# 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 section XII.II) of roughly nine billion newtons.

# P. Rule . An atom, the smallest unit of matter retaining the chemical properties of its element, 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 ~1.69 × 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 are composed of quarks - specifically, two up quarks and one down quark.

The up quark has a charge of +(2/3)e, while the down quark has a charge of -(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 nucleus of an atom would repel

The electron, being an elementary particle, 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 = Elementary Charge

# 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 . 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, the force produced by the Electric Charge inherent to all matter.

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.

# A. 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.

# A. 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 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.

# A. 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.

# A. 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.

# A. Rule . Conservation of Charge:

The total electric charge of an isolated system never changes. Emphasis on 'isolated'.

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 of +6 nC, then the final charge of the conductors, once touched, will be 1.5 nC for both.

# 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 charge/charged particle of the particular point being measured.


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

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

All Electric fields experience lines of attraction that can be summarized into a simple rule of thumb - see Rule 168.

# A. Rule . Electric Field Lines:

1. The lines of attraction in an electric field always point away from the positive charge and toward the negative charge. 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.

# A. Rule . A Continuous Charge Distribution is a simple concept that elaborates the simple 'point' model of charges into one that uses calculus to account for more chargetype possibilities, furthering our 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 167), 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.

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 170 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.

# A. Rule . The given integral equation for CCDs (continuous charge distributions, see Rule 169) 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 it's 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 169).

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.

# A. 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 167)). 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.

# A. 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 170), 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.