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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?

  1. Pick one of the topics from intro physics listed below
  2. Add content to that topic or improve the quality of what is already there.
  3. 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).

Resources


Physics 1

Week 1

Help with VPython

Vectors and Units

Vectors and Units

Week 2

Week 3

Analytic Prediction with a Constant Force

Week 4

Week 5

Conservation of Momentum

Week 6

Week 7

Week 8

Work by Non-Constant Forces

Week 9

Week 10

Choice of System

Rotational and Vibrational Energy

Week 11

Different Models of a System

Models of Friction

Week 12

Week 13

Week 14

Week 15

Physics 2

Week 1

Electric force

Electric field of a point particle

Week 2

Week 3

Week 4

Field of a Charged Rod

This entire page and all its contents were created by Lukas Yoder, PHYS 2212 Class of Fall 2018

The Main Idea

Previously, we've learned about the electric field of a point particle. Often, when analyzing physical systems, it is the case that we're unable to analyze each individual particle that composes an object and need to therefore generalize collections of particles into shapes (in this case, a rod) whereby the mathematics corresponding to electric field calculations can be simplified. This can essentially be done by adding up the contributions to the electric field made by parts of an object, approximating each part of an object as a point charge.

The System in Question

As discussed in the previous section, we're considering a system abstracted from the particle model we're familiar with, therefore we will make the generalization that our rod of length L has a total charge of quantity Q. For this generalization, we will need to assume that the rod is so thin that we can ignore its thickness.

Since the electric field produced by a charge at any given location is proportional to the distance from the charge to that location, we will need to relate the observation location to the source of the charge, which we will consider the origin of the rod. To do that, we will need to divide the rod into pieces of length \delta y each containing a charge \delta Q. In the image below, you can see what this looks like and the relation that can be found between the observation location and the source, forming the distance vector \vect{r}.

By the pythagorean theorem, we can find the vector \vect{r} as follows:


And to find the unit vector in the direction of \vect{r}, \hat{r}, we do as follows:


Finding the Contribution of Each Piece to the Electric Field

Now that we've set up a model for the system, with the rod broken down into pieces, we can find the contribution of each piece to the electric field of the system. We will start from the electric field equation you learned for a point particle but plug in the parameters for the rod system into the equation.


By mathematically simplifying, we then get the following equation:


Finding the Net Contribution of all Pieces

In the previous section, we found out the contribution to the electric field at a given location of only one of the pieces constituting the rod. In order to figure out the net field at any particular location, we need to add up the electric fields produced by individual pieces along the length of the rod.

We will switch from vector notation for the electric field to the scalar notation for the x- and y-components. (From the vector in the equation above, we can see that the z-component of the electric field at any point is always 0.) The x-component of the electric field is the sum of the x-components of every \delta{y} along the rod, and the y-component of the electric field is the sum of the y-components of every \delta{y} along the rod. We can show this mathematically:


To make use of this relation, because we don't know \delta{Q}, we need to relate it to parameters that we already know about the rod system we're analyzing. We can express \delta{Q} as the charge density of the rod (which is Q/L) times the \delta{y} we've chosen for the system. Thus,


By plugging the above equation into our equations for the x- and y-components of the electric field at a point, we can find the electric field at any point in the system. This technique is called numerical integration and is typically done by computers because the computational complexity is dependant upon the size of \delta{y} with respect to L.

Simplifying

Using calculus, we can simplify a lot of the math required to compute the electric field at any given point. Notationally, all we're doing is switching from the discretely-sized \delta{y} to \textit{dy} and from the sigma notation to an integral starting from -L/2 (the lower end of the rod) and ending at L/2 (the upper end of the rod) as follows:


By evaluating the integral, we can determine that the x-component of the electric field at any point is:


Without evaluating the integral for the y-component of the electric field, we can use symmetry to determine that the y-component of the electric field at any given point is 0. Let's consider the contributions to the electric field from the top and bottom halves of the rod at any observation point.


Since the y-components of E_top and E_bottom are of equal magnitude and opposite direction, they cancel each other out, and therefore the y-component of teh electric field at any given point due to the rod is 0.


Finally, because the rod is round and can be rotated, as a convenience, we'll use d (distance from the rod) as opposed to x (distance along the x-direction) to refer to the electric field.

Thus we can simplify electric field calculations for a rod into a form that we can readily use:


Further Simplification

By noting the contributions of each variable to the equation for the electric field, we can make approximations to simplify our math by simply declaring one variable as insignificant.

For example, if we have a system in which the length of a rod is much greater than the magnitude of the distance from the rod (denoted L>>d), we can neglect some of the instances in which d is taken into account as follows:


Finding the Electric Field from a Rod with Code

Here 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.

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.

Click Here to Run the Code

The Main Idea

Previously, we've learned about the electric field of a point particle. Often, when analyzing physical systems, it is the case that we're unable to analyze each individual particle that composes an object and need to therefore generalize collections of particles into shapes (in this case, a rod) whereby the mathematics corresponding to electric field calculations can be simplified. This can essentially be done by adding up the contributions to the electric field made by parts of an object, approximating each part of an object as a point charge.

The System in Question

As discussed in the previous section, we're considering a system abstracted from the particle model we're familiar with, therefore we will make the generalization that our rod of length L has a total charge of quantity Q. For this generalization, we will need to assume that the rod is so thin that we can ignore its thickness.

[image 1]

Since the electric field produced by a charge at any given location is proportional to the distance from the charge to that location, we will need to relate the observation location to the source of the charge, which we will consider the origin of the rod. To do that, we will need to divide the rod into pieces of length \delta y each containing a charge \delta Q. In the image below, you can see what this looks like and the relation that can be found between the observation location and the source, forming the distance vector \vect{r}.

[image 2]

By the pythagorean theorem, we can find the vector \vect{r} as follows:

[image 3]

And to find the unit vector in the direction of \vect{r}, \hat{r}, we do as follows:

[image 4]


Finding the Contribution of Each Piece to the Electric Field

Now that we've set up a model for the system, with the rod broken down into pieces, we can find the contribution of each piece to the electric field of the system. We will start from the electric field equation you learned for a point particle but plug in the parameters for the rod system into the equation.

[image 5]

By mathematically simplifying, we then get the following equation:

[image 6]

Finding the Net Contribution of all Pieces

In the previous section, we found out the contribution to the electric field at a given location of only one of the pieces constituting the rod. In order to figure out the net field at any particular location, we need to add up the electric fields produced by individual pieces along the length of the rod.

We will switch from vector notation for the electric field to the scalar notation for the x- and y-components. (From the vector in the equation above, we can see that the z-component of the electric field at any point is always 0.) The x-component of the electric field is the sum of the x-components of every \delta{y} along the rod, and the y-component of the electric field is the sum of the y-components of every \delta{y} along the rod. We can show this mathematically:

[image 7]

To make use of this relation, because we don't know \delta{Q}, we need to relate it to parameters that we already know about the rod system we're analyzing. We can express \delta{Q} as the charge density of the rod (which is Q/L) times the \delta{y} we've chosen for the system. Thus,

[image 8]

By plugging the above equation into our equations for the x- and y-components of the electric field at a point, we can find the electric field at any point in the system. This technique is called numerical integration and is typically done by computers because the computational complexity is dependant upon the size of \delta{y} with respect to L.

Simplifying

Using calculus, we can simplify a lot of the math required to compute the electric field at any given point. Notationally, all we're doing is switching from the discretely-sized \delta{y} to \textit{dy} and from the sigma notation to an integral starting from -L/2 (the lower end of the rod) and ending at L/2 (the upper end of the rod) as follows:

[image 9] [image 10]

By evaluating the integral, we can determine that the x-component of the electric field at any point is:

[image 11]

Without evaluating the integral for the y-component of the electric field, we can use symmetry to determine that the y-component of the electric field at any given point is 0. Let's consider the contributions to the electric field from the top and bottom halves of the rod at any observation point.

[image 12]

Since the y-components of E_top and E_bottom are of equal magnitude and opposite direction, they cancel each other out, and therefore the y-component of teh electric field at any given point due to the rod is 0.

[image 13]

Finally, because the rod is round and can be rotated, as a convenience, we'll use d (distance from the rod) as opposed to x (distance along the x-direction) to refer to the electric field.

Thus we can simplify electric field calculations for a rod into a form that we can readily use:

[image 14]

Further Simplification

By noting the contributions of each variable to the equation for the electric field, we can make approximations to simplify our math by simply declaring one variable as insignificant.

For example, if we have a system in which the length of a rod is much greater than the magnitude of the distance from the rod (denoted L>>d), we can neglect some of the instances in which d is taken into account as follows:

[image 15]

The Main Idea

Previously, we've learned about the electric field of a point particle. Often, when analyzing physical systems, it is the case that we're unable to analyze each individual particle that composes an object and need to therefore generalize collections of particles into shapes (in this case, a rod) whereby the mathematics corresponding to electric field calculations can be simplified. This can essentially be done by adding up the contributions to the electric field made by parts of an object, approximating each part of an object as a point charge.

The System in Question

Field of a charged ring/disk/capacitor

Week 5

Potential energy

Written by Lukas Yoder, PHYS 2212 Class of Fall 2018


The Main Idea

Potential energy is the energy that an object has because of its characteristics relative to other objects within the universe. In Physics 1

Sign of a potential difference

Week 6

Electric field and potential in an insulator

Moving charges in a magnetic field

Moving charges, electron current, and conventional current

Week 7

Magnetic field of a wire

Magnetic field of a current-carrying loop

Magnetic field of a Charged Disk

Atomic structure of magnets

Week 8

Steady state current

Kirchoff's Laws

Electric fields and energy in circuits

Week 9

Electric field and potential in circuits with capacitors

Week 10

Magnetic force

Week 12

Week 13

Semiconductors

Week 14

Circuits revisited

Week 15

Electromagnetic Radiation

Sparks in the air

Physics 3

Week 1

Classical Physics

Week 2

Week 3

Week 4

Matter Waves

Week 5

Week 6

Week 7

The Hydrogen Atom

Week 8

Week 9

Molecules

Week 10

Statistical Physics

Week 11

Condensed Matter Physics

Week 12

The Nucleus

Week 13

Week 14