Capacitor with a Dielectric — Question 1

Question 1

Inserting dielectric into an isolated capacitor lowers voltage and stored energy

An isolated plate capacitor initially has capacitance C0, voltage V0 and energy U0. A linear lossless dielectric with relative permittivity 3 fills the gap quasistatically. The terminals remain disconnected.

1. Find final capacitance, charge and voltage.

2. Find final energy and external insertion work, neglecting friction and material deformation.

Question 1 - Solution

Inserting dielectric into an isolated capacitor lowers voltage and stored energy

(a) Method and calculation

$$C_f=3C_0,\quad Q_f=Q_0=C_0V_0.$$$$V_f=Q_0/C_f=V_0/3.$$

(b) Interpretation and further calculation

$$U_f=Q_0^2/(2C_f)=U_0/3.$$$$W_{\mathrm{ext}}=\Delta U=-2U_0/3.$$

Physical check

The field pulls the dielectric inward, so an agent maintaining quasistatic motion extracts energy.