Problem 2

A parallel-plate capacitor of plate area

A

and separation

d

is connected to a constant voltage

V

. A dielectric of relative permittivity

\kappa

fills a fraction

x

of the plate length, parallel to the plates; the remainder is vacuum. Neglect edge effects. Find the capacitance, positive-plate charge, and stored field energy. (a) State the assumptions and establish the governing equation for the requested quantity. (b) Solve it in terms of the given symbols, identifying the direction, sign, or physical nature of the result. (c) Check a simple limiting case and state when the model remains valid. Use SI units consistently, retain the symbols in the setup, and enforce all geometric constraints and initial conditions.

problems.proof.stepOf

problems.proof.prevStep problems.proof.nextStep

Q_+=CV=\frac{\varepsilon_0AV}{d}[1+(\kappa-1)x],\quad U=\frac12CV^2

problems.proof.analysis

Check the dimensions of the final expression and test a simple physical limit. This independent check can expose a sign error or a missing factor in the derivation.

problems.proof.pitfall. Do not discard an initial condition or stated constraint; an algebraic root outside the physical domain is not an answer.