The correct option for the energy stored in a capacitor with capacitance \( C \) and voltage \( V \) across its plates 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 \( V \) across its plates. The relationship is given by the formula: \[ C = \frac{Q}{V} \] - Rearranging this gives us: \[ Q = CV \] 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 the plates increases as more charge is added. 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: \[ V_{\text{avg}} = \frac{V}{2} \] - 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 \( Q = CV \) into this 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 option represents the charge \( Q \) stored in the capacitor, not the energy. While \( Q = CV \) is correct, it does not account for the work done in charging the capacitor. - **Option C: \( \frac{1}{2} C^2 V \)**: - This option is dimensionally incorrect. The units of energy are Joules (J), while \( \frac{1}{2} C^2 V \) does not yield the correct units. The term \( C^2 V \) would not simplify to energy units. - **Option D: \( CV^2 \)**: - This option also does not represent the energy stored in a capacitor. It incorrectly suggests that energy is directly proportional to the square of the voltage, without the necessary factor of \( \frac{1}{2} \). ### Summary of Key Points - The energy stored in a capacitor is given by the formula \( U = \frac{1}{2} CV^2 \). - This formula is derived from the work done to charge the capacitor, considering the increasing voltage as charge is added. - The other options either represent charge or are dimensionally incorrect. - Understanding the relationship between charge, voltage, and energy is crucial for solving problems related to capacitors. This thorough understanding will help you tackle questions related to capacitors effectively in your exams!