The electric potential in a region of space depends only on the x-coordinate as V = -2x³ + 3 (SI units). Determine the volume charge density rho(x) in that region.

6 * epsilon₀ * x
12 * epsilon₀ * x
6 * epsilon₀
-12 * epsilon₀

Correct answer: 12 * epsilon₀ * x

Solution

Poisson's equation in one dimension: d²V/dx² = -rho/epsilon₀. First derivative: dV/dx = -6x². Second derivative: d²V/dx² = -12x. Therefore rho = -epsilon₀ * (-12x) = 12 * epsilon₀ * x.

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