Question #42597

This question tests the coupled thermodynamic and electrostatic equilibrium of a deformable system.

The primary trap is the assumption that the sphere's radius remains constant, leading students to incorrectly apply simple error propagation $P \propto Q^2 \implies \Delta P/P = 2\alpha$.

However, because the pressure depends on the surface charge density $\sigma$, any change in $Q$ alters the pressure, which in turn changes the volume adiabatically, further modifying $\sigma$.

Students must establish a coupled proportionality by equating the electrostrictive boundary condition with the adiabatic equation of state for a gas mixture ($\gamma = 1.5$).

Failing to account for the volumetric expansion/contraction yields incorrect sensitivities like $2\alpha$ (rigid sphere assumption), $-6\alpha$ (isothermal assumption), or $10\alpha$ (incorrectly assuming a pure monoatomic gas).