The correct option for the formula for the energy stored in a capacitor with capacitance \( C \) (in farads) charged to a voltage \( V \) (in volts) is **A. \( \frac{1}{2} C V^2 \)**. ### 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 \( V \). The relationship is given by the formula: \[ C = \frac{Q}{V} \] - Rearranging this gives us: \[ Q = C V \] 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 charge \( dq \) from one plate to the other against the electric field is given by: \[ dU = V \, dq \] - However, the voltage \( V \) is not constant; it increases as the charge increases. The average voltage during the charging process can be expressed as: \[ V_{\text{avg}} = \frac{0 + V}{2} = \frac{V}{2} \] - Therefore, the total energy stored in the capacitor can be calculated by integrating the work done from \( 0 \) to \( Q \): \[ U = \int_0^Q V \, dq = \int_0^Q \frac{q}{C} \, dq \] - Substituting \( V = \frac{q}{C} \) into the integral gives: \[ U = \int_0^Q \frac{q}{C} \, dq = \frac{1}{C} \int_0^Q q \, dq \] - The integral \( \int_0^Q q \, dq \) evaluates to \( \frac{Q^2}{2} \), so: \[ U = \frac{1}{C} \cdot \frac{Q^2}{2} = \frac{Q^2}{2C} \] - Now, substituting \( Q = C V \) into the equation gives: \[ U = \frac{(C V)^2}{2C} = \frac{C V^2}{2} \] - Thus, the energy stored in a capacitor is: \[ U = \frac{1}{2} C V^2 \] ### Why Other Options Are Incorrect - **Option B: \( C V \)**: - This option represents the charge \( Q \) stored in the capacitor, not the energy. While it is true that \( Q = C V \), it does not account for the energy stored, which involves the square of the voltage. - **Option C: \( \frac{1}{2} V^2 C \)**: - This option is mathematically equivalent to option A, but it is presented in a less conventional format. While it is correct, it is not the standard way to express the energy stored in a capacitor. The standard form is \( \frac{1}{2} C V^2 \). - **Option D: \( C V^2 \)**: - This option incorrectly suggests that the energy stored is directly proportional to the square of the voltage without the factor of \( \frac{1}{2} \). This would imply that doubling the voltage would quadruple the energy, which is not accurate for capacitors. ### Summary for Revision - The energy stored in a capacitor is given by the formula \( U = \frac{1}{2} C V^2 \). - Capacitance \( C \) relates charge \( Q \) and voltage \( V \) through \( Q = C V \). - The energy calculation involves integrating the work done to charge the capacitor, considering the changing voltage. - Remember that \( C V \) gives charge, not energy, and the correct energy formula includes the factor of \( \frac{1}{2} \).