Two parallel plate capacitors X and Y are connected in series to a ...

Question

Two parallel plate capacitors X and Y are connected in series to a 6 V battery. They have the same plate area and same plate separation but capacitor X has air between its plates, whereas capacitor Y contains a material of dielectric constant 4.

(a) Calculate the capacitances of X and Y, if the equivalent capacitance of the combination of X and Y is 4 F.

(b) Calculate the potential difference across the plates of X and Y.

Accepted Answer

$X$ (air): capacitance $C_X$. $Y$ (dielectric $K=4$): $C_Y=4C_X$ (same area, same separation).

Series combination:

$$\frac{1}{C_{eq}}=\frac{1}{C_X}+\frac{1}{4C_X}=\frac{5}{4C_X} \Rightarrow C_{eq}=\frac{4C_X}{5}$$

Given $C_{eq}=4\ \mu F$:

$$C_X=\frac{5\times4}{4}=5\ \mu F,\qquad C_Y=4C_X=20\ \mu F$$

(b) In series, both capacitors carry the same charge:

$$Q=C_{eq}\times V=4\times6=24\ \mu C$$

$$V_X=\frac{Q}{C_X}=\frac{24}{5}=4.8\text{ V},\qquad V_Y=\frac{Q}{C_Y}=\frac{24}{20}=1.2\text{ V}$$

(Check: $4.8+1.2=6$ V ✓)

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