# P. Rule . The 'Coulomb' is the SI unit for Electric Charge. Specifically, it is defined as the "amount of electric charge transported by a current of one ampere flowing for one second" (see definitions of each new term, they are gone over in more depth later on).

By itself, a Coulomb is a huge amount of charge - charged particles like protons and electrons have electric charges many orders of magnitude smaller than a single Coulomb. Two objects, each with one Coulomb of charge, located one meter apart, experience an electric force (see Subsection XII.III) of roughly nine billion Newtons.

# P. Rule . An atom is the smallest unit of matter retaining the chemical properties of its element. It is composed of a dense nucleus with Protons and Neutrons, and surrounded by orbitals of Electrons.

Electrons & Protons are small particles (e.g., specks of mass) that carry an electric charge with a magnitude of ~1.602 × 10⁻¹⁹ Coulombs, which is referred to as the "Elementary Charge". Protons are positively charged, while electrons are negatively charged.

Electrons are elementary particles, indivisible by nature, while protons and neutrons are composed of quarks, the REAL smallest particles. Specifically, the proton has two up quarks and one down quark.

The up quark has a charge of +(2/3)e, while the down quark has the charge -(1/3)e. Thus, the net quark of a proton is +e.

The neutron, being composed of one up quark and two down quarks, has a net charge of 0. If this were not the case, the neutron would not be electrically neutral, and the nucleus of an atom would tear apart as the protons would repel from one another without the stabilizing force of the neutrons.

The electron, being an elementary particle (defined below), just has a net charge of -e.

# P. Rule . There is an equation relating net charge and elementary charge (or any quantifiable proportion of elementary charge), useful for determining how many excess charge carriers it would take for a specific net charge to be achieved:

q = n × e

q = Net charge on an object
n = Excess number of charge carriers (protons or electrons or whatever - it just has to be ONE of them).
e = The Elementary Charge Constant.

# P. Rule . Charge is quantized, meaning that it must come in discrete quantities, integer multiples of the elementary charge. This is because the 'net charge' of an object is the product of having more or fewer charged particles, and thus only multiples of the elementary charge are possible (since elementary particles are indivisible).

Using the net/elementary charge relation (see Rule 159), one can find that it is impossible for an object to have a net negative charge of 2.00 × 10⁻¹⁹ Coulombs, as such a net force would require 1.25 electrons, violating quantization.

# P. Rule . When the atoms of a conductor form a solid, some of their outermost (and most loosely held) electrons become free to wander about within the solid, leaving behind positively charged atoms (positive ions). These wandering electrons are known as conduction electrons. There are practically no free electrons in a nonconductor, since those electrons are tightly bound to their atoms.

Knowledge of the "mobility of charge" within conductors, and conductors alone, is fundamental for comprehending concepts in Electric Field and Charge-Flux Law type questions - see Rule [[[. [[[this will be the bottom notepad Rule]]]

# P. Rule . Law of Electric Force (Coulomb's Law): VECTOR.

Units: Newtons.


Equation:

F_e = (k × q₁ × q₂) / (r²)

F_e = The magnitude of the electric force that exists between two charged particles.
k = The Coulomb constant, equal to 8.99 × 10⁹ (N × m²) / (C²).
q₁ = The net charge on object 1, measured in Coulombs.
q₂ = The net charge on object 2, measured in Coulombs.
r = The distance between the centers of charge of the two objects.


Definition: Coulomb's Law, hereafter referred to as the Law of Electric Force, enables one to find the magnitude of the Electric Force with respect to two particles.

Note that this equation looks very similar to the Universal Law of Gravitation (see Rule 142). Accordingly, k (the Coulomb constant) functions similar to G (the Gravitational constant), being equal to (N × m²) / (a particular composition property of mass)². Clearly, since the Coulomb constant is 20 orders of magnitude stronger than the Gravitational constant, the Electric Force between objects is much more powerful than their Gravitational attraction. F_e >>> F_g.

If the calculated Electric Force has a Negative Value, then it is an attractive force. If the Electric Force has a Positive Value, then it is a repulsive force. This is quite obvious upon momentary consideration - a negative and a positive charge will always produce a negative, and thus attractive force, which matches the Law of Charges that opposite's attract. Thus also follows for repulsive like charges.

# P. Rule . With charges, the fundamental concept to understand in applying the Law of Electric Force is that in order to derive any useful information (depending on the question, but in general), you must determine the effect of the electric force on the individual point charges. Specifically, you must be able to determine the direction the charges will move in as a result of the force.

The Law of Electric Force determines the repulsive or attractive force acting on both charges, since they're a Newton's Law Force Pair (see Rule 76). However, in addition, you must utilize this information to determine the direction of the movement of each charge. If you have a proton on the left and an electron on the right, the proton will move rightward and the electron leftward as they attract toward one another. If you have two electrons, they will move in opposite directions away from one another.

# P. Rule . Permittivity of Free Space:

The Coulomb Constant, 'k', 8.99 × 10⁹ (N × m²) / (C²), used in such equations as the Law of Electric Force (see Rule 162) and all its applications, has a special fixed relationship with another constant. This other constant is the permittivity constant, also just known as the permittivity of free space.

It is represented as ε₀ ("Epsilon Naught"), and is defined using the following relation:

(1 / 4πε₀) = k

ε₀ = The Permittivity of Free Space, equal to 8.85 × 10⁻¹² (C²) / (N × m²).
k = The Coulomb constant, equal to 8.99 × 10⁹ (N × m²) / (C²).

This, of course, can be substituted in for 'k' in any applicable equation. Furthermore, ε₀ itself appears in several equations by itself, most notably the Charge-Flux Law (see Rule 187). The reasons for the usage of Coulomb's Constant at all is a matter of historical decisioning.

The actual meaning of this 'permittivity of free space' value, is that it measures how dense an electric field, in relation to an electric charge, will be "permitted" to form.

# P. Rule . Conservation of Charge:

The total electric charge of an isolated system never changes.

q_{i total} = q_{f total}

If the conductors being touched in a conservation of charge-type equation are identical (in all manners other than charge), then the excess charge will be evenly distributed between the two conductors. For example, if there are two conductors, one with a charge of -3 nC and the other with +6 nC, then the final charge of each conductor, once touched, will be 1.5 nC.

# P. Rule . GROUND.

All excess charges can quickly be diffused through the usage of a Ground. A Ground is any sort of release point (if excess negative) or gain point (if excess positive) for the excess charges of a system: an ideal ground is an infinite well (see VII.VI) of charge carriers - such a requirement is effectively met by the Earth.

An Earth Ground is created when a circuit has a physical connection to the earth, in order to sink (lose) or source (obtain) electrons through the earth itself. The Earth has a practically infinite number of electrons that can be used to balance out a circuit/system, pulling from or giving to it. Relative to very small charged systems, the human skin could serve as a ground as well.

The end result of a ground is an electrically neutral system.

In electrical engineering, all circuits require a Ground to function - it is often referred to as "GND", and has its own symbol for use in diagrams (see E.E. Rule [[[). In many electrical situations, without the availability of a physical connection to the Earth, a "Floating Ground" can be used, which simply serves as a type of '0V reference line' (see def. of Voltage) that acts as a return path for current back to the negative side of the power supply.

# P. Rule . There are two distinct categories, processes under which charge is transferred. Specific means of transfer, like charging by friction (rubbing) fall under these wider processes. These processes are Conduction, and Induction.

Conduction:

This is the transfer of charge in which current flows because of the electric field. There are only two requisites for an object to be charged by conduction, through another object:

1. The objects have to touch.

2. The objects must, after touching, have the same sign of net charge.


Induction:

This is the transfer of charge in which a changing magnetic field generates an electromotive force (see Rule [[[), resulting in an induced charge. It is not important to fully understand what an "electromotive force" is right now, just that it is the electric force that drives current through a circuit.

There are only two requisites for an object to be charged by conduction, and they are the eact opposite of conduction:

1. The objects do not touch.

2. The objects must finish with opposite signs of net charge.

# P. Rule . Polarization:

Polarization is the process through which the charges within objects (and thus their associated particles) align themselves in such a way that there becomes a net attractive force, as a result of attraction & repulsion from an object with excess charge.

Polarization doesn't change the net charge of the object, but rather has to do with how charges rearrange themselves within the object.

The process itself is simple: When an object with an excess charge approaches a neutral object, the like charges will repel and the opposite charges will attract (the electrons being the sole moving particles, moving either in front of or behind the protons). Since the opposite charges will be closer to the charges of the object than the like charges, by the Law of Electric Force, there will be a net attractive electric force, since the opposite charges have a smaller 'r' value than the like charges.

This is the reason that balloons stick to walls once they have an excess negative charge. The wall, as an insulator without free electrons, cannot "attract" the charge of the balloon. Instead, the charges in the wall rearrange themselves and end up with that net attractive force.

Electric force due to polarization is small - objects with small masses, like balloons and aluminum cans, can be held in place and rolled respectively using only a small electric force.

Electric force caused by the polarization of a conductor is typically larger than a polarized insulator. This is because electrons in insulators are bound in their atoms, while electrons in conductors are able to move to the opposite side of the object.

# P. Rule . Placement of Charge within Conductors & Nonconductors:

The nature & positioning of the charge within the thickness of an object (like a shell or thick cylinder) differs depending on whether the material of the object is a conductor or insulator; there is no other possibility.

Conductive objects (enclosed shapes) spread out all excess charge (read: electrons) over their external surface. This is because the conduction electrons of the conductor (see Rule 161) and the excess charge electrons will repel from one another, causing the excess charge electrons to spread over and uniformly distribute across the farthest possible surface.

!!!NOTE!!!: Unless the conductor is spherical, the charge will not distribute itself uniformly. This is a matter of symmetricality, an attribute that leads spheres (shell or solid) to naturally become uniformly distributed in charge, but not for any other shape. This is not to say that a question will not simply say "assume a uniform charge density" for some ridiculous, nonspherical shape, violating the laws of physics for the sake of examining the physicist. In such a case, you would do as told and use the given information appropriately. The surface charge density σ (charge per unit area) varies over the surface of any nonspherical conductor. The electric field (see Section XIII) immediately outside of the surface of the nonspherical shape can be determined quite easily however, as described in Rule 195.

This concentration of charge on the external surface is the same for both positive and negative excess charge. If there is an excess positive charge (imagine the conduction electrons have been removed), then the positive charge will then spread uniformly across the surface.

In the state of uniform charge distribution, the conductor will be in a state of electrostatic equilibrium - see Rule [[[ for a complete explanation involving Electric Fields.

Nonconductive/Insulating objects (enclosed shapes), which do not allow free movement of charge, bind electrons to their atoms and thus do not move to redistribute excess charge. When an excess charge is added, there are conduction electrons that would even allow for the charge to be moved, and so the excess charge will simply remain wherever it is placed.

Thus, the excess charge can be uniformly distributed if that is the state in which it is added, and questions oftentime instruct the physicist to assume such a condition. However, if the charge is concentrated in a particular region, it will not naturally redistribute itself as it would in a conductor.

For a continuation of this Rule, with respect to how the conductor/nonconductor dispersal of charge affects electric field distribution, see Rule 193.

# P. Rule . Shell Theorems of Electric Force:

Shells, spherical entities of a specified thickness and a hollow interior (like a ring rotated 180° along the x-axis), retain the same importance they had in Mechanics for Electromagnetism.

Akin to the Shell Theorem for Internal Gravitation (see Rule 151), there are TWO Shell Theorems for Electric Fields - yet another similarity between electric force and gravitational force (see Rule 162 for the first). They are relatively straightforward, but are only applicable under the specified conditions.

These theorems literally do not work for any shape other than a shell/sphere - there is a particular symmetry with those specific shapes that allow these theorems to be used. They are not true for just any irregular, wack shape.

Fundamental to the understanding of these theorems is the nature of charge distribution on shells, which itself depends on whether the shell is conductive or not. See Rule 169 for a full, generalized treatment of these principles. Nonconductive shells are not in a state of electrostatic equilibrium (see Rule [[[), unlike Conductive shells.

1. A charged particle outside of a shell with charge uniformly distributed on its surface (e.g., a conductor) is attracted or repelled as if the shell’s charge were concentrated as a particle at its center. This is assuming the charge on the shell is much greater than the charge of the particle, thus not interfering with the distribution of charge on the shell (an issue detailed in Rule 169).



2. A charged particle inside a shell with charge uniformly distributed on its surface (e.g., any conductor, and any nonconstructor created in such a way) will have no net force acting on it due to the shell.

These theorems can be definitively proven using the futurely-explained Charge-Flux Theorem, done in Rule 196.

# P. Rule . Electric Field: VECTOR.

Units: Newtons / Coulombs.


Equation:

E = (F_e / q)

E = The magnitude of the electric field at a particular point.
F_e = The electric force being felt by the charge at the particular point being measured.
q = The magnitude of the charge/charged particle of the particular point being measured.


Definition: An Electric Field is a field in space surrounding a charged object, in which the object's electric force has strength.

Electric fields have their direction expressed in the form of lines, the nature of which is elucidated in Rule 173. Naturally (meaning without the interference of another electric field), these lines, which originate at every point of the object's surface, will be perpendicular to the object and will point radially outward or inward (depending on the charge sign) forever without bending - the electric field of a point charge, for example, will shoot off in every direction in such a way. The direction and lines themselves, however, can be influenced as a result of the Law of Charges and can bend accordingly - see Rule 173.

Technically, the given equation dictates that the electric field is the "amount of electric force per charge at a point in space", the ratio between the electric force of the charge and the magnitude of the charge itself. All charge creates an electric field in relation to its electric force.

The reason an exact magnitude value can be determined for an electric field (an inherently emanating and changing entity), as is done in the given equation, is because the magnitude being found is that of the strength of electric field at a particular point, denoted by the electric force experienced by the charge placed at that point.

# P. Rule . The magnitude of the electric field will decrease as the test charge gets farther from the point charge, as a result of the Law of Electric Force (since the denominator distance value increases and all). The electric field surrounding (and caused by) a point charge is constant at a constant radius from the point charge.

The entire Law of Electric Force equation can be substituted into the Electric Field equation, enabling one to simplify things considerably if the circumstances allow.

# P. Rule . Electric Field Lines:

1. The lines of attraction in an electric field always point away from the positive charge (origination) and toward the negative charge (termination). Thus, the lines always point in the direction in which the test charge would experience an electric force.

2. The # of electric field lines per unit area is proportional to the electric field strength. Therefore, a higher density of electric field lines means a higher electric field strength.

3. Electric field lines always start perpendicularly to the surface of the charge, and start on a positive charge and end on a negative charge (unless there is more of one charge, in which case some lines would start/end infinitely far away).

4. Electric field lines never cross.

Use the test charge (a positive entity - see the definition) as the sample particle, for the sake of illustrating this point.

When the test charge (standardized as positive) is placed in the field of a positive point charge, the test charge will be repelled from the point charge. Electric force projects radially outward from the positive point charge, decreasing in magnitude as distance increases.

On the flip side, if the point charge is negative, then the test charge will be attracted toward the point charge. All of the arrows will point radially inward toward the negative point charge.

# P. Rule . A Continuous Charge Distribution (CCD) is a simple concept that elaborates the simple 'point' model of charges into one that uses calculus to account for more chargetype possibilities, furthering this all-consuming journey in human knowledge.

All a 'continuous charge distribution' is, is a charge that isn't a point charge; e.g., a charge with a shape and an electric charge continuously distributed throughout the object.

The electric field that exists around a continuous charge distribution can be determined through a rethinking of the standard Electric Field-Law of Electric Force combined equation (see Rule 171), using Calculus:

Since the charged object is made up of an infinite number of infinitesimally small point charges ('dq' representing them individually), the therefore infinite number of electric fields can be calculated using an integral.

$$E_{CCD} = k \int \frac{dq}{r^2} \hat{r}$$ k = The Coulomb constant, equal to 8.99 × 10⁹ (N × m²) / (C²).

dq = The infinitesimally small point charges, of which there is an infinite number of. This value can be substituted for any of the equivalent charge density derivatives (see Linear, Surface, and Volumetric), pursuant to the nature of a particular problem.

r = The distance between the infinitesimal charge dq and the point where the electric field is being calculated. Unlike the Law of Electric Force, this r is a function that varies depending on the charge, since there is technically an infinite number of charges (dq).

r̂ = The unit vector pointing from dq (whichever charge) toward the test charge - this is most applicable post-integral, when you can have a test charge to calculate the strength of the electric field with respect to. The direction points radially outward for positive dq, and inward for negative dq.



There are very specific use cases for this integral: it is not applicable everywhere. See Rule 175 for a detailed treatise on this cases and their exceptions. The limits of integration of the integral are specific to each problem and can be determined through ingenuity and through the power of the indomitable human spirit.

That pesky 'dq' thing can be switched out by taking the derivative of any of the charge densities (see linear, surface, and volumetric), depending on what one is looking for.

# P. Rule . The given integral equation for CCDs (continuous charge distributions, see Rule 174) is only applicable when every infinitesimal charge has its electric field pointing in the same direction when it is pointing toward the test point. For example: when the CCD is a flat line, and the test charge is on its same axis.

There are some workarounds in which the above stipulation is technically still respected, allowing the integral to still be used even for shapes like rings. This is when all components in all other directions cancel out, leaving only the components in a specific direction. For example, in a uniformly charged ring, the horizontal components of the field from symmetric charge elements cancel, leaving only the vertical (axial) component.

When all fails, and the direction of the electric at test charge P is simply not the same for every charge dq on the ring, then take the derivative of both sides of the equation (giving dE) and try to see if any components cancel eachother out anywhere - in order to find the components, use the direction of the electric field from or toward the test charge (remember: the direction points radially outward for positive dq, and inward for negative dq), and imagine that beyond the test charge or dq, wherever the direction points, the direction line continues - this extended line segment represents dE, which can then be broken into components (which will hopefully cancel out in one direction). If there are equal negative and positive dq's in a particular direction, then it cancels out. Ideally, the direction that doesn't cancel out should only a single sign no matter the direction/origin of the dq.

Generally, the dq electric field values in a particular direction 'cancelling out' is the result of a symmetry created by the placement of the point charge. For example, a point charge placed somewhere along the axis of the center of the ring.

If one has succeeded in cancelling out, then they would be able to proceed with all the necessary moving of sines or cosines (resultant from the components) and the general transfiguration of terms dependent on the characteristics of the problem itself. After one has been able to simplfy and reduce everything into only one real variable on one side (every other symbol just being a constant), then both sides can be integrated, creating a problem-specific version of the hyper-generalized CCD integral equation (see Rule 174).

All of this component business requires that there be some reference position, a known x-axis and y-axis positioning in relation to the given CCD and point charge that can be used to derive the components from.

# P. Rule . Note that the farther you get from a finite continuous charge distribution, the more the electric field caused by the CCD matches the electric field caused by a point charge (as in, the simplified Law of Electric Force + Electric Force equation from way back when (see Rule 172)). Of course, this requires the distance between the test charge and the CCD to be much much greater than the internal distance with the CCD itself.

# P. Rule . Always be aware of what constants/variables are much much greater than others. In end game integrals, such as in CCD-type problems (see Rule 175), this can totally simplify whole entire parts of your equation by just thinking of the lesser variable as zero (where it is left alone and not acting as a coefficient). Of course, this mandates the usage of the approximation symbol (≈) for your answer.

# P. Rule . You can never assume that the equation for the electric field of a shape, whether 2d or 3d, will be the same if the shape is a conductor or a nonconductor. The world is not such a simple, idealistic place, even in what is essentially toy model physics (classical electromagnetism).

The key factor in determining whether the electric field equation for a shape is the same or differing between conductors and nonconductors, is how the charge distributes within the shape.

The following characteristics are indicative of identical equations between the electric field equations of a conductor and nonconductor shape:

1. Infinite Reach: If the shape is infinite (like infinite planes or cylinders), then both conductors and nonconductors produce the same electric field at points sufficiently far from the charge distribution. Example: Infinite planes as explained in Rule 179.


2. Symmetric Charge Distributions: There are cases in which some form of symmetry causes the charge distribution to remain unchanged between the conductor and nonconductor.
   
   For example: an infinite line of charge with uniform charge density λ produced the same radial field for both conductors and nonconductors, since, being a line, the charge is "on the surface" regardless of the conductivity of the shape (in terms of how the charge being on the surface is an important distinction between conductors and nonconductors and alters the calculation of their electric fields - see Rule 169), and is emanating outward in the same fashion.


3. External Field Only: If only the electric field outside of an enclosed shape is being considered, then IF SYMMETRY DICTATES A FIXED CHARGE DISTRIBUTION (like shells and spheres and cylinders and whatnot, not just any wack enclosed shape), then the equation for that electrical field in particular will be the same whether the shape material is a conductor or nonconductor. This is more due to the first shell theorem (see Rule 170) than anything.

The following characteristics are indicative of differing equations between the electric field equations of a conductor and nonconductor shape:

1. Internal Charge within Enclosed Shapes: As everyone knows, the charge within an enclosed shape for conductors and nonconductors differs as a result of charge placement in the solid part of the shape (like the shell part of a hollow shell) - see Rule 169. Conductors have zero electric field within themselves, and nonconductors do have some.


2. Non-Uniform Charge Distribution: Conductors force charge to redistribute, whereas insulators allow charge to remain fixed. For example: a finite conducting "slab" (like a brick) with excess charge will concentrate charge at the edges, while a nonconductor could have uniform volume charge density.
   
   This will cause differing internal and external electric fields, both for the reason outlined in Difference Characteristic #1 (directly above) and for not meeting the symmetricality requirement of Equivalence Characteristic #3 (further above).

# P. Rule . Disks & Infinite Planes:

The Electric Field due to a (flat) uniformly charged disk and due to an infinite sheet are, in fact, closely related, the latter effectively being derived from the former.

The equation below gives the electric field magnitude on the central axis through a flat, circular, uniformly charged disk. The equation is derived in a logical and most annoying manner, found here (offsite).

$$E = \frac{\sigma}{2\varepsilon_0} \left( 1 - \frac{z}{\sqrt{z^2 + R^2}} \right)$$ σ = Surface charge density. See treatise here.

ε₀ = The Permittivity of Free Space, equal to 8.85 × 10⁻¹² (C²) / (N × m²).

z = The distance (along the central axis) between the point at which the electric field is being measured, and the center of the disk. z ≥ 0.

R = The radius of the disk.



If one were to plug in infinity for R, letting the radius rise to infinity while keeping the z value finite, then the equation would simplify to that of the following, the equation for an infinite plane: $$E = \frac{\sigma}{2\varepsilon_0}$$ Note that despite the equation specifically being for disks and circular 2d shapes, the very act of making the radius infinite will create a plane, and thus make it being a circle, irrelevant.

By virtue of the shape being infinite (passing Equivalence Characteristic #1, see Rule 178), this equation is true for both conductor and nonconductor infinite planes. Furthermore, know that this is also the equation for the electric field at the surface of a nonconductor, as explained in Rule 195.

# P. Rule . Electric Dipole:

An Electric Dipole is an arrangement of two particles (point charges) with equal charge magnitudes but opposite signs. The particles are separated by distance d and lie along the dipole axis, an axis of symmetry going through both particles.

A simple graphic of what this entails is seen below:

An example dipole moment, with the dipole axis running straight through the center.

The electric field produced by a dipole is determined using the following equation:

E = (1 / 2πε₀) × (qd / z³)

ε₀ = The Permittivity of Free Space, equal to 8.85 × 10⁻¹² (C²) / (N × m²).
q = The absolute value of the charge of either particle.
d = The distance between the particles.
z = The distance between the point P and the dipole midpoint (d/2).

There is a simple proof of this equation, done through finding the strength of the net electric field produced by the two particles at an arbitrary point P along the dipole axis. It can be found here.

The internal force on the electric dipole (F_net) will equal to zero, since the attractive forces between the particles are equal and opposite in magnitude, forming a Newton's Third Law Force Pair.

The product qd on the right side of the equation is known as the "electric dipole moment", a concept discussed in Rule 182.

# P. Rule . Because of the 1/z³ dependence, the field magnitude of an electric dipole decreases more rapidly with distance than the field magnitude of either of the individual charges forming the dipole (which depends on 1/r²). Even beyond that, this is relatively clear when considering how at distant points, the electric fields of the oppositely charged particles will increasingly cancel eachother out.

# P. Rule . Electric Dipole Moment:

The 'qd' from the Dipole Electric Field equation (see Rule 180) is known as the Electric Dipole Moment, and can have the value 'p' substituted in for it. It has the unit of the coulomb-meter, of course.

The Dipole Moment is defined as the magnitude of the dipole, purely a matter of charge and distance. This is separate from the magnitude of the electric field of the dipole at a point - that already has its own equation established, see Rule 180. The electric field generated by the dipole is proportional to the dipole moment, though they are always in opposite directions, since the electric field points from the positive charge to the negative charge.

The Electric Dipole Moment is to the Dipole Electric Field equation, what the Discriminant is to the Quadratic Formula (see Math Rule 33), perhaps even to a further degree in which the electric dipole moment can be considered even more important than its forebear.

The direction of p is simply the axis on which the dipole resides, while the sense (see Math Rule [[[) is pointing from the negative toward the positive charge of the dipole.

# P. Rule . In defining "Electric Flux", the focus of this section, one must understand a fundamental reconsideration of how one views electric fields themselves - if not, the following definitions and descriptions of Electric Flux will seem nonsensical.

First, note that the previous usage of electrical fields treated them like a structured object, a vector field, in which every point in space around it has a magnitude and a direction in relation to the field. In this mindset, an electric field is limited to just being a function that varies with spacial positioning.

The new, super ultra modern rethinking of electric fields, treats them as a measurement of an amount - something that can be summed up, through integration. By saying "the amount of electric field", what this statement is really referring to is the cumulative influence of the electric field across a given area - in relation to electric flux, the cumulative influence of the electric field passing through a surface.

This use of language is jarring at first, but can be quickly accustomed to by the young physicist. Before, electric fields were solely used in the form of a countable noun (like "apples") - now, they can also be used in the form of an uncountable noun (like "knowledge"). Without further ado:

# P. Rule . Uniform Electric Flux: SCALAR.

Units: (Newtons × Meters²) / Coulombs


Equation:

Φ_E = E × A × cos(θ)


Φ_E = The magnitude of Electric Flux over a Uniform Electric Field (see definition below).

E = The magnitude of a uniform electric field, the amount of which (see Rule 183) is measured over a surface and not just at an individual point in space (though, being uniform, it will have the same strength at any individual point in space). Given how the existing Electric Field equation (see Rule 171) only specifically calculates the magnitude of an electric force at a point, this equation is overly idealized in assuming a uniform electric field (therefore not a point charge or anything similar).

A = The magnitude of the area of the surface through which the uniform electric field is passing. AREA, not VOLUME (explained in def.).

θ = The angle between the directions of the electric field and the direction of the area (area having its direction perpendicular to the plane, and directed outward for closed surfaces). This value, given that all other values in the equation are positive magnitudes, determines whether the flux is positive or negative.


Definition: Electric flux is the measure of the amount of electric field (see Rule 183) which passes through a defined area. The given equation is specifically for uniform electric fields, something that is totally idealized and never really happens in reality.

This equation in particular is the result of a dot product between the electric field and the surface - this is why the result of the equation is a scalar, and why a cosine is used for the theta.

Because this equation is a dot product, that means the two variables are vectors, both with magnitude and direction. Area, the result of a cross product between two vectors (see Math Rule [[[), has a direction normal to the plane of the area, just like how the direction of angular velocity is normal to the plane in which the object is rotating (see Rule 53).

Note that this equation uses AREA, not VOLUME. When the electric field is passing a multi-sided object, the electric flux must be determined for each side of the object and summed thereafter, forming Φ_total.

Generally, using this equation to determine electric flux is optimal when the flux is passing through some sort of closed surface (see Rule 186).

# P. Rule . If the surface of which the electric field is passing through is perpendicular to the electric field, then since the angle created between the surface and the field will be 90° (and as cos(90) = 0), there will be a net zero electric flux acting upon that surface. This is true for both uniform (see Rule 184) and nonuniform (see Rule 187) electric fields.

NEVER, ever assume that a net zero electric flux means net zero electric field - it just means that all the electric field entering into the surface is passing through and out of it, which furthermore just means there is no inner charge. There can still be a net electric field at the individual points inside the surface.

# P. Rule . When an electric field is going into a closed surface (like an object), the electric flux is negative. When an electric field is coming out of a closed surface, the electric flux is positive. This is the rule of thumb that results from the cosine of the uniform electric flux equation (see Rule 184).

# P. Rule . Nonuniform Electric Fields are the standard form of Electric Fields: a field emanating outward (from a point charge or CCD), decreasing in strength with distance (symbolized in the 'r' of the Law of Electric Force (see Rule 162)).

A new equation, an adapted form of the Uniform Electric Flux equation (see Rule 184) will account for this previously irreconcilable difference. Again, as was done for Electric Fields with Continuous Charge Distribution (Rule 174), this adaptation/rethinking will be done using Calculus.

Infinitesimally small areas of the surface plane, dA, will replace A. Correspondingly, infinitesimally small fluxes dΦ_E (created by the electric field) will pass through dA.

Substituting these values into the original uniform equation (see Rule 184), and then integrating both sides, will produce the new, revolutionary Nonuniform Electric Flux equation (though you could use it for uniform fields if you really wanted to):
$$\Phi_E = \oint \vec{E} d\vec{A} \cos\theta = \frac{q_{\text{enclosed}}}{\varepsilon_0}$$ Φ_E = The electric flux through a closed surface.

E = The magnitude of a uniform electric field, measured in relation to its strength at the infinitesimally small area (essentially a point) dA.

dA = An infinitesimally small area of the surface plane, each of which is collectively represented by dA in the integral.

θ = The angle between the directions of the electric field and the direction of the infinitesimally small area dA (area having its direction perpendicular to the plane, and directed outward for closed surfaces). This value, given that all other values in the equation are positive magnitudes, determines whether the flux is positive or negative.

q_enclosed = The charge enclosed in the Gaussian Surface (see Rule 189).

ε₀ = The Permittivity of Free Space, equal to 8.85 × 10⁻¹² (C²) / (N × m²). (see Rule 164).


This equation is known as Gauss's Law, hereafter referred to as the Charge-Flux Law. This law, in serving nonuniform electric fields (all real ones), relates electric flux through a "Gaussian surface" to the charge enclosed by said surface.

The shapes of these surfaces can be specifically chosen for characteristics that significantly simplify calculations in specific problems. See Rule 189 for what these characteristics entail.

The loop on the integral sign denotes a closed surface integral (see Math Rule [[[), meaning that one must integrate over the entire closed surface to get the net flux through the surface.

The only major difference from this to integrating normally is that you essentially have to sum the normal integrals of each side of the object. This must be done every time!

# P. Rule . If the net charge (q_enclosed) inside a closed surface, Gaussian or otherwise, is zero, then the net electric flux through that surface/shape is zero. This is because all of the electric field entering into the shape exits right through the other side. There must be some charge coming from the inside of the object for there to be any electric flux.

Never confuse electric field for electric flux! The Charge-Flux Law can just as easily be used to isolate net electric field as it can be used to find electric flux; indeed, the net electric field of an object is as dependent on outside charges as internal ones - 'tis what the net electric field acting upon an object means.

# P. Rule . A Gaussian surface is a three-dimensional closed surface in which flux of any kind is calculated.

Generally, these surfaces, when used in problems, are imaginary, though, according to legend, they could be real physical surfaces. Such an event, apart from the surface coinciding exactly with an actual physical surface (in which case there would be no point bothering with the Gaussian surface at all), is never used or asked for ever, and thus one should always consider Gaussian surfaces as nonexistent, imagined surfaces without the burden of its own charge conflicting with that of another (as is the case with shells that have internal point charges).

In most cases, all references to a "Gaussian Surface" can be replaced by a simple description of a closed surface - there is nothing particularly unique or individual the distinguishes a ""Gaussian"" surface from any other imaginary closed surface. It is just a generic surface purpose-built as a template for flux calculations - the electromagnetic industry standard for surfaces that do not carry the usual electric properties of conductors or nonconductors.

Typically, the shapes of these surfaces are chosen such that the electric field generated by the enclosed charge is either perpendicular or parallel to the sides of the Gaussian surface. This greatly simplifies the surface integral because all the angles are multiples of 90 degrees and the cosine of those angles have a value of -1, 0, or 1. The spherical Gaussian surface is chosen so that it is concentric with the charge distribution, making the electric field constant for reasons outlined in Rule 191.

As long as the amount of charge enclosed in a Gaussian surface is constant, the total electric flux through the Gaussian surface does not depend on the size of the Gaussian surface. Gaussian Surfaces also follow the "outside charge = zero electric flux" rule, described in Rule 188.

# P. Rule . Gaussian surfaces, for all intents and purposes, are theoretical constructs and are exclusively used as such.

Unlike regular solid shells/spheres/closed surfaces, Gaussian surfaces do not interact with nor influence internal or external charges whatsoever and cannot contribute ANY charge - they only contain/reflect charge from internal/outside sources. Charge upon the surface of a Gaussian sphere from any such outer source is unimpeded, it is just Gaussian Surfaces themselves produce none of it. Conductive and nonconductive solid spheres, which do contain charge which produce electric fields, have their own developed systems for dealing with these intervening forces in calculations (see Rule 193).

This is the result of Gaussian Surfaces purely being constructs solely intended for the calculation of the magnitude of the electric field or flux produced by any charged object. As such, a "Gaussian Shell" is never used, since shells only have purpose in electric field calculations when there is an internal charge to the shell that can affect the net electric field/flux at a point, which is null and irrelevant with regard to Gaussian Surfaces, which can have all spherical distances/calculations accounted for solely using Gaussian Spheres.

In effect, Gaussian Surfaces are mere mathematical, superimposed bodies exploiting symmetry, with no effect of their own on anything: bodies in space whose only purpose is the use of their positioning, whether through symmetry or whatever (see Rule 189), to calculate flux or field at a particular distance from a charge in space, free from the interference of some internal charge.

# P. Rule . With respect to point charges, the electric field will have the same magnitude as it passes through each area dA at a constant distance from the point charge - see Rule 172. Thus, for all concentric entities surrounding a point charge, where each point on the circle or sphere are at a constant distance from the point charge, the strength of the point charge at each point along the entity will be constant.

In such cases (limited specifically to circular and circularly-shaped object), the E variable of the Nonuniform Electric Flux equation (see Rule 187) can be moved outside of the integral, since it will be a constant.

Additionally, in these cases, the cos(θ) will equal one and thus be removed, since the angle between the electric field and every area dA will equal zero, since they are both outward and in the same direction.

# P. Rule . The individual locations of the charges within an enclosed surface are irrelevant to calculating electric flux or any variable (like Electric Field) using the Charge-Flux Law. All that is needed is the net electric charge within the surface/object - positioning is meaningless, the charges could be in each corner of a cube and it wouldn't matter.

The net Electric Field, however, does require knowledge of the position of the charges, whether they are inside or outside of the enclosed object (due to the nature of the net electric field reflecting the collective influence of all electric fields surrounding the object, which is to say, every charge).

This, of course, is given that you are calculating for electric field in particular, and not for electric flux itself, in which case outside charges contribute zero net flux to an object and thus do not need to be accounted for (see Rule 188).

# P. Rule . Electric Fields, as influenced by the Placement of Charge within Conductors & Nonconductors:

The placement of charge on the surface and thickness of an object as a result of its material (conductor or nonconductor, as previously described in Rule 169), is of considerable effect on the electric field produced by the object.

Note that for the purposes of this section, an "internal electric field" is the electric field contained within the hollow part of an object, whether that object be a shell or whatever.


Conductors: Since the charge of a conductor is solely dispersed along the surface of the object (see Rule 169), the internal electric field of a charged, isolated conductor is zero, and the external field (at points nearby to the surface) is perpendicular to the surface and has a magnitude that depends on the surface charge density σ.

Within the thickness of a conducting shell, the electric field will remain zero, since the charge will remain concentrated on the surface of the charge and thus will not produce anything internally, regardless of whether "internal" means the hollow part of the shell or the material itself.

Though nonspherical conductors do not have uniform charge density, even at their surface (as described in Rule 169), a general equation for the electric field immediately outside of the surface of the nonspherical shape can be determined quite easily, elaborated upon and derived in Rule 195.


Nonconductors, which have their charge distributed in their thickness wherever it is placed (generally expressed in problems as following a particular formula involving radius, or as simply a matter of uniform volumetric charge density), will NOT (for the purposes of an idealized, freshman Physics curriculum; see below) have an internal electric field within the hollow portion of the object, regardless of the placement of the charge (whether the charge is concentrated along the surface or uniformly distributed with the thickness).

The reason for this can be visualized simply: if you were to imagine a Gaussian Surface around the hollow part (or similarly internal part of a shape) of a shell/object with uniform charge density, you will find that with there being zero electric charge on the inside of the surface, there can be zero electric flux (see Rule 188). However, as noted in Rule 185, zero electric flux does NOT mean zero electric field, just that all electric field going in is leaving. Still - the effect of the electric fields produced by each charge around the hollow portion of the object will be a net zero electric field, a result of the repulsion between charge (because electric fields cannot cross - see #4 of Rule 173) effectively preventing the existence of electric field within the hollow center. Thus, there would both be a net zero electric flux, and net zero electric field.

Of course, this also means that if the charge distribution of the object's material is "asymmetric" (defying any logical pattern of symmetry involving radius, see Rule 194), there could theoretically be a nonzero electric field within the hollow portion of a nonconducting shell, since the electric fields produced by the charge would not cancel out nor totally repel eachother.

For the reasons detailed in Rule 194, it is exceedingly unlikely to appear in a problem - note that it is possible under the laws of physics, however.

Within the shell’s thickness (between the hollow interior and the surface), the field is nonzero and varies based on the charge distribution. Outside the shell, a uniformly charged nonconducting shell behaves similarly to a conductive shell, producing an external field equivalent to that of a point charge located at the shell’s center (Shell Theorem #1 - see Rule 170).

# P. Rule . Asymmetric Charge Distributions:

An asymmetric charge distribution is an uneven distribution of electric charge within a system, which results in an uneven distribution of electrical field.

The effect of an asymmetric charge distribution is profound in objects with hollow interiors, spherical or nonspherical.

An asymmetric charge distribution cannot be produced by an equation involving radius, since even an object whose charge density changes with respect to radius is still considered a symmetric distribution (regardless of whether the object is a perfect sphere or not).

Instead, it can only be expressed formulaically by creating a relation between the density and a particular point or position rather than radius, with density depending on distance from said point (such as from one end of the object). This, however, is itself a small fringe case of possible asymmetric distributions:

Asymmetric distributions which can be described by equation, only reflect an tiny, idealized fraction of all possible asymmetric distributions - imagine a simply random pattern of charge placements and concentrations, for example.

These forms of charge distributions, when applied to a nonconducting object with a hollow center, result in a nonzero internal electric field, a special case of the nature of nonconductors as described in Rule 193, and an absolute pain to deal with. The problem must really hate you if it is going through the trouble of creating a position-based charge distribution equation just to have an internal electric field.

# P. Rule . Electric Field on an Object's Surface (Conductor & Nonconductor!):

The equations for the electric field on the surface of any object, spherical or nonspherical, can be easily determined using the Charge-Flux Law. Although the equations are derived in the same fashion, as a result of internal differences relating to charge distribution, the equations differ between conductors and nonconductors.

Imagine two objects, one a conductor and the other a nonconductor. By the laws explained in Rule 169 (and noted again in Rule 193), the conductor, if nonspherical, will not have uniform charge density at its surface, though its charge will still be concentrated at its surface under the laws outlined in Rule 169.

Through the creation of an extremely small cylinder perpendicularly bisecting the surface of either object at a particular point (the size of bisection being small enough that, there, the object's surface can be considered flat, and the electric field thus perfectly perpendicular), one can discover that the only electric flux being produced is upon the cap of the cylinder, since the curved sides of the cylinder will be perpendicular to the object (nullifying the cosine of the Charge-Flux equation - see Rule 193).

Thus, in using the uniform Charge-Flux Theorem (uniform because the cylinder is small enough and directly upon the surface that the electric field can be considered perfectly perpendicular and constant at exactly that point/scale), all one needs to do is find the electric flux occurring on the cap of the cylinder, which will lead directly to an equation for the electric field.

The flux of the cap of the cylinder, being such a small distance away from the surface itself, can be taken to be the flux of that cap-shaped portion of the surface of the object itself. From there, the derivation of the equations (for a conducting object and a nonconducting object) simply follows these logical movements:

The charge q_enc enclosed by the Gaussian surface lies on the object’s surface in an area A. If σ is the charge per unit area, then q_enc is equal to σA. In the uniform Charge-Flux Law, substituting EA in for flux (since the cosine is nullified by the perpendicular electric field - see Rule 185), and σA in for q_enc, will produce the following equations (after simplification):

Conductor: E = (σ / ε₀)

Nonconductor: E = (σ / 2ε₀)

E = The electric field on the surface of a conductor.
σ = The charge per unit area on the surface of the conductor.
ε₀ = The Permittivity of Free Space, equal to 8.85 × 10⁻¹² (C²) / (N × m²).

# P. Rule . Electric Field outside and inside a Sphere:

The electric field due to a shell within and outside of a sphere with uniform charge density can be determined using the Charge-Flux Law. In the process, the two Shell Theorems (see Rule 170) can be mathematically proven, whereas before they were simply stated without proof.

For the following equations, assume the variables are those depicted in the image below: 'b' for the full radius of the shell, center to exterior, and 'a' for the internal radius of the shell, center to interior.

An example uniformly charged spherical shell, with a full radius of b and an internal radius of a.

Outside a spherical shell a uniform charge density q, the electric field due to the shell is radial (the lines themselves point inward or outward, depending on the sign of the charge, while the strength of the lines is constant along a ring/radius, as a result of the shell have u.c.d.). For all electric fields of a radius r ≥ b, the electric field of that radius can be determined simply using the Charge-Flux Law, and has the following magnitude: $$E = \frac{1}{4\pi \varepsilon_0} \frac{q}{r^2}$$ E = The outward electric field created by a shell with uniform charge density.
ε₀ = The Permittivity of Free Space, equal to 8.85 × 10⁻¹² (C²) / (N × m²).
q = The total enclosed charge, namely that within the shell's material.
r = The distance to the point of measurement from the center of the shell, retricted to where r ≥ b.

Through a small amount of effort, one can find that this field is the very same as the field created by a shell with all of its charge concentrated in a particle of charge q in the center. Thus, the first theorem is proved.


Through an extraordinarily simple process, one can furthermore find through the Charge-Flux Law how the electric field at radius a is equal to 0, since the Gaussian surface encloses no charge. Thus, if a charged particle were enclosed by the shell, the shell would exert no net electric force on the particle. This proves the second shell theorem.


Inside a sphere with a uniform volume charge density, the electric field is radial, and any radius r (where a ≤ r ≤ b) will have an electric field of the following magnitude: $$E = \frac{1}{4\pi \varepsilon_0} \frac{q}{b^3} r$$ E = The electric field within the material of a shell, between a and b.
ε₀ = The Permittivity of Free Space, equal to 8.85 × 10⁻¹² (C²) / (N × m²).
q = The total charge WITHIN the specified radius; the charge between the inner surface of the shell and the specified radius r. All charge within the shell but outside of the radius r is irrelevant and not included in this subsphere.
b = The full radius of the shell, from center to external surface.
r = The radius from the center of the sphere to the point of measurement between a and b. This is akin to creating a smaller shell (cutting out the material between r and b) and measuring the electric field at its surface.

# P. Rule . Finding an equation to express electric potential energy is no difficult task at all. As a conservative force, the electric force can have all of the existing equations for potential energy and work (see Rule 100) applied:

F_E = -(dU_e / dr)

E.g., the electric force equals the negative of the derivative of the "electric potential energy" with respect to position (the infinitesimally small change in position dr).

By simplifying this equation slightly, as in moving variables around, integrating both sides, and substituting in values for electric force, a new and improved Change in Electric Potential equation can be developed: $$\Delta U_e = -q \int_{i}^{f} E \cos\theta \, dr$$ ∆U_e = The change in electric potential energy. It does not depend on the path taken between point A and point B - rather, all that is required is the displacement itself. Measured in Joules.

q = The magnitude of the charge/charged particle of the particular point being measured.

i = A generic starting point of a path taken in an electric field.

f = A generic ending point of a path taken in an electric field.

E = The magnitude of the electric field at a particular point.

θ = The angle between the directions of the electric field and the direction of the infinitesimally small displacement dr. When the charge is moving in the direction opposite that of the electric field, the angle will be 180°, and the negative of the resulting cosine will cancel out with the one outside the integral.

dr = The infinitesimally small displacement of the charge, essentially its position.


The potential energy of a positive charge increases as it moves opposite the direction of the electric field, while the potential energy of the negative charge will decrease as it moves opposite the direction of the field. E.g., change in potential energy will be positive for a positive charge, and negative for a negative charge.

# P. Rule . Electric Potential: SCALAR (though an attribute of a vector Electric Field).

Units: Volts, e.g. Joules / Coulombs. The symbol for the volts unit, hilariously, is the same as the symbol for electric potential: V.


Equation:

V = (U_e / q)

V = Electric Potential, the measure of electric potential energy experienced per unit charge in an electric field. Measured in Volts.
U_e = The magnitude Electric Potential Energy, explained in detail in Rule 197 and in the introduction directly above. Note that this refers to the p.e. of a single point.
q = The magnitude of the charge/charged particle of the particular point being measured.


Definition: The Electric Potential is a measure of the electric potential energy experienced per unit charge in an electric field. V and ΔU are indeed different - they have the same relationship as electric field and electric force. The electric potential energy is what creates the electric potential.

Since Electric Potential is a scalar value, the net electric potential at a particular point is found by summing each electric potential values (from each electric field) at that point.

In the same manner in which an electric field is defined by the force experienced by a small, positive test charge (see Subsection XIII.I), the electric potential is defined by the energy experienced by a small, positive test charge.

Of considerable importance and inherent relation to electric potential is Electric Potential Difference, aka, voltage. For more information, see Rule 199.

# P. Rule . Voltage/Electric Potential Difference:

The change in electric potential, derived from the Electric Potential described in Rule 198, is arguably more important and used than electric potential by itself.

Represented by ∆V, the Electric Potential Difference (also known as Voltage) is the difference in the electric potential between two points: ∆V = V_f - V_i.

Inherently, a charge does not need to move from one point to another in order for the difference to exist; ∆V's existence is only dependent on the electric field itself.

Equations representing ∆V, slightly reworked from those of Rule 198 & Rule 197, is as follows:
$$\Delta V = \frac{\Delta U_e}{q}$$ $$\Delta V = - \int_{i}^{f} \vec{E} \cdot d\vec{r}$$ ∆V = The Electric Potential Difference, the difference between the electric potentials of two points in an electric fields. Measured in Volts.

∆U_e = The change in electric potential energy. It does not depend on the path taken between point A and point B - rather, all that is required is the displacement itself. Measured in Joules.

q = The magnitude of the charge/charged particle of the particular point being measured.

i = A generic "starting" point in the electric field for measuring the electric potential difference.

f = A generic "ending" point in the electric field for measuring the electric potential difference.

E = The electric field at a particular point, the one that is generating the electric potential. When constant, it can be taken out of the integral, and the equation can resolve to ∆V = -Ed.

dr = The infinitesimally small displacement of the charge, essentially its position.

# P. Rule . Electric Potential around a Point Charge:

In circumstances in which you are calculating the Electric Potential generated by a single point charge (which, since Electric Potential is a scalar, just means ignoring the electric fields of any other variables in the equation (if any)), there is a single equation that you can use that works regardless of whether the charge is negative or positive (though you will have to factor this into the sign of the charge variable in the equation):

V_{point charge} = (kq / r)

V_{point charge} = The electric potential created and surrounded by a point charge.
k = The Coulomb constant, equal to 8.99 × 10⁹ (N × m²) / (C²).
q = The charge of the central particle, measured in Coulombs.
r = The distance between the centers of charge of the two objects.

Of course, this equation is effectively assigning zero electric potential to any points infinitely far away.

Note that the first equation of Rule 199 can be substituted into this equation to create an alternate definition for change in potential energy.

# P. Rule . Equipotential Surfaces/Lines:

An equipotential surface or line is one in which the electric potential is the same at every point along it. These lines form from (and have their shapes determined by) an electric field. Of course, this means that if the electric field is resulting from a negative charge, then each electric potential value along the equipotential line will be negative. If resulting from a positive charge, then each value will be positive. Simple as.

Equipotential Lines, also known as isolines, are always perpendicular to the electric field - thus, the electric field will have no components along the equipotential line.

In a uniform electric field, an equipotential line will be an infinite straight line. In an electric field around a charged particle, the equipotential lines will be circles concentric to the particle.

The electric potential difference between any point on one equipotential line to any point along another equipotential line will be the same.

When moving a charge along an equipotential line, the work required to do so is zero (since the electric potential difference will be zero - see Rule 203).

# P. Rule . Kinetic Energy through Charge Movement:

There are two equations that can be used to find the kinetic energy of a charged object with an electric potential difference, one for if only conservative forces are acting on it, and one for if nonconservative ones are as well.

If the charged object moves through an electric potential difference ∆V without an applied force acting on it (moving purely through electric attraction), then only conservative forces are acting upon it. Applying and simplifying the conservation of mechanical energy equation gives the following equation for change in kinetic energy:


∆K = -q × ∆V


∆K = Change in Electric Kinetic Energy.

q = The magnitude of the electric charge carried by the object, which is irrelevant to the positioning/movement of the object.

∆V = The electric potential differene between two points in an electric field.




If, instead, an external/applied force is acting on the charged object, creating nonconservative work, then the change in kinetic energy is as follows:


∆K = (-q × ∆V) + W_nonc


∆K = Change in Electric Kinetic Energy.

q = The magnitude of the electric charge carried by the object, which is irrelevant to the positioning/movement of the object.

∆V = The electric potential differene between two points in an electric field.

W_nonc = The work produced by the external, nonconservative force, breaking the conservation of mechanical energy.


A special case can emerge in which ∆K equals to zero. Then, and only then, can W_nonc be isolated in a very simple manner, just moving variables to the other side. For more details, see Rule 203.

# P. Rule . Work in moving a Charged Object:

If a charged object is moved from an initial point to a final point (differing points, obviously) via an external force, then the external force is performing work on the object. Furthermore, this work will change the electric potential energy of the charge.

There is an equation that can be used to find this work value in relation to electric potential difference - however, it requires that there be no change in the kinetic energy of the charge. Thus, it mandates the breaking of the conservation of Mechanical Energy. Since an external force is being applied (in order for there to even be work), this was already assumed and isn't much of an issue.

Therefore, under these circumstances, work equals the magnitude of the charged object's charge, multiplied by the electric potential difference the charge went through:


W_nonc = q × ∆V


W_nonc = The work produced by the external, nonconservative force, breaking the conservation of mechanical energy.

q = The magnitude of the electric charge carried by the object, which is irrelevant to the positioning/movement of the object.

∆V = The electric potential difference between two points in an electric field.


Note that this equation is the negative of the work produced by an electric field, which itself can be derived by applying Rule 100 equations for potential energy to the 1st equation found in Rule 199.

# P. Rule . Electronvolts:

A new unit, more convenient for dealing with work & electric potential/kinetic energy, is the Electronvolt. An electronvolt is defined as the work required to move a single elementary charge e (such as that of an electron or proton) through an electric potential difference ΔV of exactly one volt, and can thus be used to represent very small amounts of energy.

1 eV = 1.602 × 10⁻¹⁹ J