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Point Charge

2025-04-13T19:38:44Z

Erice34:

Claimed by Elena Rice Spring 2025

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

<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'')

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.

<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>

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.

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> Fnet = 0 </math>)

[[File:IMG_1DCC0A11C7B8-1.jpeg]]

===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.

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

===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> = 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 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/(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?''

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==
[[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://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==

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]]

Nucleus

2022-12-05T04:51:48Z

Erice34:

Elena Rice Fall 2022

==Definition==

[[File:Helium_atom_QM.svg|thumb|This is a rough (still not small enough to be to-scale) depiction of the nucleus of a helium atom. The grey gradient around the nucleus represents the electron cloud, and the red and blue balls represent protons and neutrons respectively.]]

A '''nucleus''' is a very dense, positively charged region in the center of an atom containing almost all of the atom's mass. Nuclei are comprised of protons, particles of positive charge, and neutrons, particles with roughly the same mass as a proton but with neutral charge. Protons, being positively charged, would be expected to all fly out of the nucleus due to electrostatic repulsion, but they (and neutrons) are tightly bound together by a separate force called the strong nuclear force. So named because it is strong(er than electromagnetism) and nuclear (in the nucleus of an atom). While it may be helpful to think of a nucleus as a ball made of smaller balls as in the picture on the right, in actuality, nuclei are a Schrödinger's mess of a wavefunction. A more realistic depiction would show them as all overlapping in a small probability density cloud of their own, most likely all at the very center. Just as with the electron shell, the nucleus of a helium atom is a spherically symmetric distribution.

==Discovery of the Nucleus==

[[File:Geiger-Marsden_experiment_expectation_and_result.svg|thumb|Nucleus not to scale; this is what Thompson's plum pudding model would predict vs. what Rutherford found (and then theorized).]]

If we hearken back to the days of the turn of the 20th century (a truly revolutionary time in physics history), we have to consider the '''plum-pudding model''' of an atom put forth by J. J. Thompson just a few years ago in 1904. Thomson had discovered the electron in 1897, and to account for the neutral charge of regular matter and the negative charge of electrons, he proposed that atoms were made of some positively charged *stuff* (pudding) with negatively charged electrons embedded inside (plums). Over the next several years, many experiments headed by Ernest Rutherford and Hans Geiger showed that alpha particles emitted by radioactive elements move more slowly through solid matter than through air (testing with aluminium foil and gold leaf).

So something crazy wild happened in 1909 when Rutherford told Geiger and his (undergraduate!) assistant Ernest Marsden to pay close attention to the trajectories followed by the alpha particles when shot at gold leaf. According to the plum pudding model, the positively charged ''stuff'' of the atom should have a negligible effect on the alpha particles because it is evenly distributed. Likewise, the electrons shouldn't affect the trajectory of the particles because their mass is so much smaller; they would just be flung away. So it came as a surprise to Rutherford and Geiger (and physicists everywhere!) when a substantial number of alpha particles were deflected by significant amounts (several degrees even), and many went in all sorts of directions, including backwards! See the image for a better description.

This experiment (usually named Rutherford's gold foil experiment, but I'd like to credit Geiger and Marsden too) showed that the center of an atom contained all of an atom's positive charge and mass, and within a couple years, physicists had updated their conceptual model of atoms to account for this. By 1919, Rutherford had proven that hydrogen nuclei were present in other nuclei and, with the discovery of isotopes (due to physics' newfound fascination with radioactivity), a new atomic hypothesis was formed: the nuclear electrons hypothesis. Since atomic masses are roughly in multiples of the hydrogen nuclei, and hydrogen nuclei were present in other nuclei, it was proposed that protons (hydrogen nuclei) make up the mass of nuclei and electrons are "embedded" in it to balance the charge. For example, in nitrogen-14, there would be 14 protons and 7 electrons in the nucleus to make a net charge of +7. However, there were several issues with the embedded electrons hypothesis. For one, it would cause additional spectral line splitting (at a very fine level) that was not actually observed, and it would cause some nuclei to have a net spin inconsistent with measured values.

Then, in 1930, a new type of radiation was observed when an alpha particle collided with the nucleus of a lightweight atom. This new kind of particle emitted had to be neutral, but it was most certainly not a photon, or else it would have a ridiculous amount of energy. Several experiments were conducted in 1932, mostly due to James Chadwick, that proved that this radiation could not be gamma rays, and furthermore, that this particle had roughly the mass of a proton. Some of these experimental setups were crazy: A polonium alpha-particle source poured radiation onto a lithium metal foil, which would then emit the unknown radiation, which would then collide with molecules of some gas. But this new particle must be present in nuclei! This was it! Nuclear physics came into its own, with neutrons being used to show all kinds of things that previously puzzled physicists including the spin differences of isotopes, beta radiation, and of course, neutron emission (the induced radiation caused by hitting a light nucleus with an alpha particle).

Of course, physicists were keen to make hypothesis after hypothesis and experiment after experiment. Heisenberg, for example, hypothesized that neutrons were electron-proton combinations! An early model of the nucleus include the liquid drop model, where the nucleus is treated as a rotating drop of liquid, so protons and neutrons are like molecules of that liquid (and so can move around like molecules in a liquid might) and have various quantities of energy associated with them (pictured below).

[[File:Liquid_drop_model.svg]]

Even more modern models of the nucleus (there is still no clear consensus!) include variations on the "shell model", in which nucleons occupy orbitals in the nucleus in similar fashion to how electrons occupy orbitals in the atom. This theory arose because some nucleon numbers are stable and others aren't, and protons and neutrons each have ±1/2 spin. Furthermore, even numbers of protons (and even numbers of neutrons) are more stable than odd numbers of either. For example, no stable nucleus has 5 nucleons in it, but helium-4 and lithium-6 are fully stable. No stable nucleus has 43 or 61 protons in it (regardless of neutron number), and yet 50 protons seems to make for some of the most stable nuclei. There are many things incomplete about this hypothesis, and "subshells" beyond 1s (helium-4 under the shell model would have a filled 1s of protons and a filled 1s of neutrons) are generally not comparable to electron shells. There is still so much we don't know about nuclear physics, so it's very much still an ongoing study!

==Putting Numbers on It==

Even though the typical atom is roughly 0.05 - 0.2 nanometers (0.5 - 2 × 10<sup>-9</sup>m), the typical nucleus is 25,000 to 60,000 times smaller--close to 1.7 femtometers for hydrogen and 11.7 femtometers for uranium (1.25 - 11.7 × 10<sup>-15</sup>m). A convenient formula is that the nucleus has approximate radius
<math>R=r_0A^{1/3}</math>
where <math>r_0\approx1.25\mbox{fm}</math>. Since this formula for radius is proportional to the cube root of the number of nucleons, and the radius of a sphere is also proportional to the cube root of its volume, the volume of the nucleus is roughly proportional to the number of nucleons. In other words, nuclei have a roughly constant density (and that density works out to be about 0.14 nucleons/fm<sup>3</sup>. Since a nucleon has a mass of about 1 atomic mass unit (u), which is 1.66 × 10<sup>-27</sup> kg, atomic nuclei roughly have a density of 2.3 × 10<sup>14</sup> g/cm<sup>3</sup>! That's 230 trillion times denser than water! Well, it doesn't actually mean that much when everyday matter is made of atoms, not raw nuclei. But that might give a sense as to why neutron stars, for example, are so darned small (10 or so km in diameter) despite weighing at least as much as the sun!

Some more useful numbers are below:

mass of a proton m<sub>p</sub> = 1.673 × 10<sup>-27</sup> kg

mass of a neutron m<sub>n</sub> = 1.675 × 10<sup>-27</sup> kg

==References==

Nuclear Physics--Wikipedia https://en.wikipedia.org/wiki/Nuclear_physics

Atomic Nucleus--Wikipedia https://en.wikipedia.org/wiki/Atomic_nucleus

Proton--Wikipedia https://en.wikipedia.org/wiki/Proton

Discovery of the Neutron https://en.wikipedia.org/wiki/Discovery_of_the_neutron

Atomic Nuclei--Zach Meisel https://inpp.ohio.edu/~meisel/PHYS7501/file/AtomicNuclei_Book_Draft15Aug2019_ThruCh1.pdf