Unlike Electric Charges Attract

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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 opposite polarity.

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


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


Without loss of generality, let $q_a$ be positive and $q_b$ be negative.

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

Thus $q_a q_b$ is a negative number.


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

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


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 towards $b$.


The effect of these forces is to cause $a$ and $b$ to pull together, that is, to attract each other.


Also see


Sources

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