Like Electric Charges Repel

Theorem

Let $a$ and $b$ be stationary particles, each carrying an electric charge of $q_a$ and $q_b$ respectively.

Let $q_a$ and $q_b$ be of the same polarity.

That is, let $q_a$ and $q_b$ be like charges.

Then the forces exerted by $a$ on $b$, and by $b$ on $a$, are such as to cause $a$ and $b$ to repel each other.

$\mathbf F_{a b} \propto \dfrac {q_a q_b {\mathbf r_{a b} } } {r^3}$

where:

$\mathbf F_{a b}$ is the force exerted on $b$ by the electric charge on $a$

$\mathbf r_{a b}$ is the displacement vector from $a$ to $b$

$r$ is the distance between $a$ and $b$.

Let $q_a$ and $q_b$ both be positive.

Thus $q_a q_b$ is a positive number.

Similarly, let $q_a$ and $q_b$ both be negative.

Thus $q_a q_b$ is similarly a positive number.

So, when $q_a$ and $q_b$ are like charges, $\dfrac {q_a q_b} {r^3}$ is a positive number.

Hence $\mathbf F_{a b}$ is in the same direction as the displacement vector from $a$ to $b$.

That is, the force exerted on $b$ by the electric charge on $a$ is in the direction away from $a$.

The same applies to the force exerted on $a$ by the electric charge on $b$.

That is, the force exerted on $b$ by the electric charge on $a$ is in the direction away from $b$.

The effect of these forces is to cause $a$ and $b$ to push apart, that is, to repel each other.

1958: C.A. Coulson: Electricity (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: Preliminary Survey: $\S 1$. Electrostatics
1990: I.S. Grant and W.R. Phillips: Electromagnetism (2nd ed.) ... (previous) ... (next): Chapter $1$: Force and energy in electrostatics: $1.1$ Electric Charge