Determine the capacitances of the capacitors depicted in the figure. The area of the plates is A and the distance between them is d . In each specific part of the figure, the different dielectric slabs have equal thickness, and the dielectric slabs completely fill the space between the plates.

Step-by-step solution

For figure (a), the two dielectric slabs are in series. Each slab has an area A and thickness d/2 . The capacitance of the first slab is C₁ = K₁ ₀ A d/2 = 2 K₁ ₀ A d . The capacitance of the second slab is C₂ = K₂ ₀ A d/2 = 2 K₂ ₀ A d . The equivalent capacitance C_a is given by: 1 C_a = 1 C₁ + 1 C₂ = d 2 K₁ ₀ A + d 2 K₂ ₀ A = d 2 ₀ A ( K₁ + K₂ K₁ K₂ ) C_a = 2 K₁ K₂ ₀ A d(K₁ + K₂) For figure (b), the three dielectric slabs are in series. Each slab has an area A and thickness d/3 . The capacitances are C₁ = 3 K₁ ₀ A