1. Initial State

Initially, the pistons are in equilibrium with atmospheric pressure $P_0$. $$ P_{\text{gas, initial}} = P_0 $$ Volume $V_0 = A \cdot d$.

2. Final State with Charges

The pistons are given charges $+q$ and $-q$. This creates an attractive electrostatic force between them, compressing the gas until a new equilibrium is reached at distance $d’$.

The electrostatic attractive force is: $$ F_e = \frac{q^2}{2\epsilon_0 A} $$ The new force balance equation for one piston (forces pointing outward vs inward): $$ P_{\text{gas, final}} A = P_0 A + F_e $$ $$ P_{\text{gas, final}} = P_0 + \frac{q^2}{2\epsilon_0 A^2} $$

3. Thermodynamics (Boyle’s Law)

Assuming the temperature remains constant (Isothermal process), we apply $P_1 V_1 = P_2 V_2$:

$$ P_0 (A d) = \left( P_0 + \frac{q^2}{2\epsilon_0 A^2} \right) (A d’) $$

4. Solving for New Distance $d’$

$$ d’ = d \left( \frac{P_0}{P_0 + \frac{q^2}{2\epsilon_0 A^2}} \right) $$

Rearranging the denominator:

$$ d’ = \frac{P_0 d}{\frac{2\epsilon_0 A^2 P_0 + q^2}{2\epsilon_0 A^2}} $$