# 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 Coulomb's Law (since the denominator distance value increases and all). The entire Coulomb's law 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-Coulomb's Law 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 Coulomb's law, 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 coulomb's law + 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.