A parallel plate capacitor with plate area A and distance of separation d is filled with a dielectric. What is the capacity of the capacitor when the permittivity of the dielectric varies as ε(x)=ε₀+kx for 0

Question

A parallel plate capacitor with plate area A and distance of separation d is filled with a dielectric. What is the capacity of the capacitor when the permittivity of the dielectric varies as ε(x)=ε₀+kx for 0<x≤d/2 and ε(x)=ε₀+k(d-x) for d/2≤ x≤ d?

Accepted Answer

(3) (kA)/(2 ln ((2ε₀+kd)/(2ε₀))). Slice the gap into slabs of thickness dx. Each slab is a capacitor dC=(ε(x)A)/(dx), and the slabs sit one behind the other, so they add in series. 1/C=∫₀^d(dx)/(ε(x)A). The profile is symmetric about x=d/2, so this is twice the integral over the first half. ∫₀^d/2(dx)/(ε₀+kx)=1/k ln ((ε₀+kd/2)/(ε₀))=1/k ln ((2ε₀+kd)/(2ε₀)). 1/C=2/(kA)ln ((2ε₀+kd)/(2ε₀)). C=(kA)/(2 ln ((2ε₀+kd)/(2ε₀))).

A parallel plate capacitor with plate area A and distance of separation d is filled with a dielectric. What is the capacity of the capacitor when the permittivity of the dielectric varies as ε(x)=ε₀+kx for 0<x≤d/2 and ε(x)=ε₀+k(d-x) for d/2≤ x≤ d?

Step-by-step solution

Why the other options are wrong