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 sign.

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.

Proof
By Coulomb's Law of Electrostatics:


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

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.

Then $q_a q_b$ is a positive number multiplied by a positive number.

Thus $q_a q_b$ is a positive number.

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

Then $q_a q_b$ is a negative number multiplied by a negative number.

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.

Also see

 * Unlike Electric Charges Attract