A parallel-plate capacitor has plate area A and separation d. It is filled with a dielectric whose dielectric constant varies as k(x) = K(1 + αx), where x is measured from one plate. If αd ≪ 1, the capacitance is best approximated by:

Question

G1 Physics · Accepted Answer

For a thin layer dx, dV = (Q/A)dx/[ε₀K(1+αx)]. Integrating from 0 to d gives C = AKε₀α/ln(1+αd). For αd≪1, expand ln(1+αd) to obtain C ≈ (AKε₀/d)(1+αd/2). Hence A.