Question

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?

• A. $C_2 > C_1, U_2 < U_1$
• B. $C_1 > C_2, U_1 > U_2$
• C. $C_1 > C_2, U_1 < U_2$
• D. $C_2 > C_1, U_2 > U_1$

Answer 1

Answer: (D)

Option D. $C_2 > C_1, U_2 > U_1$

Answer 2

Solution:

1. In a circuit with a battery, each capacitor charges up to a final charge given by

   $$q = C\,V.$$

2. The graph shows that the curve labeled $C_1$ reaches a higher final charge than that labeled $C_2$. Thus,

   $$C_1\,V > C_2\,V \quad\Longrightarrow\quad C_1 > C_2.$$

3. The energy stored in a capacitor is given by

   $$U = \frac{1}{2}\,C\,V^2.$$

   Since $C_1 > C_2$ (with the same battery voltage $V$), it follows that

   $$U_1 > U_2.$$

Answer: Option B. $C_1 > C_2, U_1 > U_2$

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Brief Explanation (minimal): Using $q = CV$ and $U = \frac{1}{2}CV^2$, the capacitor with the higher saturation charge (from the graph) must have a larger capacitance, and hence stores more energy.