Two condensers are in parallel and the energy of the combination is 0.1 J, when the difference of potential between terminals is 2 V. With the same two condensers in series, the energy is $1.6 \times 10^{- 2}$ J for the same difference of potential across the series combination. What are the capacities?

Open in App

Solution

Let the capacities are $C_{1}, C_{2}$
When they are in parallel, the net capacitance $C_{p} = C_{1} + C_{2}$
Thus, energy stored for this combination,
$U_{p} = \frac{1}{2} C_{p} V^{2}$
or $0.1 = 0.5 (C_{1} + C_{2}) (2^{2})$ or $C_{1} + C_{2} = 0.05 . . . (1)$
When they are in series, the net capacitance $C_{s} = \frac{C_{1} C_{2}}{C_{1} + C_{2}}$
Thus, energy stored for this combination,
$U_{s} = \frac{1}{2} C_{s} V^{2}$
or $1.6 \times 10^{- 2} = 0.5 \frac{C_{1} C_{2}}{C_{1} + C_{2}} (2^{2})$
or $C_{1} + C_{2} = 125 C_{1} C_{2} . . . (2)$
From (1) and (2), $C_{1} + (0.05 - C_{1}) = 125 C_{1} (0.05 - C_{1})$
or $0.05 = 6.25 C_{1} - 125 C_{1}^{2}$
or $1 = 125 C_{1} - 2500 C_{1}^{2}$ or $2500 C_{1}^{2} - 125 C_{1} + 1 = 0$
so, $C_{1} = \frac{125 \pm \sqrt{125^{2} - 4 (2500) (1)}}{2 (2500)} = \frac{125 \pm 75}{5000}$
So, $C_{1} = 40 m J$ or $10 m J$
Thus, from (1) $C_{2} = 10 m J$ or $40 m J$

Suggest Corrections

Similar questions

Q. Two condensers of capacity

0.3 μ F

and

0.6 μ F

respectively are connected in series. The combination is connected across a potential of

6

volts. The ratio of energies stored by the condensers will be:

Q. identical condensers are joined in parallel and are charged to potential V . Now they are separated and joined in series. Then the total energy and potential difference of the combination will be

Q. Two condensers of capacity

0.3 μ F

and

0.6 μ F

respectively are connected in series. The combination is connected across a potential of 6V. The ratio of energies stored by the condenser will be:

Q. Two capacitors of capacities

1 μ F

and

C μ F

are connected in series and the combination is charged to a potential difference of

120 V

. If the charge on the combination is

80 μ C

, the energy stored in the capacitor of capacity

C

μ J

is :

Two condensers of capacitance $4 μ F$ and $5 μ F$ are joined in series. If the potential difference across $5 μ F$ is $10 V$ , then the potential difference across $4 μ F$ condenser is :

Join BYJU'S Learning Program

Two condensers are in parallel and the energy of the combination is 0.1 J, when the difference of potential between terminals is 2 V. With the same two condensers in series, the energy is 1.6×10−2J for the same difference of potential across the series combination. What are the capacities?