Physics Book - User contributions [en]

Superposition Principle

2022-07-24T03:05:57Z

Wnguyen32: /* The Wave Model */

'''Claimed By: Jared Nation'''

== The Big Picture ==
The superposition principle is based on the idea that in a closed system, an object receives a net force equal to the sum of all outside forces acting on it. This is commonly used to calculate the net electric field or magnetic field on an object. More specifically, the Superposition Principle states that the net result of multiple vectors acting on a given place and time is equal to the vector sum of each individual vector. For example, think about two waves colliding. When the waves collide, the height of the combined wave is the sum of the height of the two waves. This principle can be seen throughout nature and theory. A simple example can be seen below where Vector 3 is the sum of Vectors 1 and 2.

[[File:69832.jpg|400px|Imaage: 400 pixels]]
===The Wave Model===

Superposition is easy to think about in the form of waves. Let's say, you are at the beach watching the ocean. A huge 5 foot wave comes towards the land and is about to crash. However, just as this happens, a smaller 1 foot tall wave forms due to the undercurrent and goes out to the ocean. Before the waves crash, they meet. At this meeting point, the combined waves become 6 feet tall (1 + 5). Why is this? This is because of superposition.

[[File:Wav sup.jpeg|500px|center]]

Let's now think about an orchestra. Before the concert starts, individual instruments practice on stage. Sitting in the audience, you can hear the individual instruments. However, the sound of that one instrument does not nearly compare to the sound of the entire orchestra. When the entire orchestra begins to play, the volume is much louder. This is because of the superposition principle. Each instrument produces sound waves. When an entire orchestra performs at once, the sound waves from each instrument are all summed together. The superposition principle is not only seen in waves. The superposition principle can be seen in forces, electric fields, magnetic fields, etc. However, it is easier to understand when looking at waves that everyone notices.

[[File:Standing_wave_2.gif]]

This gif depicts the superposition principle. The peaks and the troughs (the bottom of the waves) are summed together as the different waves overlap. Just as the waves of the ocean and the sound waves of the orchestra summed together, the waves in this gif sum together.

===A Mathematical Model===

The Superposition Principle is derived from the properties of additivity and homogeneity for linear systems which are defined in terms of a scalar value of a by the following equations:

<math>F(x_1 + x_2) = F(x_1) + F(x_2)\quad Additivity</math>

<math>aF(x) = F(ax)\quad Homogeneity</math>
[[File:Superposition.gif|thumb|Example of the Superposition of Electric Fields using charges with changing locations]]

The principle can be applied to any linear system and can be used to find the net result of functions, vectors, vector fields, etc. For the topic of introductory physics, it will mainly apply to vectors and vector fields such as electric forces and fields.

If given a number of vectors passing through a certain point, the resultant vector is given by simply adding all the the vectors at that point. For example, for a number of uniform electric fields passing though a single point, the resulting electric field at that point is given by:

<math>\vec{E} = \vec{E}_{1} + \vec{E}_{2} +...+ \vec{E}_{n} = \sum_{i=1}^n\vec{E}_{i}</math>

This same concept can be applied to electric and magnetic fields and forces. This is helpful when dealing with scenarios in which the effect that multiple point charges have on each other is in an area void of other fields. By adding together each of the contributing field vectors at a specific point you can figure out the net contribution that all of the present charges have on the overall field experienced at that location.

You can calculate the electric field of a single point charge by using the following equation:
<math>\vec{E} = \frac{1}{4 \pi \epsilon_0}\sum_{i=1}^n\frac{q_i}{r_i^2}\hat{r_i}</math>

Where:

<math>\frac{1}{4 \pi \epsilon_0}</math> is a constant that equals <math>9e9</math>.

<math>q_i</math> is the charge of the particle you are studying. The sign of the particle matters (positive for proton, negative for electron.)

<math>r_i</math> is the magnitude of the distance between the particle and the observation location.

<math>\hat{r_i}</math> is the vector pointing from the particle to the observation location.

Conceptually is is also important to note that an object cannot exert a force on itself, so the sum of all of the forces does not include the object itself.

==Examples==

When attempting to solve these problems be sure to show all steps in your solution and include diagrams whenever possible.

===Simple - Two Point Charges and One Dimension===
[[File:SuperpositionExample.png]]

If <math>Q_1</math> equals 1e-4 C and <math>Q_2</math> equals 1e-5C, what is the net electric field at the midpoint of both charges?

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

'''''Click for Solution'''''
<div class="mw-collapsible-content">
We know that the distance of each charge to the midpoint is <math>1m</math>. Since this value is the magnitude of the distance it will serve as our <math>r_i</math> value for both of the electric field formulas.

Our <math>\hat{r_1}</math> value for <math>E_1</math> will be <math>1</math> and the <math>\hat{r_2}</math> value for <math>E_2</math> will be <math>-1</math>, if you use the midpoint as your observation point in the x-direction.

Now that we have all the necessary values to use the electric field formula, we can plug them in.

<math>\vec{E_1} = \frac{1}{4 \pi \epsilon_0}\frac{q_1}{r_1^2}\hat{r_1} = (9e9)\frac{1e-4C}{1m^2}(1m) = 9e5\frac{N}{C}</math>
<math>\vec{E_2} = \frac{1}{4 \pi \epsilon_0}\frac{q_2}{r_2^2}\hat{r_2} = (9e9)\frac{1e-5C}{1m^2}(-1m) = -9e4\frac{N}{C}</math>

By this point, you have the individual contributions of each charge. To get the net contribution of all the charges, you only need to add them.

<math>\vec{E_1}+\vec{E_2}=9e5\frac{N}{C}-9e4\frac{N}{C}= 8.1e5\frac{N}{C}</math>

This means that the net electric field will point towards the right with a magnitude of <math>8.1e5\frac{N}{C}</math>.
</div>
</div>
=== Medium - Three Point charges and two Dimensions ===
[[File:Superposfile2.JPG]]

----
Using k for the constant term
<math> E = Q1 * rhat/ (r)^2 </math>

<math> F = q3 * E </math>

<math> r = (4d, -3d,0> - (0,3d,)) = (4d,-6d,0) </math>

<math> k * Q3 * Q1 * rhat/ (r)^2 </math>

<math> k* Q^2 * 1/(52d^2) * (2/13 * sqrt(13),-3/13 sqrt(13),0) </math>

=== Medium - Three Point charges and two Dimensions===
[[File:Superpos file.JPG]]

----

What is the magnitude of the net force on the charge -q'?

To begin, we recognize that this problem will be using the superposition principle. It is additionally
important to recognize that objects can't exert forces on themselves. Keeping these in mind, The force exerted on -q' will be a combination of the forces exerted by 3q and negative q.

For this problem, let's consider -q' to be q3, 3q to be q1, and -1 to be q2 and, For the sake of simplicity, we are going to call our constant 'k'

Let's find the forces separately then add them together:

<math> k * (q1*q3)/(r)^2 * rhat </math>

Now that we have the general form of the equation, we plug in variables.

<math> F1 = k * 3q(-q')/(sqrt(3) * l)^2 * (sin(theta),cos(theta),0) </math>
<math>

= k * (-q*q')/ (l^2) * (sin(theta),cos(theta),0) </math>

Great! Now we have the first force from q1, now we need to find the force from q2

<math> F2 = k * (q2*q3)/ (r)^2 * rhat </math>

<math> = k * (q*q)/(l^2) * (-sin(theta), cos(theta),0) </math>

Now that we've found both forces, we add them together.

<math> Fnet = F1+F2 </math>
<math> = k * (-q*q')/ (l^2) * (sin(theta),cos(theta),0) + k * (q*q)/(l^2) * (-sin(theta), cos(theta),0) </math>
<math> = k * (q*q')/(l^2) * (-2sin(theta),0,0) </math>

Once we were able to recognize our problem, it became as simple as plugging in variables and using algebra to solve the problem.

Additionally, if we were asked "What is the direction of the force on the charge -q'?"

Looking back at our work, we can see that our vertical components cancel out, leaving
<math> -2sin(theta) </math>
Indicating our direction is negative x.

We could get to this conclusion without looking at the math by understanding that electric field falls off in a way that resembles
<math> 1/(r)^2 </math>

For q1, it is sqrt(3) times farther than q2, and 3 times the charge. Which is equivalent to
<math> 3*q/(sqrt(3))^2 </math>
Which is equivalent to q2 in the vertical direction. In the horizontal direction, q2 is slightly closer, meaning that the total net charge should be in the negative x direction.

===Medium - Two Point Charges and Two Dimensions===
[[File:BrooksEx1.png]]

If <math>Q_1</math> and <math>Q_2</math> are positive point charges with a charge of e, what is the net electric field at point P?

<div class="toccolours mw-collapsible mw-collapsed">
'''''Click for Solution'''''
<div class="mw-collapsible-content">
To begin this problem, the first step is to find <math>r_1</math> and <math>r_2</math>, the vectors from the charges to point P as well as their magnitudes:

<math>\vec{r_1} = 3\hat{i}+2\hat{j}-(0\hat{i}+0\hat{j})\Rightarrow\vec{r_1} = 3\hat{i}+2\hat{j}</math>

<math>||\vec{r_1}|| = \sqrt{3^2 + 2^2} =\sqrt{13}</math>

<math>\vec{r_2} = 3\hat{i}+2\hat{j}-(4\hat{i}+0\hat{j})\Rightarrow\vec{r_2} = -1\hat{i}+2\hat{j}</math>

<math>||\vec{r_2}|| = \sqrt{-1^2 + 2^2} =\sqrt{5}</math>

Using these in the equation for an electric field from a point charge, you get:

<math>\vec{E_1} = \frac{1}{4 \pi \epsilon_0}\frac{Q_1}{||r_1||^2}\hat{r_1}=\frac{1}{4 \pi \epsilon_0}\frac{e}{13}<\frac{3}{\sqrt{13}}\hat{i}+\frac{2}{\sqrt{13}}\hat{j}> = <9.21E-11\hat{i}+6.14E-11\hat{j}> </math>N/C

<math>\vec{E_2} = \frac{1}{4 \pi \epsilon_0}\frac{Q_2}{||r_2||^2}\hat{r_2}=\frac{1}{4 \pi \epsilon_0}\frac{e}{5}<\frac{-1}{\sqrt{5}}\hat{i}+\frac{2}{\sqrt{5}}\hat{j}> = <-1.29E-10\hat{i}+2.58E-10\hat{j}> </math>N/C

Then, simply add the two electric fields together:

<math>\vec{E}=\vec{E_1}+\vec{E_2} = <-3.69E-11\hat{i}+3.19E-10\hat{j}></math>
</div>
</div>

===Difficult - Five Point Charges and Three Dimensions===

[[File:brooksEx2.png]]

If all point charges have a charge of e, what the the net electric field present at point L?
<div class="toccolours mw-collapsible mw-collapsed">
'''''Click for Solution'''''
<div class="mw-collapsible-content">
This problem is similar to the previous examples but this one includes the z axis and more points. Since there are 5 points and you have already had some practice, you will only see the procedure for the first one. We will still work through the whole problem. Again, first each vector and magnitude:

<math>\vec{d_5} = 0\hat{i}+0\hat{j}+0\hat{k}-(2\hat{i}-1\hat{j}-1\hat{k})-\Rightarrow\vec{d_5} = -2\hat{i}+1\hat{j}+1\hat{k}</math>

<math>||\vec{d_5}|| = \sqrt{(-2)^2 + 1^2 + 1^2} = 2</math>

Do the same for each of the other point charges and plug them into the electric field formula:

<math>\vec{E_5} = \frac{1}{4 \pi \epsilon_0}\frac{Q_5}{||d_5||^2}\hat{d_5}=\frac{1}{4 \pi \epsilon_0}\frac{e}{4}<\frac{-2}{2}\hat{i}+\frac{1}{2}\hat{j}+\frac{1}{2}\hat{k}> = <-3.6E-10\hat{i}+1.8E-10\hat{j}+1.8E-10\hat{k}></math>N/C

Follow the same steps for the other electric fields and add them all together to get your final answer:

<math>\vec{E_1} = <1.44E-9\hat{i}+0\hat{j}+0\hat{k}></math>N/C

<math>\vec{E_2} = <0\hat{i}-1.44E-9\hat{j}+0\hat{k}></math>N/C

<math>\vec{E_3} = <-1.29E-10\hat{i}+0\hat{j}+2.58E-10\hat{k}></math>N/C

<math>\vec{E_4} = <0\hat{i}+0\hat{j}-1.44E-9\hat{k}></math>N/C

<math>\vec{E} = \vec{E_1}+\vec{E_2}+\vec{E_3}+\vec{E_4}+\vec{E_5}=<9.51E-10\hat{i}-1.26E-9\hat{j}+1.00E-9\hat{k}></math>N/C
</div>
</div>

==Connectedness==
[[File:Superposition Example.PNG|thumb|left|A diagram of a circuit you will be able to analyze further on by using the superposition of currents.]]
The superposition principle is critical as you progress to more complicated topics in Physics. This principle simplifies many topics that you will encounter further ahead, by assuming that their net contribution will be equal to the addition of all the individual ones. You will use superposition with electric fields and forces, as well as magnetic fields and forces. In addition, if in a problem you are only given the final net contribution of any type of field, you can inversely solve it by constructing equations that relate the individual formulas. You can then plug in values, and subtract in order to find the magnitudes of each individual contributor. This will also enable you to better analyze different types of circuits.

However, the superposition principle also aids outside of the world of abstract electromagnetism. It is also useful for static problems, which you might encounter if you are a Civil or Mechanical Engineer, or systems and circuitry, which you might encounter if you are a Biomedical engineer.

==History==
[[File:Danielbernoulli.jpg|thumb|right|Daniel Bernoulli]]
The superposition principle was supposedly first stated by [[Daniel Bernoulli]]. Bernoulli was a famous scientist whose primary work was in fluid mechanics and statistics. If you recognize the name, he is primarily remembered for discovering the Bernoulli Principle. In 1753, he stated that, "The general motion of a vibrating system is given by a superposition of its proper vibrations." This idea was at first rejected by some other popular scientists until it became widely accepted due to the work of Joseph Fourier. Fourier was a famous scientist known for his work on the Fourier Series, which is used in heat transfer and vibrations, as well as his discovery of the Greenhouse Effect. Fournier’s support legitimized Bernoulli’s claims and has made a huge impact on history.

== See also ==

===Next Topics===
Now that you have a solid understanding of the Superposition Principle, you can move on to these topics:
*[[Electric Field]]
**[[Point Charge]]
**[[Electric Dipole]]
*[[Electric Force]]
*[[Magnetic Field]]
*[[Magnetic Force]]
*[[Superposition Principle]]
===Further reading===

Books, Articles or other print media on this topic

[http://www.acoustics.salford.ac.uk/feschools/waves/super.php Superposition of Waves]

[http://whatis.techtarget.com/definition/Schrodingers-cat Schrodingers' Cat from a Superposition Point of View]

===External Links===

[https://www.youtube.com/watch?v=p3Xugztl9Ho Visual Explanation of Superposition in Electric Fields]

[https://www.youtube.com/watch?v=mdulzEfQXDE Electric Fields and Superposition Principle Crash Course Video]

[https://www.youtube.com/watch?v=S1TXN1M9t18 Instructional video on how to calculate the net electric field using the superposition principle]

[https://www.youtube.com/watch?v=lCM5dql_ul0 Additional Video on calculating superposition principle]

[https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1479978&tag=1 Using superposition principle to analyze solar cells]

==References==
[http://physics-help.info/physicsguide/electricity/electric_field.shtml Electric Fields]

[https://en.wikipedia.org/wiki/Superposition_principle Superposition Principle]

[https://en.wikipedia.org/wiki/Daniel_Bernoulli Daniel Bernoulli Biography]

[https://en.wikipedia.org/wiki/Joseph_Fourier Joseph Fourier Biography]
[[Category:Fields]]

Point Charge

2022-07-24T02:54:43Z

Wnguyen32: /* Middling */

This page is all about the [[Electric Field]] due to a Point Charge.

==The Main Idea ==
(Ch 13.1 in ''Matter & Interactions Vol. 2: Modern Mechanics, 4th Edition by R. Chabay & B. Sherwood'')

'''Point Charge/Particle''' - an object with a radius that is very small compared to the distance between it and any other objects of interest in the system. Since it is very small, the object can be treated as if all of its charge and mass are concentrated at a single "point".
*Electrons and Protons are always considered to be point particles unless stated otherwise

<u> 2 types of point charges: </u>
*Protons (e) --> positive point charges, ( q = 1.6e-19 Coulombs)
*Electrons (-e) --> negative point charges, (q = -1.6e-19 Coulombs)

''Like'' point charges ''attract'', ''opposite'' point charges ''repel''.
ex.<table border> <tr>
<th> Point Charges </th>
<th> Result </th>
<th>Diagram</th>
</tr>
<tr>
<td> 1 proton, 1 electron</td>
<td> Attract </td>
<td>[[File:Proton_electron_attraction.png]]</td>
</tr>
<tr>
<td>2 protons </td>
<td> Repel </td>
<td>[[File:Proton_repulsion.png]]</td>
</tr>
<tr>
<td>2 electrons </td>
<td> Repel </td>
<td>[[File:Electron_repulsion.png]]</td>
</tr>
</table border>

===The Electric Field===
(Ch 13.3 in ''Matter & Interactions Vol. 2: Modern Mechanics, 4th Edition by R. Chabay & B. Sherwood'')

The electric field created by a charge is present throughout space at all times, whether or not there is another charge around to feel its effects.

Electric Field of a Charge Observed at a location: F = Eq
*F = Force on particle 2
*E = electric field at source location
*q = charge of particle 2
The magnitude of the electric field decreases with increasing distance from the point charge.

<table border>
<tr>
<td>The electric field of a positive point charge points radially outward</td>
<td>The electric field of a negative point charge points radially inward</td>
</tr>
<tr>
<td>[[File:Proton_electric_field.png]] </td>
<td> [[File:Electron_electric_field.png]] </td>
</tr>
</table border>

===A Mathematical Model===

====Electric Field due to Point Charge====
(Ch 13.4 in ''Matter & Interactions Vol. 2: Modern Mechanics, 4th Edition by R. Chabay & B. Sherwood'')

Electric Field of a Point Charge (<math>\vec E</math>):

<math>\vec E=\frac{1}{4 \pi \epsilon_0 } \frac{q}{\mid\vec r\mid ^2} \hat r</math> (Newtons/Coulomb)

*<math>\frac{1}{4 \pi \epsilon_0 } </math> is Coulomb's Constant and is approximately <math>8.987*10^{9}\frac{N m^2}{C^2} </math>
*'''''q''''' is the charge of the particle
*'''''r''''' is the magnitude of the distance between the observation location and the source location
*<math>\hat r </math> is the unit vector in the direction of the distance from the source location to the observation point.

The direction of the electric field at the observation location depends on the both the direction of <math>\hat r </math> and the sign of the source charge.
*If the source charge is positive, the field points away from the source charge.
*If the source charge is negative, the field points toward the source charge.

====Coulomb Force Law for Point Charges====
(Ch 13.2 in ''Matter & Interactions Vol. 2: Modern Mechanics, 4th Edition by R. Chabay & B. Sherwood'')

<math>\mid\vec F\mid=\frac{1}{4 \pi \epsilon_0 } \frac{\mid Q_1Q_2 \mid}{r^2}</math>

Coulomb's law is one of the four fundamental physical interactions, and it describes the magnitude of the electric force between two point-charges.
<math>Q_1, Q_2</math>= The charge of two particles of interest

*<math>\mid\vec F\mid=\frac{1}{4 \pi \epsilon_0 }</math> = constant,
* <math>Q_1, Q_2</math> = the magnitudes of the point charges
*r = The distance between the two particles

====Connection Between Electric Field and Force====
The force on a source charge is determined by <math> F = Eq </math> where '''''E''''' is the electric field and '''''q''''' is the charge of a test charge in Coulombs.

By solving for the electric field in <math> F = Eq </math>, with F modeled by Coulomb's Law, you get the equation for the electric field of the point charge:

<math> E = \frac{F}{q_2} = \frac{1}{4 \pi \epsilon_0 } \frac{q_1q_2}{r^2}\frac{1}{q_2}\hat r = \frac{1}{4 \pi \epsilon_0 } \frac{q_1}{r^2} \hat r </math>

===A Computational Model===

Below is a link to a code which can help visualize the Electric Field at various observation locations due to a proton. Notice how the arrows decrease in size by a factor of <math> \frac{1}{r^{2}} </math> as the observation location gets farther from the proton. The magnitude of the electric field decreases as the distance to the observation location increases.

[[File:First code.gif]]

Two adjacent point charges of opposite sign exhibit an electric field pattern that is characteristic of a dipole. This interaction is displayed in the code below. Notice how the electric field points towards the negatively charged point charge (blue) and away from the positively charged point charge (red).

[[File:Code_2.png]]

==Examples==

===Simple===
There is an electron at the origin. Calculate the electric field at <4, -3, 1> m.

<table border>
<tr>
<td> <b>Step 1.</b> Find <math>\hat r</math>
Find <math>\vec r_{obs} - \vec r_{electron}:

((4,-3,1) - (0,0,0) = <4,-3,1> </math>m.

Calculate the magnitude of <math>\vec r</math>:

(<math>\sqrt{4^2+(-3)^2+1^2}=\sqrt{26}</math> From <math>\vec r</math>, find the unit vector <math>\hat{r}.</math> <math> <\frac{4}{\sqrt{26}},\frac{-3}{\sqrt{26}},\frac{1}{\sqrt{26}}> </math>
</td></tr>

<tr><td>'''Step 2:''' Find the magnitude of the Electric Field

<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r^2} = \frac{1}{4 \pi \epsilon_0 } \frac{-1.6 * 10^{-19}}{26} </math> </td>
<td> </td></tr>

<td>'''Step 3:''' Multiply the magnitude of the Electric Field by <math>\hat{r}</math> to find the Electric Field
<math>E = \frac{1}{4 \pi \epsilon_0 } \frac{-1.6 * 10^{-19}}{26}*<\frac{4}{\sqrt{26}},\frac{-3}{\sqrt{26}},\frac{1}{\sqrt{26}}>

= <-4.34*10^{-11},3.26*10^{-11},-1.09*10^{-11}> N/C </math></td>
</table border>

===Middling===
A particle of unknown charge is located at <-0.21, 0.02, 0.11> m. Its electric field at point <-0.02, 0.31, 0.28> m is <math><0.124, 0.188, 0.109> </math> N/C. Find the magnitude and sign of the particle's charge.

Given both an observation location and a source location, one can find both r and <math>\hat{r}</math> Given the value of the electric field, one can also find the magnitude of the electric field. Then, using the equation for the magnitude of electric field of a point charge,<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r^2} </math> one can find the magnitude and sign of the charge.

<table border>
<tr>
<td> <b>Step 1.</b> Find <math>\vec r_{obs} - \vec r_{particle} </math>:

<math>\vec r = <-0.02, 0.31, 0.28> m - <-0.21, 0.02, 0.11> m = <0.19,0.29,0.17> m </math>

To find <math>\vec r_{mag} </math>, find the magnitude of <math><0.19,0.29,0.17></math>

<math>\sqrt{0.19^2+0.29^2+0.17^2}=\sqrt{0.1491}= 0.39</math>
</td></tr>

<tr><td><b>Step 2:</b> Find the magnitude of the Electric Field:

<math>E= <0.124, 0.188, 0.109> N/C</math>

<math>E_{mag} = (\sqrt{0.124^2+0.188^2+0.109^2}=\sqrt{0.0626}=0.25</math> </td></tr>

<td>'''Step 3:''' Find '''''q''''' by rearranging the equation for <math>E_{mag}</math>

<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r^2} </math>

By rearranging this equation we get

<math> q= {4 pi * epsilon;_0 } *{r^2}*E_{mag} </math>

<math> q= {1/(9*10^9)} *{0.39^2}*0.25 </math>

<math> q= + 4.3*10^{-12} C </math></td>

</table border>

===Difficult===
The electric force on a -2mC particle at a location (3.98 , 3.98 , 3.98) m due to a particle at the origin is <math>< -5.5*10^{3} , -5.5*10^{3}, -5.5*10^{3}></math> N. What is the charge on the particle at the origin?

Given the force and charge on the particle, one can calculate the surrounding electric field. With this variable found, this problem becomes much like the last one.
<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r_{mag}^2} </math> to find the rmag value. To find <math>\hat r</math> we can find the direction of the electric field as that is obviously going to be in the same direction as <math>\hat r</math>. Then, once we find <math>\hat r</math>, all that is left to do is multiply <math>\hat r</math> by rmag and that will give us the <math> r</math> vector. We can then find the location of the particle as we know <math>r=r_{observation}-r_{particle}</math>

<table border>
<tr>
<td> <b>Step 1.</b> Find the magnitude of the Electric field:
<math> F = Eq </math>
<math> <5.5e3, -7.6e3, 0> = E * -2mC </math>
<math> E = \frac{< -5.5e3 , -5.5e3, -5.5e3>}{-2mC}

= <2.75e6 , 2.75e6, 2.75e6> </math> N/C
</td></tr>

<tr><td><b>Step 2:</b> Find <math>\vec r_{obs} - \vec r_{particle} </math>.

<math>\vec r = <3.98 , 3.98 , 3.98> m - <0 , 0 , 0> m = <3.98 , 3.98 , 3.98> m </math>

To find <math>\vec r_{mag} </math>, find the magnitude of <math><3.98 , 3.98 , 3.98></math>

<math>\sqrt{3.98^2+3.98^2+3.98^2}=\sqrt{47.52}= 6.9</math>
</td></tr>

<td> '''Step 4:''' Find '''''q''''' by rearranging the equation for <math>E_{mag}</math>

<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r^2} </math>

By rearranging this equation we get

<math> q= {4 pi * ε_0 } *{r^2}*E_{mag} </math>

<math> q= {1/(9e9)} *{6.9^{2}}*4.76e6} </math>

<math> q= + 0.253 C </math></td>

</table border>

==Connectedness==
''1. How is this topic connected to something that you are interested in?''

I think the topic is interesting because electric fields are inside the human body and we self-create these fields constantly. It's really cool to think about how your body might be responding and creating these fields continuously and what happens when the balances are thrown off by something (i.e. illness, injury).

''2. How is it connected to your major?''

I'm a business major, but I'm pre-med so it's interesting to think about point charges/electric fields in the macroscopic lens and how they can combine with other forces (magnetic --> electromagnetic forces) to affect your internal systems and processes that occur without you consciously thinking about them.

''3. Is there an interesting industrial application?''

PEMF stands for pulsed electricmagnetic field, which is used by athletes during recovery or physical therapy. There is a difference in cell voltage between healthy cells and diseased cells. While healthy cells maintain a voltage around 70-100 mV, cells with illness have diminished voltage around 40mV. PEMF restores that optimal voltage in damaged cells. Through the utilization of low frequency pulsed electromagnetic fields at a high intensity, voltage in the damaged cells is increased. This stimulates cellular repair and recharges the body’s cells to optimize their performance.

PEMF therapy creates a magnetic field, which increases the movement of ions and electrolytes in the tissues and fluids of the body. The magnetic field helps cells increase ATP production, which restores and / or maintains normal cellular function, speeding up the tissues healing process, repairing damaged tissue, improving circulation, and increasing cellular energy.

==History==
[[File:CoulombCharles300px.jpg]]
''Charles de Coulomb''

Charles de Coulomb was born in June 14, 1736 in central France. He spent much of his early life in the military and was placed in regions throughout the world. He only began to do scientific experiments out of curiously on his military expeditions. However, when controversy arrived with him and the French bureaucracy coupled with the French Revolution, Coulomb had to leave France and thus really began his scientific career.

Between 1785 and 1791, de Coulomb wrote several key papers centered around multiple relations of electricity and magnetism. This helped him develop the principle known as Coulomb's Law, which confirmed that the force between two electrical charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This is the same relationship that is seen in the electric field equation of a point charge.

== See also ==

[[Electric Field]] <br>
[[Electric Force]] <br>
[[Superposition Principle]] <br>
[[Electric Dipole]]

===Further reading===

Principles of Electrodynamics by Melvin Schwartz
ISBN: 9780486134673

Electricity and Magnetism: Edition 3 , Edward M. Purcell David J. Morin

===External links===

Some more information:

*http://hyperphysics.phy-astr.gsu.edu/hbase/electric/epoint.html
*http://www.physics.umd.edu/courses/Phys260/agashe/S10/notes/lecture18.pdf
*https://www.reliantphysicaltherapy.com/services/pulsed-electromagnetic-field-pemf

==References==

Chabay. (2000-2018). ''Matter & Interactions'' (4th ed.). John Wiley & Sons.

PY106 Notes. (n.d.). Retrieved November 27, 2016, from http://physics.bu.edu/~duffy/py106.html

Retrieved November 28, 2016, from http://www.biography.com/people/charles-de-coulomb-9259075#controversy-and-absolution

[[Category:Fields]]