Question

In figure, charge on the capacitor is plotted against potential difference across the capacitor. The capacitance and energy stored in the capacitor are respectively.

• A. 12 μF, 1200 μJ
• B. 12 μF, 600 μJ
• C. 24 μF, 600 μJ
• D. 24 μF, 1200 μJ

Answer 1

The Correct Option is B

1. Capacitance:

The capacitance of a capacitor is defined as the ratio of the charge stored to the potential difference across it:

$C = \frac{Q}{V}$

From the graph, we can see that when $V = 10\,\text{V}$, $Q = 120\,\mu\text{C}$. Therefore:

$C = \frac{120\,\mu\text{C}}{10\,\text{V}} = 12\,\mu\text{F}$

2. Energy Stored:

The energy stored in a capacitor can be calculated using the following formula:

$U = \frac{1}{2}CV^2 = \frac{1}{2}QV$

Using the values from the graph ($Q = 120\,\mu\text{C}$ and $V = 10\,\text{V}$):

$U = \frac{1}{2}(120 \times 10^{-6}\,\text{C})(10\,\text{V}) = 600\,\mu\text{J}$

Alternatively, using the calculated capacitance:

$U = \frac{1}{2}(12 \times 10^{-6}\,\text{F})(10\,\text{V})^2 = 600\,\mu\text{J}$

The correct answer is (B) 12 μF, 600 μJ.

Answer 2

Step 1: Determine Charge and Voltage from the Graph:

From the graph, we can observe a point where the potential difference (V) is 10V and the corresponding charge (Q) is 120 $\mu$C.

Step 2: Calculate Capacitance (C):

The relationship between charge, capacitance, and voltage for a capacitor is given by the formula:

$Q = CV$

To find the capacitance (C), we can rearrange the formula:

$C = \frac{Q}{V}$

Substitute the values from the graph:

$C = \frac{120 \ \mu C}{10 \ V}$

$C = 12 \ \mu F$

Step 3: Calculate Energy Stored (E):

The energy stored in a capacitor can be calculated using the formula:

$E = \frac{1}{2} QV$

Substitute the values of Q and V from the graph and the calculated capacitance:

$E = \frac{1}{2} \times (120 \ \mu C) \times (10 \ V)$

$E = \frac{1}{2} \times (120 \times 10^{-6} \ C) \times (10 \ V)$

$E = \frac{1}{2} \times 1200 \times 10^{-6} \ J$

$E = 600 \times 10^{-6} \ J$

$E = 600 \ \mu J$

Step 4: Match with the Options:

Comparing our calculated values with the given options:

(A) 12 $\mu$F, 1200 $\mu$J (Capacitance is correct, Energy is incorrect)

(B) 12 $\mu$F, 600 $\mu$J (Capacitance is correct, Energy is correct)

(C) 24 $\mu$F, 600 $\mu$J (Capacitance is incorrect, Energy is correct)

(D) 24 $\mu$F, 1200 $\mu$J (Capacitance is incorrect, Energy is incorrect)

Final Answer: Option (B) matches calculated capacitance and energy stored.