# Electrostatics and Inverse-Square Laws

## Equations

Wapplied = ∆ U = q∆V

V = Ed

## Explanation of the derivation of the equations

- We start with Coulomb’s Law at the upper left
- Normalizing by the value of the test particle’s charge (E = F/q), we can figure out Electric Field strength measured in (N/C), which lets us predict the Force that any test particle would feel if it were placed at that position relative to the source charge.

- Notice that by substituting in (bottom left), we have literally eliminated the test charge from the equation. Thus Electric Fields are functions of the source charge and your distance from it only.

- Just like Gravitational Potential Energy Ug = mgh = Fg∆y, we can say that Electrostatic Potential Energy Ues = Fes∆r. Thus, to get to the right side of the table, we multiply the left column by r, the distance between the source and test charge.

- This is good to know, but unfortunately, this formula still takes into account the test charge, so we cannot yet make predictions for all possible test charges.

- Voltage is the Electrostatic Potential Energy normalized for all test charges.

- Voltage lets us predict the potential energy of any test particle at that distance from the source charge
- Notice that while Force and Electric Fields are Inverse-Square Laws, Potential Energy, and thus Voltage, decrease linearly with distance (r1 in the denominator, not r2).
- Voltage is the crown jewel of these equations.

- Knowing Voltage at one spot, we can easily calculate Voltage at a different radius away from the same test charge
- Knowing Voltage at those radii, we can figure out the Potential Energy any test charge we choose would have at that distance
- Knowing the difference in those potential energies, we can calculate how much energy we would have to invest or release to move the particle between them, or in other words, the Work needed to fight the Electrostatic Force.

## Why forces that generate fields are governed by inverse-square laws

Imagine the source of the field is a spherical transmitter (like the button on a radio tower). A wave leaves it, and propagates out in 3-d space as an expanding sphere. Every bit of energy passing through the button is now diluted by being spread out across the surface area of that larger sphere, which is 4πr2. That’s why, in the full statement of Coulomb’s Law, 4πr2 appears in the denominator (Magnetic and Gravitational Fields show this effect, too!) |