Capacitor with a Dielectric — Question 4

Question 4

Side-by-side dielectrics share the same voltage and add their charges

Two dielectrics of relative permittivities 2 and 6 each fill half the plate area A and the full gap d. Their interface runs along the field direction. Neglect fringing.

1. Find the capacitance relative to $C_0=\epsilon_0A/d$.

2. At fixed voltage V find each region charge and its fraction of the total.

Question 4 - Solution

Side-by-side dielectrics share the same voltage and add their charges

(a) Method and calculation

$$C_1=\frac{2\epsilon_0(A/2)}d=C_0,\quad C_2=\frac{6\epsilon_0(A/2)}d=3C_0.$$$$C=C_1+C_2=4C_0.$$

(b) Interpretation and further calculation

$$Q_1=C_0V,\quad Q_2=3C_0V,\quad E_1=E_2=V/d.$$$$Q_1/Q=1/4,\quad Q_2/Q=3/4.$$

Physical check

The same material fractions give a different capacitance when their orientation relative to the field changes.