The potential difference that must be applied across the parallel and series combination of three identical capacitors such that the energy stored in them becomes the same. The ratio of potential diff

Question

The potential difference that must be applied across the parallel and series combination of three identical capacitors such that the energy stored in them becomes the same. The ratio of potential difference in parallel to series combination is

TestHub · Accepted Answer

Let the capacitance of each identical capacitor be $C$ .

For the parallel combination, the equivalent capacitance is $C_{p} = 3 C$ .

The energy stored in the parallel combination when a potential difference $V_{p}$ is applied is $U_{p} = \frac{1}{2} C_{p} V_{p}^{2} = \frac{1}{2} (3 C) V_{p}^{2}$ .

For the series combination, the equivalent capacitance is $C_{s} = \frac{C}{3}$ .

The energy stored in the series combination when a potential difference $V_{s}$ is applied is $U_{s} = \frac{1}{2} C_{s} V_{s}^{2} = \frac{1}{2} (\frac{C}{3}) V_{s}^{2}$ .

Given that the energy stored in both combinations is the same, $U_{p} = U_{s}$ .

$\frac{1}{2} (3 C) V_{p}^{2} = \frac{1}{2} (\frac{C}{3}) V_{s}^{2}$

$3 V_{p}^{2} = \frac{V _{s}^{2}}{3}$

$\frac{V _{p}^{2}}{V _{s}^{2}} = \frac{1}{9}$

$\frac{V _{p}}{V _{s}} = \frac{1}{3}$

The ratio of potential difference in parallel to series combination is $1 : 3$ .

Answer: $1 : 3$

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