Physic Labs

Problem 2

A parallel-plate capacitor of plate area AA and separation dd is connected to a constant voltage VV. A dielectric of relative permittivity κ\kappa fills a fraction xx 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.prevStepproblems.proof.nextStep
A1=xA,A2=(1−x)A,C1=κε0xAd,C2=ε0(1−x)Ad,C=C1+C2=ε0Ad[1+(κ−1)x],Q+=CV=ε0AVd[1+(κ−1)x],U=12CV2\boxed{A_1=xA,\quad A_2=(1-x)A,\quad C_1=\frac{\kappa\varepsilon_0xA}{d},\quad C_2=\frac{\varepsilon_0(1-x)A}{d},\quad C=C_1+C_2=\frac{\varepsilon_0A}{d}[1+(\kappa-1)x],\quad Q_+=CV=\frac{\varepsilon_0AV}{d}[1+(\kappa-1)x],\quad U=\frac12CV^2}
problems.proof.analysis

The results are listed in the order requested and use one consistent sign convention. Their dimensions, initial conditions, and model constraints must all agree.