The potential difference that must be applied across the series and parallel combination of 4 identical capacitors is such that the energy stored in them becomes the same. The ratio of potential diffe

Question

The potential difference that must be applied across the series and parallel combination of 4 identical capacitors is such that the energy stored in them becomes the same. The ratio of potential difference in series to parallel combination is

TestHub · Accepted Answer

Let the capacitance of each of the 4 identical capacitors be $C$ .

The equivalent capacitance in series combination is $C_{s} = \frac{C}{4}$ .

The equivalent capacitance in parallel combination is $C_{p} = 4 C$ .

Let $V_{s}$ and $V_{p}$ be the potential differences applied across the series and parallel combinations respectively.

The energy stored in the series combination is $U_{s} = \frac{1}{2} C_{s} V_{s}^{2} = \frac{1}{2} (\frac{C}{4}) V_{s}^{2}$ .

The energy stored in the parallel combination is $U_{p} = \frac{1}{2} C_{p} V_{p}^{2} = \frac{1}{2} (4 C) V_{p}^{2}$ .

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

$\frac{1}{2} (\frac{C}{4}) V_{s}^{2} = \frac{1}{2} (4 C) V_{p}^{2}$

$\frac{V _{s}^{2}}{4} = 4 V_{p}^{2}$

$V_{s}^{2} = 16 V_{p}^{2}$

$\frac{V _{s}}{V _{p}} = \frac{4}{1}$

The ratio of potential difference in series to parallel combination is $4 : 1$ .

Answer: $4 : 1$

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