Question #42947

This question tests the fundamental definition of electrostatic energy stored in an electric field.

A common trap for advanced students is to calculate the volume charge density using Poisson's equation ($\rho = -\epsilon_0 \nabla^2 V$).

For this specific quadrupole potential, $\nabla^2 V = 2\alpha + 2\alpha - 4\alpha = 0$, meaning the region is completely charge-free ($\rho = 0$).

Students relying on the formula $U = \frac{1}{2} \int \rho V d\tau$ will incorrectly conclude the stored energy is zero (Option B).

However, that formula represents the total energy of a complete charge distribution over all space, not the energy localized within a specific volume.

To find the energy stored in a specific region, one must strictly integrate the local energy density $u = \frac{1}{2}\epsilon_0 E^2$ over that volume.

Furthermore, the non-spherical nature of the potential gradient forces students to perform a full 3D volume integral in spherical coordinates, preventing algebraic shortcuts and penalizing those who incorrectly assume $|\vec{E}|^2 \propto r^2$ symmetrically (Option C).