The correct option for the formula for calculating the energy stored in a capacitor with capacitance \( C \) (in farads) charged to a voltage \( V \) (in volts) is **A. \( \frac{1}{2} CV^2 \)**. ### Detailed Explanation 1. **Understanding Capacitance**: - A capacitor is a device that stores electrical energy in an electric field. The capacitance \( C \) of a capacitor is defined as the amount of charge \( Q \) it can store per unit voltage \( V \). This relationship is given by the formula: \[ C = \frac{Q}{V} \] - Rearranging this gives us: \[ Q = CV \] - This means that the charge stored in the capacitor is directly proportional to both the capacitance and the voltage across it. 2. **Energy Stored in a Capacitor**: - The energy \( U \) stored in a capacitor can be derived from the work done to charge it. When charging a capacitor, the voltage across it increases as it accumulates charge. The work done to move a small amount of charge \( dq \) against the voltage \( V \) is given by: \[ dU = V \, dq \] - Since the voltage \( V \) is not constant while charging, we need to express \( V \) in terms of \( q \): \[ V = \frac{q}{C} \] - Substituting this into the work done gives: \[ dU = \frac{q}{C} \, dq \] - To find the total energy stored, we integrate from \( 0 \) to \( Q \): \[ U = \int_0^Q \frac{q}{C} \, dq \] - Performing the integration: \[ U = \frac{1}{C} \int_0^Q q \, dq = \frac{1}{C} \left[ \frac{q^2}{2} \right]_0^Q = \frac{1}{C} \cdot \frac{Q^2}{2} = \frac{Q^2}{2C} \] - Now, substituting \( Q = CV \) into the equation: \[ U = \frac{(CV)^2}{2C} = \frac{C V^2}{2} \] - Thus, the energy stored in a capacitor is: \[ U = \frac{1}{2} CV^2 \] ### Why Other Options Are Incorrect - **Option B: \( CV \)**: - This formula represents the charge \( Q \) stored in the capacitor, not the energy. While it is true that \( Q = CV \), it does not account for the work done in charging the capacitor. - **Option C: \( \frac{1}{2} C^2V \)**: - This option is incorrect because it incorrectly combines the terms. The energy stored is not dependent on \( C^2 \) but rather on \( C \) and \( V^2 \). The correct relationship involves \( V^2 \) rather than \( C^2 \). - **Option D: \( V^2/C \)**: - This option is also incorrect. It suggests a relationship that does not correspond to the energy stored in a capacitor. The energy should be proportional to \( V^2 \) but divided by \( 2 \) and multiplied by \( C \), not divided by \( C \). ### Summary for Revision - The energy stored in a capacitor is given by the formula \( U = \frac{1}{2} CV^2 \). - Capacitance \( C \) relates charge \( Q \) and voltage \( V \) through \( Q = CV \). - The work done to charge a capacitor is integrated to find the energy stored. - Common pitfalls include confusing charge with energy and misapplying the relationships between \( C \), \( V \), and \( Q \). This thorough understanding of the energy stored in a capacitor will help you in both theoretical and practical applications in physics.