Coulomb’s law

Just as there is a gravitational force between any two masses, there is an electric force between any two charged objects. Unlike gravity, this force can repel as well as attract. The charge on an object is either positive or negative—opposites charges attract, and similar charges repel. The magnitude of the force is given by Coulomb’s law:

$\left|{\overrightarrow{F}}_{\text{el}}\right|=k\frac{q_1{q}_2}{r^2}\text{.}$

The $q$ values are the charges on the objects, measured in coulombs (C). Their product is divided by radius squared (from centre to centre, measured in metres), and then multiplied by Coulomb’s constant ($k$), which has a value of 9.00 × 10⁹ N⋅m²/C².

Like with gravity, we will sometimes want to look at how the force changes in terms of ratios. To do that, we can use the following equation:

$\frac{F_{\text{el,2}}}{F_{\text{el,1}}}=\left(\frac{q_{\text{A,2}}}{q_{\text{A,1}}}\right)\left(\frac{q_{\text{B,2}}}{q_{\text{B,1}}}\right){\left(\frac{r_1}{r_2}\right)}^2\text{.}$

In some electric force questions, you will be given a diagram of several charged objects and you must find the net force on one particular object. To do this, you must find the values of $F_{\text{el}}$ between that object and each other one, determine the direction for each force, and add them up. This also works in two dimensions.

Example

A positively charged object of 1.95 × 10⁻⁶ C is beside a negatively charged object of 2.75 × 10⁻⁶ C. They are separated by 0.124 m. What is the electric force, and does it attract or repel?

All we have to do is plug in the values into Coloumb’s law:

$\left|{\overrightarrow{F}}_{\text{el}}\right|=(9.00\times {10}^9\frac{{\text{Nm}}^2}{{\text{C}}^2})\frac{(1.95\times {10}^{-6}\text{C})(2.75\times {10}^{-6}\text{C})}{(0.124\text{m}{)}^2}\text{.}$

Evaluating this gives us 3.14 N, and this force attracts the two objects together because they have opposite charges.