A capacitor with plate area A and separation d is filled with a dielectric κ. Provide the capacitance.

A capacitor with plate area A and separation d is filled with a dielectric κ. Provide the capacitance.

• A. C = ε0 A / d
• B. C = ε0 A / (κ d)
• C. C = κ ε0 A / d
• D. C = ε0 κ A d

Answer: (C)

The key idea is that the capacitance of a parallel-plate capacitor depends on the permittivity of the material between the plates. For a uniform field, C equals ε times the plate area over the separation: C = ε A / d. A dielectric with relative permittivity κ means the actual permittivity between the plates is ε = κ ε0, where ε0 is the vacuum permittivity. Substituting gives C = κ ε0 A / d. This is why the correct form scales the vacuum result by κ. If κ were 1, you’d recover the vacuum case C = ε0 A / d. The other expressions either use vacuum permittivity incorrectly, or place κ in the wrong place or even mix in the distance in a way that isn’t dimensionally correct.

The key idea is that the capacitance of a parallel-plate capacitor depends on the permittivity of the material between the plates. For a uniform field, C equals ε times the plate area over the separation: C = ε A / d. A dielectric with relative permittivity κ means the actual permittivity between the plates is ε = κ ε0, where ε0 is the vacuum permittivity. Substituting gives C = κ ε0 A / d. This is why the correct form scales the vacuum result by κ. If κ were 1, you’d recover the vacuum case C = ε0 A / d. The other expressions either use vacuum permittivity incorrectly, or place κ in the wrong place or even mix in the distance in a way that isn’t dimensionally correct.