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Claimed by Sam Bureau Spring 2024
This page is all about the [[Electric Field]] due to a Point Charge.
This page is all about the [[Electric Field]] due to a Point Charge.


== Electric Field==


A Work In Progress by Brandon Weiner:  [[User:bweiner6|bweiner6]] ([[User talk:bweiner6|talk]])


===A Mathematical Model of Electric Field due to Point Charge===
==The Main Ideas ==
(Ch 13.1 in ''Matter & Interactions Vol. 2: Modern Mechanics, 4th Edition by R. Chabay & B. Sherwood'')


The Electric Field of a Point Charge can be found by the formula:
'''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>
   
   
<math>\vec E=\frac{1}{4 \pi \epsilon_0 } \frac{q}{r^2} \hat r,\text{where } \frac{1}{4 \pi \epsilon_0 } \text{is approximately }  9*10^{9} \frac{N m^2}{C^2} \text{, q is the charge of the particle,}


\text{r is the magnitude of the distance}
===The Electric Field===
\text{ between the point charge and the observation point, and }</math>  
(Ch 13.3 in ''Matter & Interactions Vol. 2: Modern Mechanics, 4th Edition by R. Chabay & B. Sherwood'')
<math>\hat r \text { is the direction of the distance from } 
 
\text{the point charge to the observation point.}
In general, the electric field evaluates the affect of the source on the surrounding objects and area. 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. Therefore, the concept of the electric field by a point charge describes the interactions that can happen at a distance, due to these affects caused by this point charge.
\text{ This equation becomes Coulomb's Law when multiplied by a second particles's charge. } </math>
 
Important to differentiate that Electric Force does not equal the Electric Field.
 
Electric Field of a Charge Observed at a location: F = Eq
*F = Force on particle
*E = electric field at source location
*q = magnitude of the charge of particle (assume q= 1.6 x 10^-19 unless stated otherewise)
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'')
 
The magnitude of the electric field decreases with increasing distance from the point charge. This is described by the equation below:
 
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, in the same direction as <math>\hat r </math>.
*If the source charge is negative, the field points toward the source charge, in the opposite direction as <math>\hat r </math>.
 
====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 charges of the 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===
The direction of electric force also depends on the direction of the electric field too:
 
*If the source charge is positive, the field points away from the source charge, in the same direction as the electric force.
*If the source charge is negative, the field points toward the source charge, in the opposite direction as the electric force.
 
====Electric Field Superposition (Point Charges)====
 
When there are multiple point charges present, the total net electric field <math> Enet </math>, is equal to the sum of the electric field of each independent point charge present.
 
This is due to  concept of Superposition which is when the total effect is the sum of the effects of each part.


Here is a link to some code which shows the Electric Field due to an Proton at different points.
When it comes to the Electric Field Superposition of Point Charges, be sure to remember that:


<html> <iframe src="https://trinket.io/embed/glowscript/cf036f65f7?start=result" width="100%" height="600" frameborder="0" marginwidth="0" marginheight="0" allowfullscreen></iframe> </html>
*A charge cannot exert a force on itself
*Assume that the source charges do not move. (Therefore <math> Fnet = 0 </math>)


==Examples==




'''Problem 1:''' There is a proton at <1,2,3>. Calculate the electric field at <2,-1,3>.  
[[File:IMG_1DCC0A11C7B8-1.jpeg]]


'''Step 1:''' Find <math>\hat r</math>
===A Computational Model===


Find <math>\vec r_{obs} - \vec r_{proton}   (<2,-1,3> - <1,2,3> = <1,-3,0>) </math>
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.


Calculate the magnitude of r.  (<math>\sqrt{1^2+(-3)^2+0^2}=\sqrt{10}</math>


From r, find the unit vector <math>\hat{r}.</math> <math>  <\frac{1}{\sqrt{10}},\frac{-3}{\sqrt{10}},\frac{0}{\sqrt{10}}> </math>
[[File:First code.gif]]


'''Step 2:''' Find the magnitude of the Electric Field
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).


<math> E= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r^2} = \frac{1}{4 \pi \epsilon_0 } \frac{1.6 * 10^{-19}}{10} </math>
[[File:Code_2.png]]


'''Step 3:''' Multiply the magnitude by <math>\hat{r}</math> to find the Electric Field
==Examples==


E= <math>  \frac{1}{4 \pi \epsilon_0 } \frac{1.6 * 10^{-19}}{10}*<\frac{1}{\sqrt{10}},\frac{-3}{\sqrt{10}},\frac{0}{\sqrt{10}}>=<4.554*10^{-11},-1.366*10^{-10},0>  N/C    </math>
===Simple===
There is an electron at the origin. Calculate the electric field at <4, -3, 1> m.  


[[File:Screenshot_2024-04-13_160839.png]]


'''Problem 2:'''
===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.


Once the basic equation and execution of Coulomb’s Law is understood, it can seem that problems with point charges cannot possibly get any harder. The three variables are Electric Field, charge, and distance between the charges. Any problem must provide two of the three, and the student “plugs and chugs” to get the third. It may seem that these problems could not possibly be made harder without looking at a bunch of point charges together in the shape of a disk or ring or rod. However, this is not true.
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.  


The variable that one often does not consider is that the third variable, distance between charges, is actually the negative summation of two variables, observation location and location of charge (or source location). Meaning, to know the distance between the charges (or the charge and wherever you are calculating the field), you must know both the observation location and where the charge is. The example problem below elucidates how this can become quite tricky.  
<table border>
  <tr>
    <td> <b>Step 1.</b> Find <math>\vec r_{obs} - \vec r_{particle} </math>:


'''Problem''': There is a charged particle at an unknown location. Its charge is <math> Q </math> Coulombs. At location (<math>obs_{x}, obs_{y}, obs_{z}) </math> meters, an Electric Field of <math>(E_{x},E_{y},E_{z}) N/C </math> is observed. At what location is the charged particle? Assume the Electric Field is caused only by the single charged particle, no other charges are close.
<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>
You could use your own made up values for the three givens, but for the solution we will us <math> Q = 2*10^{-8} </math> Coulombs, location is <math> (.26, .75, .09) </math> meters, and Electric Field is <math> (400,800,500) N/C </math>.


'''Solution''':
To find <math>\vec r_{mag} </math>, find the magnitude of <math><0.19,0.29,0.17></math>
First, I will summarize the steps. Then I will give a detailed explanation of why each step is the correct one, plugging in the numbers for the student to follow along with. The last part of the solution guide is a Matlab code that solves any inputs to this problem so the student can visualize the solving of the problem in a slightly different way, as well as generate nir own problems.  


'''Summary of Steps''':
<math>\sqrt{0.19^2+0.29^2+0.17^2}=\sqrt{0.1491}= 0.39</math>
</td></tr>


Step 1: Find Magnitude of Electric Field by using Pythagorean Theorem on the components
    <tr><td><b>Step 2:</b> Find the magnitude of the Electric Field:


Step 2: Find Magnitude of  <math>\vec r </math> (which is distance between charge and observation location) by rearranging Coulomb’s Law
<math>E= <0.124, 0.188, 0.109> N/C</math>  


Step 3: Find unit vector of Electric Field
<math>E_{mag} = (\sqrt{0.124^2+0.188^2+0.109^2}=\sqrt{0.0626}=0.25</math>  </td></tr>


Step 4: Using sign of the charge and unit vector of Electric Field, find unit vector of <math>\vec r </math>
    <td>'''Step 3:''' Find '''''q''''' by rearranging the equation for <math>E_{mag}</math>


Step 5: Find the vector <math>\vec r </math> by multiplying the unit vector and magnitude of <math>\vec r </math>
<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r^2} </math>


Step 6: Solve for position of charge
By rearranging this equation we get


Detail Explanation with numbers:
<math> q= {4 \pi * \epsilon_0 } *{r^2}*E_{mag} </math>


First, let’s clarify what <math>\vec r </math> is. We observe an Electric Field, <math>\vec E </math>, at a given point; thus, this is the observation location. The distance from the charged particle to this observation location is <math>\vec r </math>. So for this problem, we know the observation location, but not where the charge is. Observation location is called <math> (obs_{x}, obs_{y}, obs_{z} </math>).
<math> q= {1/(9*10^9)} *{0.39^2}*0.25 </math>  


So:
<math> q= + 4.3*10^{-12} C </math></td>
<math> r = (obs_{x}, obs_{y}, obs_{z}) – (x, y, z)  </math>


Or in this case:
</table border>


<math> r = (.26,.75, .09) – (x,y,z) </math>
===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?


The problem is asking us what (x,y,z) is, so to solve it, we must find <math>\vec r </math>.  
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>


However, the issue is Coulomb’s Law doesn’t use r in vector form. It uses the magnitude of r and the unit vector of r, which from now on will be referred to as <math> r_{mag} </math> and <math>\hat{r} </math> respectively.
<table border>
  <tr>
    <td> <b>Step 1.</b> Find the magnitude of the Electric field:
<math> F = Eq </math>  
<math> = E * -2mC </math>  


So to find r vector, we need both <math> r_{mag} </math> and <math>\hat{r} </math>
<math> E = \frac{< -5.5e3 , -5.5e3, -5.5e3>}{-2mC}


Magnitude is easy because we can just use the magnitude version of Coulomb’s Law:


<math> E_{mag} = \frac{1}{4 \pi \epsilon_0} * \frac{q}{(rmag)^{2}} </math>
= <2.75e6 , 2.75e6, 2.75e6> </math> N/C
</td></tr>


We are given <math> q </math>, and <math> E_{mag} </math> can easily be calculated from the given <math>\vec E </math> using Pythagorean Theorem. And then we can rearrange Coulomb’s Law from above so that rmag is on one side of the equation.
    <tr><td><b>Step 2:</b> Find <math>\vec r_{obs} - \vec r_{particle} </math>.


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


<math> r_{mag} = \sqrt{\frac{1}{4 \pi \epsilon_0} * \frac{q}{E_{mag}}} </math>
To find <math>\vec r_{mag} </math>, find the magnitude of <math><3.98 , 3.98 , 3.98></math>  


And with our numbers:
<math>\sqrt{3.98^2+3.98^2+3.98^2}=\sqrt{47.52}= 6.9</math>
  </td></tr>


<math> r_{mag} = \sqrt{9*10^{9} * \frac{2*10^{-8}}{1024.7}} = .42 </math>
    <td> '''Step 3:''' Find '''''q''''' by rearranging the equation for <math>E_{mag}</math>


But how do we find <math>\hat{r} </math> so we can finish calculating the <math>\vec r </math>?
<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r^2} </math>


Well, think about this. A unit vector is simply a way of mathematically describing direction. All of the direction of the Electric Field comes from the direction of the vector between the two locations, or <math>\vec r </math>. <math> Q </math> is a scalar and alters direction in no way, with one exception. All of the direction of <math>\vec E </math> comes from <math>\vec r </math>,then. Said mathematically, <math>\hat{E} </math> equals the absolute value of <math>\hat{r} </math>. Why absolute value? It is necessary because of the exception mentioned above. The charge can flip the direction of the field 180 degrees depending on its sign. Replacing a charged particle with a particle of the opposite sign results in an <math>\vec E </math> in the exact opposite direction, despite <math>\hat{r} </math> remaining the same.
By rearranging this equation we get


So:
<math> q= {4 pi * \epsilon_0 } *{r^2}*E_{mag} </math>  
<math>\hat{E} = \frac{\vec E}{E_{mag}} = (400,800,500)/1024.7 = (.39,.78,.488) </math>
And q is positive:
<math>\hat{E} = (.39,.78,.488) </math>
<math> \vec r = \hat{r} * r_{mag} = (.39,.78,.488) * .42 = (.1638, .3276, .2058) </math>


Going back to our original equation:
<math> q= {{1/(9e9)} *{6.9^{2}}*4.76e6} </math>  
<math> (.1638, .3276, .2058) = (.26,.75,.09) – (x,y,z) </math>
So the location of the charge is <math> (.096, .422, -.115) </math>


This problem can be solved instantaneously with this Matlab code, which take the inputs of E and observation location as six different scalars:
<math> q= + 0.253 C </math></td>


    function [positionOfCharge] = Elec(Ex, Ey, Ez, obs_x, obs_y, obs_z, q)
 
    Emag = (Ex^2 + Ey^2 + Ez^2)^(1/2);
</table border>
    rmag = abs((9e9*q/Emag)^(1/2));
    Ehatx = Ex./Emag;
    Ehaty = Ey./Emag;
    Ehatz = Ez./Emag;
        if q < 0
            rhatx = -(Ehatx);
            rhaty = -(Ehaty);
            rhatz = -(Ehatz);
        elseif q > 0
            rhatx = Ehatx;
            rhaty = Ehaty;
            rhatz = Ehatz;
        end
    rx = rhatx * rmag;
    ry = rhaty * rmag;
    rz = rhatz * rmag;
    xpos = obs_x - rx;
    ypos = obs_y - ry;
    zpos = obs_z - rz;
    positionOfCharge = [xpos ypos zpos];
    end


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


I am very interested in the idea of forces and how objects interact with each other. After you calculate the Electric Field you can easily find the Electric Force one particle exerts on another.
The topic is fascinating because electric fields not only exist within the human body but also in the realms of aerospace. It's captivating to think about the parallels between biological systems and aerospace systems, where electric field management is crucial for functions such as propulsion and navigation. Analyzing how bodies maintain and adapt their electric fields and applying this knowledge to aerospace engineering can lead to innovative approaches to energy management and system efficiency, particularly in the face of perturbations such as equipment failure or external environmental factors.


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


I am a CompE major and so Electric Fields have to do with my major because when you integrate them with respect to dL, and swap the sign, you get potential difference(voltage), which is very important in circuits. As ECE majors take circuits classes, this topic is relevant to me.
As an aerospace engineer, my interest lies in the application of principles from various sciences, including biochemistry, to improve and innovate within the field of aviation and space exploration. Understanding the role of electric fields in cellular behavior provides insights into potential analogs in aerospace technology. For instance, similar to how cells adjust their electric fields for optimal function, aerospace systems must regulate their onboard electric fields for navigation, communication, and operational efficiency. Such interdisciplinary knowledge can be the foundation for advancements in aerospace materials and systems that are responsive and adaptive to their environments.  


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


An interesting application is that electric fields of point charges can be used to find forces. Then you can predict the motion of various particles by the forces acting on them.
PEMF (Pulsed Electromagnetic Field) therapy's principle of restoring optimal voltage in damaged cells through electromagnetic fields has interesting parallels in aerospace engineering. For example, the management of electromagnetic fields is critical in spacecraft and aircraft to protect onboard electronics and enhance communication signals. The concept of PEMF can inspire the development of systems that can self-regulate and optimize electrical potential across different components of an aircraft or spacecraft, thereby enhancing performance and resilience. Such systems could prove vital in long-duration space missions, where human intervention is limited, and the machine's ability to self-heal and maintain operational integrity can be a game-changer.


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


In the 1780s a French scientist named Charles Coulomb published many scientific papers on electricity and magnetism. While doing experiments, Coulomb had discovered an inverse square relationship between the amount of electric field and the distance between two particles and the electric field pointed in a line between the particles.  
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.  


Also, he discovered that the charge of an particle (ie. positive or negative) determined the direction of the electric field (either a repulsion or attraction).  
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.  
 
From these observations, as well as the use of fundamental constants, the equation of the electric field due to a point charge was created.


== See also ==
== See also ==




[[Electric Field]] More general ideas about electric fields <br>
[[Electric Field]] <br>
[[Electric Force]] One application of electric fields due to point charges deals with finding electric force
[[Electric Force]] <br>
[[Superposition Principle]] <br>
[[Electric Dipole]]


===Further reading===
===Further reading===
Line 174: Line 255:
Principles of Electrodynamics by Melvin Schwartz
Principles of Electrodynamics by Melvin Schwartz
ISBN: 9780486134673
ISBN: 9780486134673
Electricity and Magnetism: Edition 3 , Edward M. Purcell David J. Morin


===External links===
===External links===


Some more information : http://hyperphysics.phy-astr.gsu.edu/hbase/electric/epoint.html
Some more information:
 
*http://www.physics.umd.edu/courses/Phys260/agashe/S10/notes/lecture18.pdf
*https://www.reliantphysicaltherapy.com/services/pulsed-electromagnetic-field-pemf
*https://www.youtube.com/watch?v=HG9KxDZ-qwI&t=1s
*https://www.youtube.com/watch?v=8GJf-Fj-qoI&t=3s


==References==
==References==


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


Charles-Augustin de Coulomb. (n.d.). Retrieved December 3, 2015, from https://nationalmaglab.org/education/magnet-academy/history-of-electricity-magnetism/pioneers/charles-augustin-de-coulomb
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


Shech, E., & Hatleback, E. (n.d.). The Material Intricacies of Coulomb’s 1785 Electric Torsion Balance Experiment. Retrieved December 3, 2015, from http://philsci-archive.pitt.edu/11048/1/The_Material_Intricacies_of_Coulomb's_1785_Electric_Torsion_Balance_Experiment_(EV).pdf




[[Category:Fields]]
[[Category:Fields]]

Latest revision as of 16:46, 13 April 2024

Claimed by Sam Bureau Spring 2024

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


The Main Ideas

(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

2 types of point charges:

  • 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.

Point Charges Result Diagram
1 proton, 1 electron Attract
2 protons Repel
2 electrons Repel


The Electric Field

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

In general, the electric field evaluates the affect of the source on the surrounding objects and area. 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. Therefore, the concept of the electric field by a point charge describes the interactions that can happen at a distance, due to these affects caused by this point charge.

Important to differentiate that Electric Force does not equal the Electric Field.

Electric Field of a Charge Observed at a location: F = Eq

  • F = Force on particle
  • E = electric field at source location
  • q = magnitude of the charge of particle (assume q= 1.6 x 10^-19 unless stated otherewise)

The magnitude of the electric field decreases with increasing distance from the point charge.

The electric field of a positive point charge points radially outward The electric field of a negative point charge points radially inward

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)

The magnitude of the electric field decreases with increasing distance from the point charge. This is described by the equation below:

Electric Field of a Point Charge ([math]\displaystyle{ \vec E }[/math]):

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

  • [math]\displaystyle{ \frac{1}{4 \pi \epsilon_0 } }[/math] is Coulomb's Constant and is approximately [math]\displaystyle{ 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]\displaystyle{ \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]\displaystyle{ \hat r }[/math] and the sign of the source charge.

  • If the source charge is positive, the field points away from the source charge, in the same direction as [math]\displaystyle{ \hat r }[/math].
  • If the source charge is negative, the field points toward the source charge, in the opposite direction as [math]\displaystyle{ \hat r }[/math].

Coulomb Force Law for Point Charges

(Ch 13.2 in Matter & Interactions Vol. 2: Modern Mechanics, 4th Edition by R. Chabay & B. Sherwood)

[math]\displaystyle{ \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]\displaystyle{ Q_1, Q_2 }[/math]= The charges of the two particles of interest
  • [math]\displaystyle{ \mid\vec F\mid=\frac{1}{4 \pi \epsilon_0 } }[/math] = constant,
  • [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ F = Eq }[/math], with F modeled by Coulomb's Law, you get the equation for the electric field of the point charge:

[math]\displaystyle{ 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]


The direction of electric force also depends on the direction of the electric field too:

  • If the source charge is positive, the field points away from the source charge, in the same direction as the electric force.
  • If the source charge is negative, the field points toward the source charge, in the opposite direction as the electric force.

Electric Field Superposition (Point Charges)

When there are multiple point charges present, the total net electric field [math]\displaystyle{ Enet }[/math], is equal to the sum of the electric field of each independent point charge present.

This is due to concept of Superposition which is when the total effect is the sum of the effects of each part.

When it comes to the Electric Field Superposition of Point Charges, be sure to remember that:

  • A charge cannot exert a force on itself
  • Assume that the source charges do not move. (Therefore [math]\displaystyle{ Fnet = 0 }[/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]\displaystyle{ \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.


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).

Examples

Simple

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

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]\displaystyle{ \lt 0.124, 0.188, 0.109\gt }[/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]\displaystyle{ \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]\displaystyle{ E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r^2} }[/math] one can find the magnitude and sign of the charge.

Step 1. Find [math]\displaystyle{ \vec r_{obs} - \vec r_{particle} }[/math]:

[math]\displaystyle{ \vec r = \lt -0.02, 0.31, 0.28\gt m - \lt -0.21, 0.02, 0.11\gt m = \lt 0.19,0.29,0.17\gt m }[/math]

To find [math]\displaystyle{ \vec r_{mag} }[/math], find the magnitude of [math]\displaystyle{ \lt 0.19,0.29,0.17\gt }[/math]

[math]\displaystyle{ \sqrt{0.19^2+0.29^2+0.17^2}=\sqrt{0.1491}= 0.39 }[/math]

Step 2: Find the magnitude of the Electric Field:

[math]\displaystyle{ E= \lt 0.124, 0.188, 0.109\gt N/C }[/math]

[math]\displaystyle{ E_{mag} = (\sqrt{0.124^2+0.188^2+0.109^2}=\sqrt{0.0626}=0.25 }[/math]
Step 3: Find q by rearranging the equation for [math]\displaystyle{ E_{mag} }[/math]

[math]\displaystyle{ E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r^2} }[/math]

By rearranging this equation we get

[math]\displaystyle{ q= {4 \pi * \epsilon_0 } *{r^2}*E_{mag} }[/math]

[math]\displaystyle{ q= {1/(9*10^9)} *{0.39^2}*0.25 }[/math]

[math]\displaystyle{ q= + 4.3*10^{-12} C }[/math]

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]\displaystyle{ \lt -5.5*10^{3} , -5.5*10^{3}, -5.5*10^{3}\gt }[/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]\displaystyle{ E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r_{mag}^2} }[/math] to find the rmag value. To find [math]\displaystyle{ \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]\displaystyle{ \hat r }[/math]. Then, once we find [math]\displaystyle{ \hat r }[/math], all that is left to do is multiply [math]\displaystyle{ \hat r }[/math] by rmag and that will give us the [math]\displaystyle{ r }[/math] vector. We can then find the location of the particle as we know [math]\displaystyle{ r=r_{observation}-r_{particle} }[/math]

Step 1. Find the magnitude of the Electric field:

[math]\displaystyle{ F = Eq }[/math] [math]\displaystyle{ = E * -2mC }[/math]

[math]\displaystyle{ E = \frac{\lt -5.5e3 , -5.5e3, -5.5e3\gt }{-2mC} = \lt 2.75e6 , 2.75e6, 2.75e6\gt }[/math] N/C

Step 2: Find [math]\displaystyle{ \vec r_{obs} - \vec r_{particle} }[/math].

[math]\displaystyle{ \vec r = \lt 3.98 , 3.98 , 3.98\gt m - \lt 0 , 0 , 0\gt m = \lt 3.98 , 3.98 , 3.98\gt m }[/math]

To find [math]\displaystyle{ \vec r_{mag} }[/math], find the magnitude of [math]\displaystyle{ \lt 3.98 , 3.98 , 3.98\gt }[/math]

[math]\displaystyle{ \sqrt{3.98^2+3.98^2+3.98^2}=\sqrt{47.52}= 6.9 }[/math]

Step 3: Find q by rearranging the equation for [math]\displaystyle{ E_{mag} }[/math]

[math]\displaystyle{ E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r^2} }[/math]

By rearranging this equation we get

[math]\displaystyle{ q= {4 pi * \epsilon_0 } *{r^2}*E_{mag} }[/math]

[math]\displaystyle{ q= {{1/(9e9)} *{6.9^{2}}*4.76e6} }[/math]

[math]\displaystyle{ q= + 0.253 C }[/math]

Connectedness

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

The topic is fascinating because electric fields not only exist within the human body but also in the realms of aerospace. It's captivating to think about the parallels between biological systems and aerospace systems, where electric field management is crucial for functions such as propulsion and navigation. Analyzing how bodies maintain and adapt their electric fields and applying this knowledge to aerospace engineering can lead to innovative approaches to energy management and system efficiency, particularly in the face of perturbations such as equipment failure or external environmental factors.

2. How is it connected to your major?

As an aerospace engineer, my interest lies in the application of principles from various sciences, including biochemistry, to improve and innovate within the field of aviation and space exploration. Understanding the role of electric fields in cellular behavior provides insights into potential analogs in aerospace technology. For instance, similar to how cells adjust their electric fields for optimal function, aerospace systems must regulate their onboard electric fields for navigation, communication, and operational efficiency. Such interdisciplinary knowledge can be the foundation for advancements in aerospace materials and systems that are responsive and adaptive to their environments.

3. Is there an interesting industrial application?

PEMF (Pulsed Electromagnetic Field) therapy's principle of restoring optimal voltage in damaged cells through electromagnetic fields has interesting parallels in aerospace engineering. For example, the management of electromagnetic fields is critical in spacecraft and aircraft to protect onboard electronics and enhance communication signals. The concept of PEMF can inspire the development of systems that can self-regulate and optimize electrical potential across different components of an aircraft or spacecraft, thereby enhancing performance and resilience. Such systems could prove vital in long-duration space missions, where human intervention is limited, and the machine's ability to self-heal and maintain operational integrity can be a game-changer.

History

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
Electric Force
Superposition Principle
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:

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

Retrieved from "http://www.physicsbook.gatech.edu/index.php?title=Point_Charge&oldid=46255"