Two capacitors C_1 and C_2 are connected in parallel to a battery....

Two capacitors

C_1

and

C_2

are connected in parallel to a battery. Charge-time graph is shown below for the two capacitors. The energy stored with them are

U_1

and

U_2

, respectively. Which of the given statements is true? Two capacitors C_1 and C_2 are connected in parallel to a battery. Charge-time graph is shown below for the two

Two capacitors C_1 and C_2 are connected in parallel to a battery. Charge-time graph is shown below for the two

Options

A $\mathrm{C}_2\gt\mathrm{C}_1, \mathrm{U}_2 \lt \mathrm{U}_1$
B $\mathrm{C}_1\gt\mathrm{C}_2, \mathrm{U}_1\gt\mathrm{U}_2$
C $\mathrm{C}_1\gt\mathrm{C}_2, \mathrm{U}_1 \lt \mathrm{U}_2$
D $\mathrm{C}_2\gt\mathrm{C}_1, \mathrm{U}_2\gt\mathrm{U}_1$

Correct answer

D. $\mathrm{C}_2\gt\mathrm{C}_1, \mathrm{U}_2\gt\mathrm{U}_1$

Step-by-step solution

For a capacitor at steady state

q=C V \text{

and

} U=\frac{1}{2} C V^2

Since

C_1 and C_2

are connected in parallel,

V_1=V_2

. Also from graph

q_1 < q_2

\begin{aligned} & \Rightarrow \quad C_1 V_1 < C_2 V_2 \\ & \text{ i.e. } C_1 < C_2 \text{ or } C_2>C_1 \\ & \qquad \frac{U_1}{U_2}=\frac{C_1 V_1^2}{C_2 V_2^2}=\frac{C_1}{C_2} < 1 \end{aligned}

or U_1 < U_2 or U_2>U_1

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