Question 481: Electrostatics & Capacitance

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

Accepted Answer

To find the ratio of the potential difference required for equal energy storage in both series and parallel combinations of four identical capacitors, let's analyze the equivalent capacitance for each configuration.
Series Combination
The equivalent capacitance for capacitors in series is given by:
$ C_1 = \frac{C}{4} $
The potential energy stored in this series configuration is:
$ U_1 = \frac{1}{2} C_1 V_1^2 $
Parallel Combination
The equivalent capacitance for capacitors in parallel is:
$ C_2 = 4C $
The potential energy stored in the parallel configuration is:
$ U_2 = \frac{1}{2} C_2 V_2^2 $
Equating Energy
Since the energy stored in both configurations is equal, we have:
$ U_1 = U_2 $
This implies:
$ C_1 V_1^2 = C_2 V_2^2 $
Substituting the expressions for $ C_1 $ and $ C_2 $, we get:
$ \frac{C}{4} V_1^2 = 4C V_2^2 $
To find the ratio of the potential differences, solve for:
$ \frac{V_1^2}{V_2^2} = \frac{4C 	imes 16}{C} $
Simplifying:
$ \frac{V_1^2}{V_2^2} = \frac{16}{1} $
So, the ratio of the potential differences is:
$ \frac{V_1}{V_2} = \sqrt{\frac{16}{1}} = \frac{4}{1} $
Thus, the ratio of potential difference in series to parallel combination is 4:1.

Answer

$1: 2$

Answer

$1: 4$

Answer

$4: 1$

Answer

$2: 1$

Electrostatics & Capacitance

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

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