Question

what happens as you move them closer? further?

Explanation:

Step1: Recall electric - potential - energy formula

The electric - potential energy between two point charges $q_1$ and $q_2$ separated by a distance $r$ is given by $U = k\frac{q_1q_2}{r}$, where $k=\frac{1}{4\pi\epsilon_0}\approx9\times 10^{9}\ N\cdot m^{2}/C^{2}$.

Step2: Analyze when moving charges closer

When we move the two charges closer together (decrease $r$), if $q_1$ and $q_2$ have the same sign (both positive or both negative), the potential energy $U$ increases because $U$ is inversely proportional to $r$ and the product $q_1q_2> 0$. If $q_1$ and $q_2$ have opposite signs (one positive and one negative), the potential energy $U$ decreases as $r$ decreases since $U$ is negative and its magnitude increases (but the value of $U$ gets less negative).

Step3: Analyze when moving charges further

When we move the two charges further apart (increase $r$), if $q_1$ and $q_2$ have the same sign, the potential energy $U$ decreases. If $q_1$ and $q_2$ have opposite signs, the potential energy $U$ increases (becomes more negative).

Answer:

If the charges have the same sign: moving them closer increases potential energy and moving them further decreases potential energy. If the charges have opposite signs: moving them closer decreases potential energy and moving them further increases potential energy.