Consider the situation depicted in the figure. Each plate has a width of b . The capacitor plates are rigidly clamped in the laboratory and connected to a battery of emf E . Assume all surfaces are frictionless. Determine the value of M for which the dielectric slab remains in equilibrium.

Step-by-step solution

Let x be the length of the dielectric slab inside the capacitor plates. The system can be modeled as two capacitors in parallel, one with air and one with the dielectric. The equivalent capacitance is: C = ₀ b (l - x) d + K ₀ b x d = ₀ b d [l + x(K - 1)] Since the battery remains connected, the potential difference across the plates is constant at E . The electric force pulling the dielectric slab into the capacitor is given by: F_e = 1 2 E ^2 dC dx Differentiating C with respect to x gives: dC dx = ₀ b (K - 1) d S