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. \( C_1>C_2, U_1<U_2 \)
• B. \( C_1>C_2, U_1>U_2 \)
• C. \( C_2>C_1, U_2<U_1 \)
• D. \( C_2>C_1, U_2>U_1 \)

Hint:

Capacitance affects charging time and the energy stored; a higher capacitance results in slower charging and more energy storage at the same voltage.

Answer 1

The Correct Option is B

Capacitors $ C_1 $ and $ C_2 $ are connected in parallel to a battery. The provided charge-time graph illustrates the charge on each capacitor over time.

Observation from the graph indicates that as $ t \to \infty $, $ C_1 $ accumulates a greater charge than $ C_2 $.
Due to their parallel connection, both capacitors experience the same voltage $ V $.

The relationship between charge ($ Q $), capacitance ($ C $), and voltage ($ V $) is defined by $ Q = CV $. 
Given that $ C_1 $ holds a higher charge ($ Q_1 > Q_2 $) at the identical voltage $ V $, it logically follows that $ C_1 $ possesses a larger capacitance ($ C_1 > C_2 $).

The energy stored in a capacitor is calculated using $ U = \frac{1}{2} CV^2 $. 
Since $ C_1 $ is greater than $ C_2 $ and both capacitors share the same voltage $ V $, the energy stored in $ C_1 $ ($ U_1 $) exceeds that stored in $ C_2 $ ($ U_2 $), meaning $ U_1 > U_2 $.

Consequently, we conclude that $ C_1 > C_2 $ and $ U_1 > U_2 $.

Final Answer:
The final answer is $ \ C_1 > C_2,\ U_1 > U_2 $.