb|Figure 5: apparatus diagram with -q]]

Again the can will also stay still, but this time it is because the polarization force between two objects is always attractive.

So in what scenario will the can move, and what time of movement will the can exhibit? Considering the first setup, imagine this time we initially touch the negatively charged rod and the can for a brief moment. Holding the rods at equal distance on either side of the can, the can will now roll toward the positively charged rod. This is because the can acquires a net negative charge after being touched, so it is then attracted to the positively charged rod.

'''Charged Rod and Pith Ball'''

Another DIY experiment that demonstrates the effects of an electric field is shown in the video embedded below, which depicts the interaction between an initially neutral pith ball hanging on a string from a stand, and a charged rod.

[http://www.youtube.com/watch?v=aeiqw81kGio Interaction between a Charged Rod and Pith Ball]

== History ==

Physicists and scientists make use of electric fields and charged objects all the time. Many times, we may need to know which objects are contributing how much charge in certain areas. Charged objects may attract or repel (depending on the signs of their charge), so we often need to know how objects will interact with each other based on their charges. The phenomenon of this interaction, or electric force between charged particles, was finally confirmed and stated as a law in 1785 by French physicist Charles-Augustin de Coulomb, hence "Coulomb's Law."

== See also ==

The equation for the electric field of a charged rod was derived from the equation for the electric field of a charged particle. See the article "[[Electric Field]]" for more information.

=== Further Reading ===

The page on electric fields: [[Electric Field]]

=== External Links ===
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elelin.html

http://online.cctt.org/physicslab/content/phyapc/lessonnotes/Efields/EchargedRods.asp

https://pages.uncc.edu/phys2102/online-lectures/chapter-02-electric-field/2-4-electric-field-of-charge-distributions/example-1-electric-field-of-a-charged-rod-along-its-axis/

http://dev.physicslab.org/Document.aspx?doctype=3&filename=Electrostatics_ContinuousChargedRod.xml

== References ==

https://www.glowscript.org/#/

https://rhettallain_gmail_com.trinket.io/intro-to-electric-and-magnetic-fields#/electric-fields/multiple-charges

https://www.youtube.com/watch?v=BBWd0zUe0mI

(For the above reference, the textbook's method is followed in that the charge distribution was left undefined, and assumed to be constant)

Chabay and Sherwood: Matter and Interactions, Fourth Edition, Chapter 15

[[Category: Electric Field]]

File:SajanThomas2.png

2024-04-15T04:27:04Z

Sthomas386:

Field of a Charged Rod

2024-04-15T04:14:13Z

Sthomas386:

'''Sajan Thomas, Spring 2024'''

== The Main Idea ==

Previously, we've learned about the electric field of a point particle. While this skill does build the fundamentals of E&M physics, the real world is not simply made up of isolated particles. In fact, the world is consisted of many geometric shapes our one equation for the electric field of a particle cannot account for. One such example of these geometric shapes is a '''rod'''.

For the purposes of this course's scope, we assume that the charge is uniformly distributed along the rod. This is an important assumption to make since if it weren't the case, a lot of the calculations would change. This in mind, let's start with what we know: the electric field of a point particle. By itself, we can't represent a rod. But, if we stack a lot of charged particles together closely in a line, what do we get? a rod. That is the basis of how we calculate the electric field of a charged rod.

That's the basis of it but for now, we'll keep that for later when we're actually deriving how to find it. First, let's discuss the factors that can contribute to the electric field of the charged rod. Firstly, as with any electric field, we have to know the charge of the source-- in this case, the rod. Secondly, we have to know the length of the rod. Thirdly, we also should know the observation location to find the radius from the rod.

The process of finding the electric field due to charge distributed over an object has four steps:

1. Consider the rod, as previously discussed, as a collection of many, many, infinitesimally small charged particles. Make a diagram and draw the electric field <math>\Delta \vec{E}</math> contributed by one of the pieces.

2. By drawing the vectors for the <math>\Delta \vec{E}</math> contributed to one piece, and repeating it along the rod, you should realize by symmetry that several of the vector components will cancel out.

3. Add up the contributions of all pieces, either numerically or symbolically. While there is a formula we can just apply simply later, this step requires integration as you are adding all of the <math>\Delta \vec{E}</math> contributions of many, many, charged particles.

4. Check that the result is physically correct.

=== A Mathematical Model ===
The process of calculating a uniformly charged rod's electric field is tedious, but breaking the process down into several steps makes the task at hand easier. Consider a uniformly charged thin rod of length <math>L</math> and positive charge <math>Q</math> centered on and lying along the x-axis. The rod is being observed from above at a point on the y-axis.

[[Image:SajanThomas1.png|800px|center|thumb|Figure 2: Problem 2]]

'''Step 1: Divide the Distribution into Pieces; Draw <math>\Delta \vec{E}</math>'''

Imagine dividing the rod into a series of very thin slices, each with the same charge <math>\Delta Q</math>. This charge <math>\Delta Q</math> is a small part of the overall charge. Picture it as a point charge. Each slice contributes its own electric field, <math>\Delta E</math>. Summing all these individual slices of <math>E</math> gives you the total electric field of the rod. This process approaches taking an integral, as each thickness approaches 0 and the the number of slices approaches infinity. Note that in this example, the variable that is changing for each slice is its x-coordinate.

'''Step 2: Write an Expression for the Electric Field Due to One Piece'''

The second step is to write a mathematical expression for the field <math>\Delta E</math> contributed by a single slice of the rod. We use the formula of the electric field for a point charge because we are imagining each slice as a point charge. First, determine <math>r</math>, the vector pointing from the source to the observation location. For our example, this is <math> r = obs - source = <0,y,0> - < x,0,0> = <-x,y,0></math>. Now use this to calculate the magnitude and direction of <math>r</math>. So <math>|\vec{r}| = \sqrt{(-x)^2 + y^2} = \sqrt{x^2 + y^2}</math> and <math>\hat{r} = \frac{\vec{r}}{\hat{r}} = \frac{< -x,y,0>}{\sqrt{x^2 + y^2}} </math>. <math> \hat{r}</math> is the vector portion of the expression for the field. The scalar portion is <math> \frac{1}{4\pi\epsilon_0} \cdot \frac{\Delta Q}{|\vec{r}|^2}</math>. Thus the expression for one slice of the rod is:
<math> \Delta \vec{E} = \frac{1}{4\pi\epsilon_0} \cdot \frac{\Delta Q}{(x^2+y^2)^{3/2}} \cdot < -x,y,0> </math>.

'''Determining <math>\Delta Q</math> and the integration variable'''

In the first step, we determined that the changing variable for this rod was its x-coordinate. This means the integration variable is <math> dx</math>. We need to put this integration variable into our expression for the electric field. More specifically, we need to express <math>\Delta Q</math> in terms of the integration variable. Recall that the rod is uniformly charged, so the charge on any single slice of it is: <math>
\Delta Q = (\frac{\Delta x}{L})\cdot Q</math>. This quantity can also be expressed in terms of the charge density.

'''Expression for <math> \Delta \vec{E}</math>

Substitute the expression for the integration variable into the formula for the electric field of one slice. Separating the equation into separate x and y components, we get <math> \Delta \vec{E_x} = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{L} \cdot \frac{-x}{(x^2+y^2)^{3/2}} \cdot dx </math> and <math> \Delta \vec{E_y} = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{L} \cdot \frac{y}{(x^2+y^2)^{3/2}} \cdot dx </math>. Note that we have replaced <math> \Delta x </math> with <math> dx</math> in preparation for integration.

'''Step 3: Add Up the Contributions of All the Pieces'''

The third step is to sum all of our slices. We can go about this in two ways. One way is with numerical summation, or separating the object into a finite number of small pieces, calculating the individual contributing electric fields, and then summing them. Another, more precise method is to integrate. Most of the work of finding the field of a uniformly charged object is setting up this integral. If you have reached the correct expression to integrate, the rest is simple math. The bounds for integration are the coordinates of the start and stop of the rod. In this example the bounds are from <math>-L/2</math> to <math>+L/2</math>. So the expression is <math> \int\limits_{-L/2}^{L/2}\ \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{L} \cdot \frac{y}{(x^2+y^2)^{3/2}} \cdot dx. </math> Solving this gives the final expression <math> E_y = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{x} \cdot \frac{1}{(\sqrt{x^2+ (L/2)^2})} </math>. Note that the field parallel to the x axis is zero. This can be observed due to the symmetry of the problem.
This equation can be written more generally as <math> E = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{r} \cdot \frac{1}{(\sqrt{r^2+ (L/2)^2})} </math> where r represents the distance from the rod to the observation location.

'''Step 4:Checking the Result'''

Finally, the fourth step is to check the result. The units should be the same as the units of the expression for the electric field for a single point particle <math> E = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{r^2} </math>.
Our answer has the right units, since <math> \frac{1}{(\sqrt{x^2+ (L/2)^2})} </math>. has the same units of <math>\frac{Q}{r^2}</math>

Is the direction qualitatively correct? We have the electric field pointing straight away from the midpoint of the rod, which is correct, given the symmetry of the situation. The vertical component of the electric field should indeed be zero.

=== Computational Models ===

'''Finding the Electric Field from a Rod with Code'''
While computation done by hand does have its merits, and certainly was the
methodology used when these ideas were conceived, there are much more efficient,
powerful, and most importantly, pretty ways to go about finding and showing the
electric field. This is of course referring to computers, specifically computers
running glowscript code in the case of Physics 2212. With that idea in mind,
here are some demonstrations of said methodology:

This is some code that you can run which shows the electric field vector
at a given distance from the rod along its length. The rod is shown as a
series of green balls to help emphasize that when using the numerical
integrations mentioned on this page, you are measuring the field produced
by discrete parts of the rod being analyzed. At each point of analysis,
eight field arrows are shown so as to visualize the electric field.

Notice the edge-effects of the electric field of the rod. For reasons
discussed above, if we used the long rod approximation (L>>d), these
effects would be negligible.

[http://www.glowscript.org/#/user/yoderlukas/folder/Public/program/ElectricFieldAlongRodLength '''Click Here to Run the Code''']

For a more precise model, the two links below lead to code that generates a
forty element line of charge with given magnitude, and length, and then
iterates vectors representing the electric field in the space all around said
line of charge. Note that the vectors are small, but for the positively charged
rod, they lead radially outward, and for the negative, radially inward. This is
due to the fact that the rod(s) are treated as lines of positive or negative
charge, and the electric field behaves as such.

Try zooming in and out! You can really see the symmetry of the field far out,
and the edge effects when zoomed in.

[https://www.glowscript.org/#/user/michaelwise/folder/Public/program/LineofCharge-Positive '''Positive Charge''']

[https://www.glowscript.org/#/user/michaelwise/folder/Public/program/LineofCharge-Negative '''Negative Charge''']

==Examples==

Although this is not a very difficult topic, some reasonably difficult conceptual questions can be asked about it.

===Simple===

[[Image:LukasYoderNo.jpg|800px|center|thumb|Figure1: Problem 1]]

===Middling===

[[Image:LukasYoderMaybe.jpg|800px|center|thumb|Figure 2: Problem 2]]

===Difficult===

[[Image:LukasYoderYes.jpg|800px|center|thumb|Figure 3: Problem 3]]

== Connectedness ==
The following two DIY experiments are shown to better understand and visualize the physical consequences of electric fields.

'''Charged Rod and Aluminum Can'''

In our first example we set up an experiment using two charged rods placed to the left and right of an aluminum can, distanced by a length d. If one of the rods is positively charged and the other is negatively charged, what will the can do? Because the positively charged rod induces a negative charge on the left side of the can, creating an attractive force between the rod and the can, and the negatively charged rod induces an equal positive charge on the right side of the can, which creates an attractive force between the can and that rod, the net force on the can is zero. Thus the can will stay still. The setup is depicted in the image below.

[[File:plusq.png|300px|center|thumb|Figure 4: apparatus diagram with +q]]

Next, lets consider a setup but with both rods having equal positive charge, as shown in the image below. What will the can do in this situation?

[[File:minusq.png|300px|center|thumb|Figure 5: apparatus diagram with -q]]

Again the can will also stay still, but this time it is because the polarization force between two objects is always attractive.

So in what scenario will the can move, and what time of movement will the can exhibit? Considering the first setup, imagine this time we initially touch the negatively charged rod and the can for a brief moment. Holding the rods at equal distance on either side of the can, the can will now roll toward the positively charged rod. This is because the can acquires a net negative charge after being touched, so it is then attracted to the positively charged rod.

'''Charged Rod and Pith Ball'''

Another DIY experiment that demonstrates the effects of an electric field is shown in the video embedded below, which depicts the interaction between an initially neutral pith ball hanging on a string from a stand, and a charged rod.

[http://www.youtube.com/watch?v=aeiqw81kGio Interaction between a Charged Rod and Pith Ball]

== History ==

Physicists and scientists make use of electric fields and charged objects all the time. Many times, we may need to know which objects are contributing how much charge in certain areas. Charged objects may attract or repel (depending on the signs of their charge), so we often need to know how objects will interact with each other based on their charges. The phenomenon of this interaction, or electric force between charged particles, was finally confirmed and stated as a law in 1785 by French physicist Charles-Augustin de Coulomb, hence "Coulomb's Law."

== See also ==

The equation for the electric field of a charged rod was derived from the equation for the electric field of a charged particle. See the article "[[Electric Field]]" for more information.

=== Further Reading ===

The page on electric fields: [[Electric Field]]

=== External Links ===
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elelin.html

http://online.cctt.org/physicslab/content/phyapc/lessonnotes/Efields/EchargedRods.asp

https://pages.uncc.edu/phys2102/online-lectures/chapter-02-electric-field/2-4-electric-field-of-charge-distributions/example-1-electric-field-of-a-charged-rod-along-its-axis/

http://dev.physicslab.org/Document.aspx?doctype=3&filename=Electrostatics_ContinuousChargedRod.xml

== References ==

https://www.glowscript.org/#/

https://rhettallain_gmail_com.trinket.io/intro-to-electric-and-magnetic-fields#/electric-fields/multiple-charges

https://www.youtube.com/watch?v=BBWd0zUe0mI

(For the above reference, the textbook's method is followed in that the charge distribution was left undefined, and assumed to be constant)

Chabay and Sherwood: Matter and Interactions, Fourth Edition, Chapter 15

[[Category: Electric Field]]

File:SajanThomas1.png

2024-04-15T04:11:46Z

Sthomas386:

Field of a Charged Rod

2024-04-15T03:56:30Z

Sthomas386:

'''Sajan Thomas, Spring 2024'''

== The Main Idea ==

Previously, we've learned about the electric field of a point particle. While this skill does build the fundamentals of E&M physics, the real world is not simply made up of isolated particles. In fact, the world is consisted of many geometric shapes our one equation for the electric field of a particle cannot account for. One such example of these geometric shapes is a '''rod'''.

For the purposes of this course's scope, we assume that the charge is uniformly distributed along the rod. This is an important assumption to make since if it weren't the case, a lot of the calculations would change. This in mind, let's start with what we know: the electric field of a point particle. By itself, we can't represent a rod. But, if we stack a lot of charged particles together closely in a line, what do we get? a rod. That is the basis of how we calculate the electric field of a charged rod.

That's the basis of it but for now, we'll keep that for later when we're actually deriving how to find it. First, let's discuss the factors that can contribute to the electric field of the charged rod. Firstly, as with any electric field, we have to know the charge of the source-- in this case, the rod. Secondly, we have to know the length of the rod. Thirdly, we also should know the observation location to find the radius from the rod.

The process of finding the electric field due to charge distributed over an object has four steps:

1. Consider the rod, as previously discussed, as a collection of many, many, infinitesimally small charged particles. Make a diagram and draw the electric field <math>\Delta \vec{E}</math> contributed by one of the pieces.

2. By drawing the vectors for the <math>\Delta \vec{E}</math> contributed to one piece, and repeating it along the rod, you should realize by symmetry that several of the vector components will cancel out.

3. Add up the contributions of all pieces, either numerically or symbolically. While there is a formula we can just apply simply later, this step requires integration as you are adding all of the <math>\Delta \vec{E}</math> contributions of many, many, charged particles.

4. Check that the result is physically correct.

=== A Mathematical Model ===
The process of calculating a uniformly charged rod's electric field is tedious, but breaking the process down into several steps makes the task at hand easier. Consider a uniformly charged thin rod of length <math>L</math> and positive charge <math>Q</math> centered on and lying along the x-axis. The rod is being observed from above at a point on the y-axis.

'''Step 1: Divide the Distribution into Pieces; Draw <math>\Delta \vec{E}</math>'''

Imagine dividing the rod into a series of very thin slices, each with the same charge <math>\Delta Q</math>. This charge <math>\Delta Q</math> is a small part of the overall charge. Picture it as a point charge. Each slice contributes its own electric field, <math>\Delta E</math>. Summing all these individual slices of <math>E</math> gives you the total electric field of the rod. This process approaches taking an integral, as each thickness approaches 0 and the the number of slices approaches infinity. Note that in this example, the variable that is changing for each slice is its x-coordinate.

'''Step 2: Write an Expression for the Electric Field Due to One Piece'''

The second step is to write a mathematical expression for the field <math>\Delta E</math> contributed by a single slice of the rod. We use the formula of the electric field for a point charge because we are imagining each slice as a point charge. First, determine <math>r</math>, the vector pointing from the source to the observation location. For our example, this is <math> r = obs - source = <0,y,0> - < x,0,0> = <-x,y,0></math>. Now use this to calculate the magnitude and direction of <math>r</math>. So <math>|\vec{r}| = \sqrt{(-x)^2 + y^2} = \sqrt{x^2 + y^2}</math> and <math>\hat{r} = \frac{\vec{r}}{\hat{r}} = \frac{< -x,y,0>}{\sqrt{x^2 + y^2}} </math>. <math> \hat{r}</math> is the vector portion of the expression for the field. The scalar portion is <math> \frac{1}{4\pi\epsilon_0} \cdot \frac{\Delta Q}{|\vec{r}|^2}</math>. Thus the expression for one slice of the rod is:
<math> \Delta \vec{E} = \frac{1}{4\pi\epsilon_0} \cdot \frac{\Delta Q}{(x^2+y^2)^{3/2}} \cdot < -x,y,0> </math>.

'''Determining <math>\Delta Q</math> and the integration variable'''

In the first step, we determined that the changing variable for this rod was its x-coordinate. This means the integration variable is <math> dx</math>. We need to put this integration variable into our expression for the electric field. More specifically, we need to express <math>\Delta Q</math> in terms of the integration variable. Recall that the rod is uniformly charged, so the charge on any single slice of it is: <math>
\Delta Q = (\frac{\Delta x}{L})\cdot Q</math>. This quantity can also be expressed in terms of the charge density.

'''Expression for <math> \Delta \vec{E}</math>

Substitute the expression for the integration variable into the formula for the electric field of one slice. Separating the equation into separate x and y components, we get <math> \Delta \vec{E_x} = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{L} \cdot \frac{-x}{(x^2+y^2)^{3/2}} \cdot dx </math> and <math> \Delta \vec{E_y} = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{L} \cdot \frac{y}{(x^2+y^2)^{3/2}} \cdot dx </math>. Note that we have replaced <math> \Delta x </math> with <math> dx</math> in preparation for integration.

'''Step 3: Add Up the Contributions of All the Pieces'''

The third step is to sum all of our slices. We can go about this in two ways. One way is with numerical summation, or separating the object into a finite number of small pieces, calculating the individual contributing electric fields, and then summing them. Another, more precise method is to integrate. Most of the work of finding the field of a uniformly charged object is setting up this integral. If you have reached the correct expression to integrate, the rest is simple math. The bounds for integration are the coordinates of the start and stop of the rod. In this example the bounds are from <math>-L/2</math> to <math>+L/2</math>. So the expression is <math> \int\limits_{-L/2}^{L/2}\ \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{L} \cdot \frac{y}{(x^2+y^2)^{3/2}} \cdot dx. </math> Solving this gives the final expression <math> E_y = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{x} \cdot \frac{1}{(\sqrt{x^2+ (L/2)^2})} </math>. Note that the field parallel to the x axis is zero. This can be observed due to the symmetry of the problem.
This equation can be written more generally as <math> E = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{r} \cdot \frac{1}{(\sqrt{r^2+ (L/2)^2})} </math> where r represents the distance from the rod to the observation location.

'''Step 4:Checking the Result'''

Finally, the fourth step is to check the result. The units should be the same as the units of the expression for the electric field for a single point particle <math> E = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{r^2} </math>.
Our answer has the right units, since <math> \frac{1}{(\sqrt{x^2+ (L/2)^2})} </math>. has the same units of <math>\frac{Q}{r^2}</math>

Is the direction qualitatively correct? We have the electric field pointing straight away from the midpoint of the rod, which is correct, given the symmetry of the situation. The vertical component of the electric field should indeed be zero.

=== Computational Models ===

'''Finding the Electric Field from a Rod with Code'''
While computation done by hand does have its merits, and certainly was the
methodology used when these ideas were conceived, there are much more efficient,
powerful, and most importantly, pretty ways to go about finding and showing the
electric field. This is of course referring to computers, specifically computers
running glowscript code in the case of Physics 2212. With that idea in mind,
here are some demonstrations of said methodology:

This is some code that you can run which shows the electric field vector
at a given distance from the rod along its length. The rod is shown as a
series of green balls to help emphasize that when using the numerical
integrations mentioned on this page, you are measuring the field produced
by discrete parts of the rod being analyzed. At each point of analysis,
eight field arrows are shown so as to visualize the electric field.

Notice the edge-effects of the electric field of the rod. For reasons
discussed above, if we used the long rod approximation (L>>d), these
effects would be negligible.

[http://www.glowscript.org/#/user/yoderlukas/folder/Public/program/ElectricFieldAlongRodLength '''Click Here to Run the Code''']

For a more precise model, the two links below lead to code that generates a
forty element line of charge with given magnitude, and length, and then
iterates vectors representing the electric field in the space all around said
line of charge. Note that the vectors are small, but for the positively charged
rod, they lead radially outward, and for the negative, radially inward. This is
due to the fact that the rod(s) are treated as lines of positive or negative
charge, and the electric field behaves as such.

Try zooming in and out! You can really see the symmetry of the field far out,
and the edge effects when zoomed in.

[https://www.glowscript.org/#/user/michaelwise/folder/Public/program/LineofCharge-Positive '''Positive Charge''']

[https://www.glowscript.org/#/user/michaelwise/folder/Public/program/LineofCharge-Negative '''Negative Charge''']

==Examples==

Although this is not a very difficult topic, some reasonably difficult conceptual questions can be asked about it.

===Simple===

[[Image:LukasYoderNo.jpg|800px|center|thumb|Figure1: Problem 1]]

===Middling===

[[Image:LukasYoderMaybe.jpg|800px|center|thumb|Figure 2: Problem 2]]

===Difficult===

[[Image:LukasYoderYes.jpg|800px|center|thumb|Figure 3: Problem 3]]

== Connectedness ==
The following two DIY experiments are shown to better understand and visualize the physical consequences of electric fields.

'''Charged Rod and Aluminum Can'''

In our first example we set up an experiment using two charged rods placed to the left and right of an aluminum can, distanced by a length d. If one of the rods is positively charged and the other is negatively charged, what will the can do? Because the positively charged rod induces a negative charge on the left side of the can, creating an attractive force between the rod and the can, and the negatively charged rod induces an equal positive charge on the right side of the can, which creates an attractive force between the can and that rod, the net force on the can is zero. Thus the can will stay still. The setup is depicted in the image below.

[[File:plusq.png|300px|center|thumb|Figure 4: apparatus diagram with +q]]

Next, lets consider a setup but with both rods having equal positive charge, as shown in the image below. What will the can do in this situation?

[[File:minusq.png|300px|center|thumb|Figure 5: apparatus diagram with -q]]

Again the can will also stay still, but this time it is because the polarization force between two objects is always attractive.

So in what scenario will the can move, and what time of movement will the can exhibit? Considering the first setup, imagine this time we initially touch the negatively charged rod and the can for a brief moment. Holding the rods at equal distance on either side of the can, the can will now roll toward the positively charged rod. This is because the can acquires a net negative charge after being touched, so it is then attracted to the positively charged rod.

'''Charged Rod and Pith Ball'''

Another DIY experiment that demonstrates the effects of an electric field is shown in the video embedded below, which depicts the interaction between an initially neutral pith ball hanging on a string from a stand, and a charged rod.

[http://www.youtube.com/watch?v=aeiqw81kGio Interaction between a Charged Rod and Pith Ball]

== History ==

Physicists and scientists make use of electric fields and charged objects all the time. Many times, we may need to know which objects are contributing how much charge in certain areas. Charged objects may attract or repel (depending on the signs of their charge), so we often need to know how objects will interact with each other based on their charges. The phenomenon of this interaction, or electric force between charged particles, was finally confirmed and stated as a law in 1785 by French physicist Charles-Augustin de Coulomb, hence "Coulomb's Law."

== See also ==

The equation for the electric field of a charged rod was derived from the equation for the electric field of a charged particle. See the article "[[Electric Field]]" for more information.

=== Further Reading ===

The page on electric fields: [[Electric Field]]

=== External Links ===
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elelin.html

http://online.cctt.org/physicslab/content/phyapc/lessonnotes/Efields/EchargedRods.asp

https://pages.uncc.edu/phys2102/online-lectures/chapter-02-electric-field/2-4-electric-field-of-charge-distributions/example-1-electric-field-of-a-charged-rod-along-its-axis/

http://dev.physicslab.org/Document.aspx?doctype=3&filename=Electrostatics_ContinuousChargedRod.xml

== References ==

https://www.glowscript.org/#/

https://rhettallain_gmail_com.trinket.io/intro-to-electric-and-magnetic-fields#/electric-fields/multiple-charges

https://www.youtube.com/watch?v=BBWd0zUe0mI

(For the above reference, the textbook's method is followed in that the charge distribution was left undefined, and assumed to be constant)

Chabay and Sherwood: Matter and Interactions, Fourth Edition, Chapter 15

[[Category: Electric Field]]

Field of a Charged Rod

2024-04-15T03:49:08Z

Sthomas386:

'''Sajan Thomas, Spring 2024'''

== The Main Idea ==

Previously, we've learned about the electric field of a point particle. While this skill does build the fundamentals of E&M physics, the real world is not simply made up of isolated particles. In fact, the world is consisted of many geometric shapes our one equation for the electric field of a particle cannot account for. One such example of these geometric shapes is a '''rod''.

For the purposes of this course's scope, we assume that the charge is uniformly distributed along the rod. This is an important assumption to make since if it weren't the case, a lot of the calculations would change. This in mind, let's start with what we know: the electric field of a point particle. By itself, we can't represent a rod. But, if we stack a lot of charged particles together closely in a line, what do we get? a rod. That is the basis of how we calculate the electric field of a charged rod.

That's the basis of it but for now, we'll keep that for later when we're actually deriving how to find it. First, let's discuss the factors that can contribute to the electric field of the charged rod. Firstly, as with any electric field, we have to know the charge of the source-- in this case, the rod. Secondly, we have to know the length of the rod. Thirdly, we also should know the observation location to find the radius from the rod.

The process of finding the electric field due to charge distributed over an object has four steps:

1. Divide the charged object into small pieces. Make a diagram and draw the electric field <math>\Delta \vec{E}</math> contributed by one of the pieces.

2. Choose an origin and the axes. Write an algebraic expression for the electric field <math>\Delta \vec{E}</math> due to one piece.

3. Add up the contributions of all pieces, either numerically or symbolically.

4. Check that the result is physically correct.

=== A Mathematical Model ===
The process of calculating a uniformly charged rod's electric field is tedious, but breaking the process down into several steps makes the task at hand easier. Consider a uniformly charged thin rod of length <math>L</math> and positive charge <math>Q</math> centered on and lying along the x-axis. The rod is being observed from above at a point on the y-axis.

'''Step 1: Divide the Distribution into Pieces; Draw <math>\Delta \vec{E}</math>'''

Imagine dividing the rod into a series of very thin slices, each with the same charge <math>\Delta Q</math>. This charge <math>\Delta Q</math> is a small part of the overall charge. Picture it as a point charge. Each slice contributes its own electric field, <math>\Delta E</math>. Summing all these individual slices of <math>E</math> gives you the total electric field of the rod. This process approaches taking an integral, as each thickness approaches 0 and the the number of slices approaches infinity. Note that in this example, the variable that is changing for each slice is its x-coordinate.

'''Step 2: Write an Expression for the Electric Field Due to One Piece'''

The second step is to write a mathematical expression for the field <math>\Delta E</math> contributed by a single slice of the rod. We use the formula of the electric field for a point charge because we are imagining each slice as a point charge. First, determine <math>r</math>, the vector pointing from the source to the observation location. For our example, this is <math> r = obs - source = <0,y,0> - < x,0,0> = <-x,y,0></math>. Now use this to calculate the magnitude and direction of <math>r</math>. So <math>|\vec{r}| = \sqrt{(-x)^2 + y^2} = \sqrt{x^2 + y^2}</math> and <math>\hat{r} = \frac{\vec{r}}{\hat{r}} = \frac{< -x,y,0>}{\sqrt{x^2 + y^2}} </math>. <math> \hat{r}</math> is the vector portion of the expression for the field. The scalar portion is <math> \frac{1}{4\pi\epsilon_0} \cdot \frac{\Delta Q}{|\vec{r}|^2}</math>. Thus the expression for one slice of the rod is:
<math> \Delta \vec{E} = \frac{1}{4\pi\epsilon_0} \cdot \frac{\Delta Q}{(x^2+y^2)^{3/2}} \cdot < -x,y,0> </math>.

'''Determining <math>\Delta Q</math> and the integration variable'''

In the first step, we determined that the changing variable for this rod was its x-coordinate. This means the integration variable is <math> dx</math>. We need to put this integration variable into our expression for the electric field. More specifically, we need to express <math>\Delta Q</math> in terms of the integration variable. Recall that the rod is uniformly charged, so the charge on any single slice of it is: <math>
\Delta Q = (\frac{\Delta x}{L})\cdot Q</math>. This quantity can also be expressed in terms of the charge density.

'''Expression for <math> \Delta \vec{E}</math>

Substitute the expression for the integration variable into the formula for the electric field of one slice. Separating the equation into separate x and y components, we get <math> \Delta \vec{E_x} = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{L} \cdot \frac{-x}{(x^2+y^2)^{3/2}} \cdot dx </math> and <math> \Delta \vec{E_y} = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{L} \cdot \frac{y}{(x^2+y^2)^{3/2}} \cdot dx </math>. Note that we have replaced <math> \Delta x </math> with <math> dx</math> in preparation for integration.

'''Step 3: Add Up the Contributions of All the Pieces'''

The third step is to sum all of our slices. We can go about this in two ways. One way is with numerical summation, or separating the object into a finite number of small pieces, calculating the individual contributing electric fields, and then summing them. Another, more precise method is to integrate. Most of the work of finding the field of a uniformly charged object is setting up this integral. If you have reached the correct expression to integrate, the rest is simple math. The bounds for integration are the coordinates of the start and stop of the rod. In this example the bounds are from <math>-L/2</math> to <math>+L/2</math>. So the expression is <math> \int\limits_{-L/2}^{L/2}\ \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{L} \cdot \frac{y}{(x^2+y^2)^{3/2}} \cdot dx. </math> Solving this gives the final expression <math> E_y = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{x} \cdot \frac{1}{(\sqrt{x^2+ (L/2)^2})} </math>. Note that the field parallel to the x axis is zero. This can be observed due to the symmetry of the problem.
This equation can be written more generally as <math> E = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{r} \cdot \frac{1}{(\sqrt{r^2+ (L/2)^2})} </math> where r represents the distance from the rod to the observation location.

'''Step 4:Checking the Result'''

Finally, the fourth step is to check the result. The units should be the same as the units of the expression for the electric field for a single point particle <math> E = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{r^2} </math>.
Our answer has the right units, since <math> \frac{1}{(\sqrt{x^2+ (L/2)^2})} </math>. has the same units of <math>\frac{Q}{r^2}</math>

Is the direction qualitatively correct? We have the electric field pointing straight away from the midpoint of the rod, which is correct, given the symmetry of the situation. The vertical component of the electric field should indeed be zero.

=== Computational Models ===

'''Finding the Electric Field from a Rod with Code'''
While computation done by hand does have its merits, and certainly was the
methodology used when these ideas were conceived, there are much more efficient,
powerful, and most importantly, pretty ways to go about finding and showing the
electric field. This is of course referring to computers, specifically computers
running glowscript code in the case of Physics 2212. With that idea in mind,
here are some demonstrations of said methodology:

This is some code that you can run which shows the electric field vector
at a given distance from the rod along its length. The rod is shown as a
series of green balls to help emphasize that when using the numerical
integrations mentioned on this page, you are measuring the field produced
by discrete parts of the rod being analyzed. At each point of analysis,
eight field arrows are shown so as to visualize the electric field.

Notice the edge-effects of the electric field of the rod. For reasons
discussed above, if we used the long rod approximation (L>>d), these
effects would be negligible.

[http://www.glowscript.org/#/user/yoderlukas/folder/Public/program/ElectricFieldAlongRodLength '''Click Here to Run the Code''']

For a more precise model, the two links below lead to code that generates a
forty element line of charge with given magnitude, and length, and then
iterates vectors representing the electric field in the space all around said
line of charge. Note that the vectors are small, but for the positively charged
rod, they lead radially outward, and for the negative, radially inward. This is
due to the fact that the rod(s) are treated as lines of positive or negative
charge, and the electric field behaves as such.

Try zooming in and out! You can really see the symmetry of the field far out,
and the edge effects when zoomed in.

[https://www.glowscript.org/#/user/michaelwise/folder/Public/program/LineofCharge-Positive '''Positive Charge''']

[https://www.glowscript.org/#/user/michaelwise/folder/Public/program/LineofCharge-Negative '''Negative Charge''']

==Examples==

Although this is not a very difficult topic, some reasonably difficult conceptual questions can be asked about it.

===Simple===

[[Image:LukasYoderNo.jpg|800px|center|thumb|Figure1: Problem 1]]

===Middling===

[[Image:LukasYoderMaybe.jpg|800px|center|thumb|Figure 2: Problem 2]]

===Difficult===

[[Image:LukasYoderYes.jpg|800px|center|thumb|Figure 3: Problem 3]]

== Connectedness ==
The following two DIY experiments are shown to better understand and visualize the physical consequences of electric fields.

'''Charged Rod and Aluminum Can'''

In our first example we set up an experiment using two charged rods placed to the left and right of an aluminum can, distanced by a length d. If one of the rods is positively charged and the other is negatively charged, what will the can do? Because the positively charged rod induces a negative charge on the left side of the can, creating an attractive force between the rod and the can, and the negatively charged rod induces an equal positive charge on the right side of the can, which creates an attractive force between the can and that rod, the net force on the can is zero. Thus the can will stay still. The setup is depicted in the image below.

[[File:plusq.png|300px|center|thumb|Figure 4: apparatus diagram with +q]]

Next, lets consider a setup but with both rods having equal positive charge, as shown in the image below. What will the can do in this situation?

[[File:minusq.png|300px|center|thumb|Figure 5: apparatus diagram with -q]]

Again the can will also stay still, but this time it is because the polarization force between two objects is always attractive.

So in what scenario will the can move, and what time of movement will the can exhibit? Considering the first setup, imagine this time we initially touch the negatively charged rod and the can for a brief moment. Holding the rods at equal distance on either side of the can, the can will now roll toward the positively charged rod. This is because the can acquires a net negative charge after being touched, so it is then attracted to the positively charged rod.

'''Charged Rod and Pith Ball'''

Another DIY experiment that demonstrates the effects of an electric field is shown in the video embedded below, which depicts the interaction between an initially neutral pith ball hanging on a string from a stand, and a charged rod.

[http://www.youtube.com/watch?v=aeiqw81kGio Interaction between a Charged Rod and Pith Ball]

== History ==

Physicists and scientists make use of electric fields and charged objects all the time. Many times, we may need to know which objects are contributing how much charge in certain areas. Charged objects may attract or repel (depending on the signs of their charge), so we often need to know how objects will interact with each other based on their charges. The phenomenon of this interaction, or electric force between charged particles, was finally confirmed and stated as a law in 1785 by French physicist Charles-Augustin de Coulomb, hence "Coulomb's Law."

== See also ==

The equation for the electric field of a charged rod was derived from the equation for the electric field of a charged particle. See the article "[[Electric Field]]" for more information.

=== Further Reading ===

The page on electric fields: [[Electric Field]]

=== External Links ===
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elelin.html

http://online.cctt.org/physicslab/content/phyapc/lessonnotes/Efields/EchargedRods.asp

https://pages.uncc.edu/phys2102/online-lectures/chapter-02-electric-field/2-4-electric-field-of-charge-distributions/example-1-electric-field-of-a-charged-rod-along-its-axis/

http://dev.physicslab.org/Document.aspx?doctype=3&filename=Electrostatics_ContinuousChargedRod.xml

== References ==

https://www.glowscript.org/#/

https://rhettallain_gmail_com.trinket.io/intro-to-electric-and-magnetic-fields#/electric-fields/multiple-charges

https://www.youtube.com/watch?v=BBWd0zUe0mI

(For the above reference, the textbook's method is followed in that the charge distribution was left undefined, and assumed to be constant)

Chabay and Sherwood: Matter and Interactions, Fourth Edition, Chapter 15

[[Category: Electric Field]]

Main Page

2024-04-15T01:10:28Z

Sthomas386: /* Superposition */

__NOTOC__
= '''Georgia Tech Student Wiki for Introductory Physics.''' =

This resource was created so that students can contribute and curate content to help those with limited or no access to a textbook. When reading this website, please correct any errors you may come across. If you read something that isn't clear, please consider revising it for future students!

Looking to make a contribution?
#Pick one of the topics from intro physics listed below
#Add content to that topic or improve the quality of what is already there.
#Need to make a new topic? Edit this page and add it to the list under the appropriate category. Then copy and paste the default [[Template]] into your new page and start editing.

Please remember that this is not a textbook and you are not limited to expressing your ideas with only text and equations. Whenever possible embed: pictures, videos, diagrams, simulations, computational models (e.g. Glowscript), and whatever content you think makes learning physics easier for other students.

== Source Material ==
All of the content added to this resource must be in the public domain or similar free resource. If you are unsure about a source, contact the original author for permission. That said, there is a surprisingly large amount of introductory physics content scattered across the web. Here is an incomplete list of intro physics resources (please update as needed).
* A physics resource written by experts for an expert audience [https://en.wikipedia.org/wiki/Portal:Physics Physics Portal]
* A wiki written for students by a physics expert [http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes MSU Physics Wiki]
* A wiki book on modern physics [https://en.wikibooks.org/wiki/Modern_Physics Modern Physics Wiki]
* A collection of 26 volumes of lecture notes by Prof. Wheeler of Reed College [https://rdc.reed.edu/c/wheeler/home/]
* The MIT open courseware for intro physics [http://ocw.mit.edu/resources/res-8-002-a-wikitextbook-for-introductory-mechanics-fall-2009/index.htm MITOCW Wiki]
* An online concept map of intro physics [http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics]
* Interactive physics simulations [https://phet.colorado.edu/en/simulations/category/physics PhET]
* OpenStax intro physics textbooks: [https://openstax.org/details/books/university-physics-volume-1 Vol1], [https://openstax.org/details/books/university-physics-volume-2 Vol2], [https://openstax.org/details/books/university-physics-volume-3 Vol3]
* The Open Source Physics project is a collection of online physics resources [http://www.opensourcephysics.org/ OSP]
* A resource guide compiled by the [http://www.aapt.org/ AAPT] for educators [http://www.compadre.org/ ComPADRE]
* The Feynman lectures on physics are free to read [http://www.feynmanlectures.caltech.edu/ Feynman]
* Final Study Guide for Modern Physics II created by a lab TA [https://docs.google.com/document/d/1_6GktDPq5tiNFFYs_ZjgjxBAWVQYaXp_2Imha4_nSyc/edit?usp=sharing Modern Physics II Final Study Guide]

== Resources ==
* Commonly used wiki commands [https://en.wikipedia.org/wiki/Help:Cheatsheet Wiki Cheatsheet]
* A guide to representing equations in math mode [https://en.wikipedia.org/wiki/Help:Displaying_a_formula Wiki Math Mode]
* A page to keep track of all the physics [[Constants]]
* A listing of [[Notable Scientist]] with links to their individual pages

<div style="float:left; width:30%; padding:1%;">

==Physics 1==
===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====GlowScript 101====
<div class="mw-collapsible-content">
*[[Python Syntax]]
*[[GlowScript]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====VPython====

<div class="mw-collapsible-content">
*[[VPython]]
*[[VPython basics]]
*[[VPython Common Errors and Troubleshooting]]
*[[VPython Functions]]
*[[VPython Lists]]
*[[VPython Loops]]
*[[VPython Multithreading]]
*[[VPython Animation]]
*[[VPython Objects]]
*[[VPython 3D Objects]]
*[[VPython Reference]]
*[[VPython MapReduceFilter]]
*[[VPython GUIs]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Vectors and Units====
<div class="mw-collapsible-content">
*[[Vectors]]
*[[SI Units]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Interactions====
<div class="mw-collapsible-content">
*[[Types of Interactions and How to Detect Them]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Velocity and Momentum====
<div class="mw-collapsible-content">
*[[Newton's First Law of Motion]]
*[[Mass]]
*[[Velocity]]
*[[Speed]]
*[[Speed vs Velocity]]
*[[Relative Velocity]]
*[[Derivation of Average Velocity]]
*[[2-Dimensional Motion]]
*[[3-Dimensional Position and Motion]]
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Momentum and the Momentum Principle====
<div class="mw-collapsible-content">
*[[Linear Momentum]]
*[[Newton's Second Law: the Momentum Principle]]
*[[Impulse and Momentum]]
*[[Net Force]]
*[[Inertia]]
*[[Acceleration]]
*[[Relativistic Momentum]]

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Iterative Prediction with a Constant Force====
<div class="mw-collapsible-content">
*[[Iterative Prediction]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">

====Analytic Prediction with a Constant Force====
<div class="mw-collapsible-content">

*[[Kinematics]]
*[[Projectile Motion]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Iterative Prediction with a Varying Force====
<div class="mw-collapsible-content">
*[[Fundamentals of Iterative Prediction with Varying Force]]
*[[Spring_Force]]
*[[Simple Harmonic Motion]]


*[[Iterative Prediction of Spring-Mass System]]
*[[Terminal Speed]]
*[[Predicting Change in multiple dimensions]]
*[[Two Dimensional Harmonic Motion]]
*[[Determinism]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Fundamental Interactions====
<div class="mw-collapsible-content">
*[[Gravitational Force]]
*[[Gravitational Force Near Earth]]
*[[Gravitational Force in Space and Other Applications]]
*[[3 or More Body Interactions]]

*[[Electric Force]]
*[[Introduction to Magnetic Force]]
*[[Strong and Weak Force]]
*[[Reciprocity]]
*[[Conservation of Momentum]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Properties of Matter====
<div class="mw-collapsible-content">
*[[Kinds of Matter]]
*[[Ball and Spring Model of Matter]]
*[[Density]]
*[[Length and Stiffness of an Interatomic Bond]]
*[[Young's Modulus]]
*[[Speed of Sound in Solids]]
*[[Malleability]]
*[[Ductility]]
*[[Weight]]
*[[Hardness]]
*[[Boiling Point]]
*[[Melting Point]]
*[[Change of State]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Identifying Forces====
<div class="mw-collapsible-content">
*[[Free Body Diagram]]
*[[Inclined Plane]]
*[[Compression or Normal Force]]
*[[Tension]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Curving Motion====
<div class="mw-collapsible-content">
*[[Curving Motion]]
*[[Centripetal Force and Curving Motion]]
*[[Perpetual Freefall (Orbit)]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====Energy Principle====
<div class="mw-collapsible-content">
*[[Energy of a Single Particle]]
*[[Kinetic Energy]]
*[[Work/Energy]]
*[[The Energy Principle]]
*[[Conservation of Energy]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Work by Non-Constant Forces====
<div class="mw-collapsible-content">
*[[Work Done By A Nonconstant Force]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Potential Energy====
<div class="mw-collapsible-content">
*[[Potential Energy]]
*[[Potential Energy of Macroscopic Springs]]
*[[Spring Potential Energy]]
*[[Ball and Spring Model]]
*[[Gravitational Potential Energy]]
*[[Energy Graphs]]
*[[Escape Velocity]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Multiparticle Systems====
<div class="mw-collapsible-content">
*[[Center of Mass]]
*[[Multi-particle analysis of Momentum]]
*[[Potential Energy of a Multiparticle System]]
*[[Work and Energy for an Extended System]]
*[[Internal Energy]]
**[[Potential Energy of a Pair of Neutral Atoms]]
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">
====Choice of System====
<div class="mw-collapsible-content">
*[[System & Surroundings]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Thermal Energy, Dissipation, and Transfer of Energy====
<div class="mw-collapsible-content">
*[[Thermal Energy]]
*[[Specific Heat]]
*[[Calorific Value(Heat of combustion)]]
*[[First Law of Thermodynamics]]
*[[Second Law of Thermodynamics and Entropy]]
*[[Temperature]]
*[[Transformation of Energy]]
*[[The Maxwell-Boltzmann Distribution]]
*[[Air Resistance]]
*[[The Third Law of Thermodynamics]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Rotational and Vibrational Energy====
<div class="mw-collapsible-content">
*[[Translational, Rotational and Vibrational Energy]]
*[[Rolling Motion]]
</div>
</div>

===Week 11===
<div class="toccolours mw-collapsible mw-collapsed">
====Different Models of a System====
<div class="mw-collapsible-content">
*[[Point Particle Systems]]
*[[Real Systems]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Friction====
<div class="mw-collapsible-content">
*[[Friction]]
*[[Static Friction]]
*[[Kinetic Friction]]
</div>
</div>

===Week 12===
<div class="toccolours mw-collapsible mw-collapsed">
====Conservation of Momentum====
<div class="mw-collapsible-content">
*[[Conservation of Momentum]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Collisions====
<div class="mw-collapsible-content">
*[[Newton's Third Law of Motion]]
*[[Collisions]]
*[[Elastic Collisions]]
*[[Inelastic Collisions]]
*[[Maximally Inelastic Collision]]
*[[Head-on Collision of Equal Masses]]
*[[Head-on Collision of Unequal Masses]]
*[[Scattering: Collisions in 2D and 3D]]
*[[Rutherford Experiment and Atomic Collisions]]
*[[Coefficient of Restitution]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Rotations====
<div class="mw-collapsible-content">
*[[Rotational Kinematics]]
*[[Eulerian Angles]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Angular Momentum====
<div class="mw-collapsible-content">
*[[Total Angular Momentum]]
*[[Translational Angular Momentum]]
*[[Rotational Angular Momentum]]
*[[The Angular Momentum Principle]]
*[[Angular Impulse]]
*[[Predicting the Position of a Rotating System]]
*[[The Moments of Inertia]]
*[[Right Hand Rule]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Analyzing Motion with and without Torque====
<div class="mw-collapsible-content">
*[[Torque]]
*[[Torque 2]]
*[[Systems with Zero Torque]]
*[[Systems with Nonzero Torque]]
*[[Torque vs Work]]
*[[Gyroscopes]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
====Introduction to Quantum Concepts====
<div class="mw-collapsible-content">
*[[Bohr Model]]
*[[Energy graphs and the Bohr model]]
*[[Quantized energy levels]]
*[[Electron transitions]]
*[[Entropy]]
</div>
</div>
</div>

<div style="float:left; width:30%; padding:1%;">

==Physics 2==
===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====3D Vectors====

<div class="mw-collapsible-content">
*[[Vectors]]
*[[Right-Hand Rule]]
*[[Right Hand Rule]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric field====
<div class="mw-collapsible-content">
*[[Electric Field]]
*[[Electric Field and Electric Potential]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric force====
<div class="mw-collapsible-content">
*[[Electric Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric field of a point particle====
<div class="mw-collapsible-content">
*[[Point Charge]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Superposition====
<div class="mw-collapsible-content">
*[[Superposition Principle]]
*[[Superposition principle]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Dipoles====
<div class="mw-collapsible-content">
*[[Electric Dipole]]
*[[Magnetic Dipole]]
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Interactions of charged objects====
<div class="mw-collapsible-content">
*[[Electric Field]]
*[[Electric Potential]]
*[[Electric Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Tape experiments====
<div class="mw-collapsible-content">
*[[Polarization]]
*[[Electric Polarization]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Polarization====
<div class="mw-collapsible-content">
*[[Polarization]]
*[[Electric Polarization]]
*[[Polarization of an Atom]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Conductors and Insulators====
<div class="mw-collapsible-content">
*[[Conductivity and Resistivity]]
*[[Insulators]]
*[[Potential Difference in an Insulator]]
*[[Conductors]]
*[[Polarization of a conductor]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Charging and Discharging====
<div class="mw-collapsible-content">
*[[Charge Transfer]]
*[[Electrostatic Discharge]]
*[[Charged Conductor and Charged Insulator]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged rod====
<div class="mw-collapsible-content">
*[[Field of a Charged Rod|Charged Rod]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Field of a charged ring/disk/capacitor====
<div class="mw-collapsible-content">
*[[Charged Ring]]
*[[Charged Disk]]
*[[Charged Capacitor]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged sphere====
<div class="mw-collapsible-content">
*[[Charged Spherical Shell]]
*[[Field of a Charged Ball]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Potential energy====
<div class="mw-collapsible-content">
*[[Potential Energy]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric potential====
<div class="mw-collapsible-content">
*[[Electric Potential]]
*[[Path Independence of Electric Potential]]
*[[Potential Difference Path Independence, claimed by Aditya Mohile]]
*[[Potential Difference in a Uniform Field]]
*[[Potential Difference of Point Charge in a Non-Uniform Field]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Sign of a potential difference====
<div class="mw-collapsible-content">
*[[Sign of a Potential Difference]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Potential at a single location====
<div class="mw-collapsible-content">
*[[Electric Potential]]
*[[Potential Difference at One Location]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Path independence and round trip potential====
<div class="mw-collapsible-content">
*[[Path Independence of Electric Potential]]
*[[Potential Difference Path Independence, claimed by Aditya Mohile]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Electric field and potential in an insulator====
<div class="mw-collapsible-content">
*[[Potential Difference in an Insulator]]
*[[Electric Field in an Insulator]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Moving charges in a magnetic field====
<div class="mw-collapsible-content">
*[[Magnetic Field]]
*[[Magnetic Force]]
*[[Lorentz Force]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Biot-Savart Law====
<div class="mw-collapsible-content">
*[[Biot-Savart Law]]
*[[Biot-Savart Law for Currents]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Moving charges, electron current, and conventional current====
<div class="mw-collapsible-content">
*[[Moving Point Charge]]
*[[Current]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a wire====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Long Straight Wire]]
*[[Magnetic Field of a Curved Wire]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Magnetic field of a current-carrying loop====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Loop]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a Charged Disk====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Disk]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic dipoles====
<div class="mw-collapsible-content">
*[[Magnetic Dipole Moment]]
*[[Bar Magnet]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Atomic structure of magnets====
<div class="mw-collapsible-content">
*[[Atomic Structure of Magnets]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">

====Circuitry Basics====
<div class="mw-collapsible-content">
*[[Understanding Fundamentals of Current, Voltage, and Resistance]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Steady state current====
<div class="mw-collapsible-content">
*[[Steady State]]
*[[Non Steady State]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Kirchoff's Laws====
<div class="mw-collapsible-content">
*[[Kirchoff's Laws]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Electric fields and energy in circuits====
<div class="mw-collapsible-content">
*[[Electric Potential Difference]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Macroscopic analysis of circuits====
<div class="mw-collapsible-content">
*[[Series Circuits]]
*[[Parallel Circuits]]
*[[Parallel Circuits vs. Series Circuits*]]
*[[Loop Rule]]
*[[Node Rule]]
*[[Fundamentals of Resistance]]
*[[Problem Solving]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Electric field and potential in circuits with capacitors====
<div class="mw-collapsible-content">
*[[Charging and Discharging a Capacitor]]
*[[RC Circuit]]
*[[R Circuit]]
*[[AC and DC]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic forces on charges and currents====
<div class="mw-collapsible-content">
*[[Magnetic Force]]
*[[Lorentz Force]]
*[[Motors and Generators]]
*[[Applying Magnetic Force to Currents]]
*[[Magnetic Force in a Moving Reference Frame]]
*[[Right-Hand Rule]]
*[[Analysis of Railgun vs Coil gun technologies]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric and magnetic forces====
<div class="mw-collapsible-content">
*[[Electric Force]]
*[[Magnetic Force]]
*[[Lorentz Force]]
*[[VPython Modelling of Electric and Magnetic Forces]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Velocity selector====
<div class="mw-collapsible-content">
*[[Lorentz Force]]
*[[Combining Electric and Magnetic Forces]]
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">

====Hall Effect====
<div class="mw-collapsible-content">
*[[Hall Effect]]
*[[Right-Hand Rule]]
*[[Motional Emf]]
*[[Magnetic Force]]
*[[Magnetic Torque]]
</div>

====Magnetic force====
<div class="mw-collapsible-content">
*[[Magnetic Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic torque====
<div class="mw-collapsible-content">
*[[Magnetic Torque]]
*[[Right-Hand Rule]]
</div>
</div>

===Week 12===

<div class="toccolours mw-collapsible mw-collapsed">
====Gauss's Law====
<div class="mw-collapsible-content">
*[[Gauss's Flux Theorem]]
*[[Gauss's Law]]
*[[Magnetic Flux]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Ampere's Law====
<div class="mw-collapsible-content">
*[[Ampere's Law]]
*[[Ampere-Maxwell Law]]
*[[Magnetic Field of Coaxial Cable Using Ampere's Law]]
*[[Magnetic Field of a Long Thick Wire Using Ampere's Law]]
*[[Magnetic Field of a Toroid Using Ampere's Law]]
*[[Magnetic Field of a Solenoid Using Ampere's Law]]
*[[The Differential Form of Ampere's Law]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Semiconductors====
<div class="mw-collapsible-content">
*[[Semiconductor Devices]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Faraday's Law====
<div class="mw-collapsible-content">
*[[Faraday's Law]]
*[[Motional Emf using Faraday's Law]]
*[[Lenz's Law]]

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Maxwell's equations====
<div class="mw-collapsible-content">
*[[Gauss's Law]]
*[[Magnetic Flux]]
*[[Ampere's Law]]
*[[Faraday's Law]]
*[[Maxwell's Electromagnetic Theory]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Circuits revisited====
<div class="mw-collapsible-content">

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Inductors====
<div class="mw-collapsible-content">
*[[Inductors]]
*[[Current in an LC Circuit]]
*[[Current in an RL Circuit]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
==== Electromagnetic Radiation ====
<div class="mw-collapsible-content">
*[[Electromagnetic Radiation]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Sparks in the air====
<div class="mw-collapsible-content">
*[[Sparks in Air]]
*[[Spark Plugs]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Superconductors====
<div class="mw-collapsible-content">
*[[Superconducters]]
*[[Superconductors]]
*[[Meissner effect]]
</div>
</div>
</div>

<div style="float:left; width:30%; padding:1%;">

==Physics 3==

===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====Classical Physics====
<div class="mw-collapsible-content">
*[[Classical Physics]]
</div>
</div>

[[Category:Which Category did you place this in?]]

===Weeks 2 and 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Special Relativity and the Lorentz Transformation====
<div class="mw-collapsible-content">
*[[Frame of Reference]]

*[[Einstein's Theory of Special Relativity]]
*[[Time Dilation]]
*[[Lorentz Transformations]]
*[[Relativistic Doppler Effect]]
*[[Einstein's Theory of General Relativity]]
*[[Albert A. Micheleson & Edward W. Morley]]
*[[Magnetic Force in a Moving Reference Frame]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Photons and the Photoelectric Effect====
<div class="mw-collapsible-content">
*[[Spontaneous Photon Emission]]
*[[Light Scattering]]
*[[Lasers]]
*[[Electronic Energy Levels and Photons]]
*[[Quantum Properties of Light]]
*[[The Photoelectric Effect]]
</div>
</div>

===Weeks 5 and 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Matter Waves and Wave-Particle Duality====
<div class="mw-collapsible-content">
*[[Wave-Particle Duality]]
*[[Particle in a 1-Dimensional box]]
*[[Heisenberg Uncertainty Principle]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====Wave Mechanics====
<div class="mw-collapsible-content">
*[[Standing Waves]]
*[[Wavelength]]
*[[Wavelength and Frequency]]
*[[Mechanical Waves]]
*[[Transverse and Longitudinal Waves]]
*[[Fourier Series and Transform]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Schrödinger Equation====
<div class="mw-collapsible-content">
*[[The Born Rule]]
*[[Solution for a Single Free Particle]]
*[[Solution for a Single Particle in an Infinite Quantum Well - Darin]]
*[[Solution for a Single Particle in a Semi-Infinite Quantum Well]]
*[[Quantum Harmonic Oscillator]]
*[[Solution for Simple Harmonic Oscillator]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Quantum Mechanics====
<div class="mw-collapsible-content">
*[[Quantum Tunneling through Potential Barriers]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====The Hydrogen Atom====
<div class="mw-collapsible-content">
*[[Quantum Theory]]
*[[Atomic Theory]]
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">
====Rutherford-Bohr Model====
<div class="mw-collapsible-content">
*[[Rutherford Experiment and Atomic Collisions]]
*[[Bohr Model]]
*[[Quantized energy levels]]
*[[Energy graphs and the Bohr model]]
</div>
</div>

===Week 11===
<div class="toccolours mw-collapsible mw-collapsed">
====Many-Electron Atoms====
<div class="mw-collapsible-content">
*[[Quantum Theory]]
*[[Atomic Theory]]
*[[Pauli exclusion principle]]
</div>
</div>

===Week 12===
<div class="toccolours mw-collapsible mw-collapsed">
====The Nucleus====
<div class="mw-collapsible-content">
*[[Nucleus]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Molecules====
<div class="mw-collapsible-content">
*[[Molecules]]
*[[Covalent Bonds]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Statistical Physics====
<div class="mw-collapsible-content">
*[[Application of Statistics in Physics]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
====Statistical Physics====
<div class="mw-collapsible-content">
*[[Temperature & Entropy]]
</div>
</div>

===Additional Topics===
<div class="toccolours mw-collapsible mw-collapsed">
====Condensed Matter Physics====
<div class="mw-collapsible-content">
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Nuclear Physics====
<div class="mw-collapsible-content">
*[[Nuclear Fission]]
*[[Nuclear Energy from Fission and Fusion]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Particle Physics====
<div class="mw-collapsible-content">
*[[Elementary Particles and Particle Physics Theory]]
*[[String Theory]]
</div>
</div>
</div>