Consider a capacitor constructed from two square metal plates, each having an edge length a , with a separation distance d between them. As depicted in the figure, the space between the plates is filled with dielectrics having dielectric constants K₁ and K₂ . Determine the capacitance of this arrangement.

Step-by-step solution

Consider an elemental strip of width dx at a distance x from the left end of the capacitor. The area of the plates for this strip is dA = a , dx . This strip can be considered as two capacitors in series: one with dielectric K₁ and the other with dielectric K₂ . From the geometry of the figure, the thickness of the dielectric K₂ at distance x is y = d a x . The thickness of the dielectric K₁ at distance x is d - y = d - d a x . The capacitance of the upper part (with K₁ ) is: dC₁ = K₁ ₀ a , dx d - d a x The capacit