The gap between the plates of a parallel plate capacitor of area A and distance between plates d, is filled with a dielectric whose permittivity varies linearly from ε₁ at one plate to ε₂ at the other. The capacitance of the capacitor is :

Answer is D. The capacitance of a parallel plate capacitor with a dielectric is given by C = εA/d, where ε is the permittivity of the dielectric, A is the area of the plates, and d is the distance between the plates. In this case, the permittivity varies linearly from ε₁ to ε₂. Let's consider a small element of thickness dx at a distance x from the plate with permittivity ε₁. The permittivity at this point is given by ε(x) = ε₁ + (ε₂ - ε₁)(x/d). The capacitance of this small element is dC = ε(x)A/dx. To find the total capacitance, we integrate over the entire distance d: C = ∫₀ᵈ (ε₁ + (ε₂ - ε₁)(x/d))A/dx = A∫₀ᵈ (ε₁/dx + (ε₂ - ε₁)(x/d)/dx) dx = A[ε₁x + (ε₂ - ε₁)x²/2d] from 0 to d = A[ε₁d + (ε₂ - ε₁)d/2] = A[(ε₁ + ε₂)/2]d = ε₀A/d[dln(ε₂/ε₁)] This is not one of the options provided. Therefore, the given options may contain errors or be incomplete to answer the question.