Question

If the charge on a capacitor of capacitance 15 $\mu$F is 300 $\mu$C, then the energy stored in the capacitor is:

• A. 3 mJ
• B. 9 mJ
• C. 6 mJ
• D. 12 mJ

Hint:

To find the energy stored in a capacitor, use either $U = \frac{Q^2}{2C}$ or $U = \frac{1}{2} C V^2$, where $V = \frac{Q}{C}$. Always ensure units are consistent (e.g., convert $\mu\text{C}$ and $\mu\text{F}$ to SI units).

Answer 1

The Correct Option is A

Step 1: Identify the Formula for Energy Stored in a Capacitor
The energy $U$ stored in a capacitor with capacitance $C$ and charge $Q$ is given by the formula: \[ U = \frac{Q^2}{2C} \] Given: $Q = 300 \, \mu\text{C}$ and $C = 15 \, \mu\text{F}$. Step 2: Convert Units to SI for Consistency
Convert the given values to SI units for calculation:
Charge: \[ Q = 300 \, \mu\text{C} = 300 \times 10^{-6} \, \text{C} = 3 \times 10^{-4} \, \text{C} \] Capacitance: \[ C = 15 \, \mu\text{F} = 15 \times 10^{-6} \, \text{F} = 1.5 \times 10^{-5} \, \text{F} \] Step 3: Compute the Energy
Substitute the values into the energy formula: \[ U = \frac{(3 \times 10^{-4})^2}{2 \times (1.5 \times 10^{-5})} \] Calculate the numerator: \[ (3 \times 10^{-4})^2 = 9 \times 10^{-8} \] Calculate the denominator: \[ 2 \times (1.5 \times 10^{-5}) = 3 \times 10^{-5} \] Thus: \[ U = \frac{9 \times 10^{-8}}{3 \times 10^{-5}} = 3 \times 10^{-3} \, \text{J} \] Convert to millijoules: \[ 3 \times 10^{-3} \, \text{J} = 3 \, \text{mJ} \] Step 4: Verify Using an Alternative Formula
The voltage across the capacitor is: \[ V = \frac{Q}{C} = \frac{3 \times 10^{-4}}{1.5 \times 10^{-5}} = 20 \, \text{V} \] Use the alternative energy formula $U = \frac{1}{2} C V^2$: \[ U = \frac{1}{2} \times (1.5 \times 10^{-5}) \times (20)^2 = \frac{1}{2} \times (1.5 \times 10^{-5}) \times 400 = 3 \times 10^{-3} \, \text{J} = 3 \, \text{mJ} \] This confirms the result. Step 5: Analyze Options
Option (1): 3 mJ. Correct, as it matches our calculated energy.
Option (2): 9 mJ. Incorrect, as the energy is 3 mJ.
Option (3): 6 mJ. Incorrect, as the energy is 3 mJ.
Option (4): 12 mJ. Incorrect, as the energy is 3 mJ.