The correct option for the formula for the energy (U) stored in a capacitor with capacitance (C) and voltage (V) across its plates is **B. U = 0.5 CV^2**. ### Detailed Explanation 1. **Understanding Capacitance**: - Capacitance (C) is defined as the ability of a capacitor to store charge per unit voltage. It is measured in farads (F). The relationship can be expressed as: \[ C = \frac{Q}{V} \] where \( Q \) is the charge stored in the capacitor and \( V \) is the voltage across the capacitor. 2. **Energy Stored in a Capacitor**: - The energy (U) stored in a capacitor is derived from the work done to charge it. When a capacitor is charged, work is done against the electric field created by the charge on the plates. - The energy can be calculated by integrating the work done to move charge from one plate to the other as the voltage increases. 3. **Derivation of the Energy Formula**: - The work done (W) to move a small charge \( dq \) to a potential \( V \) is given by: \[ dW = V \, dq \] - Since the voltage across the capacitor changes as it charges, we can express \( V \) in terms of charge \( Q \) and capacitance \( C \): \[ V = \frac{Q}{C} \] - Therefore, the work done to charge the capacitor from 0 to a charge \( Q \) is: \[ W = \int_0^Q V \, dq = \int_0^Q \frac{q}{C} \, dq \] - Evaluating this integral: \[ W = \frac{1}{C} \int_0^Q q \, dq = \frac{1}{C} \left[ \frac{q^2}{2} \right]_0^Q = \frac{Q^2}{2C} \] - Now, using the relationship \( Q = CV \), we can substitute \( Q \) into the equation: \[ W = \frac{(CV)^2}{2C} = \frac{C V^2}{2} \] - Thus, the energy stored in the capacitor is: \[ U = \frac{1}{2} C V^2 \] ### Why Other Options Are Incorrect - **Option A: U = CV** - This option suggests that the energy stored is directly proportional to the capacitance and voltage. However, this does not account for the fact that energy is related to the square of the voltage, not just the product of capacitance and voltage. This formula does not represent the energy stored in a capacitor. - **Option C: U = C/V** - This option implies that the energy stored decreases with increasing voltage, which is incorrect. The energy stored in a capacitor increases with the square of the voltage, not inversely proportional to it. This formula does not reflect the relationship between energy, capacitance, and voltage. - **Option D: U = V^2 / C** - This option suggests that energy is proportional to the square of the voltage divided by capacitance. While it has the correct form of voltage squared, it incorrectly places capacitance in the denominator. The correct relationship shows that energy is proportional to the capacitance multiplied by the square of the voltage, not divided by it. ### Summary of Key Points - The energy stored in a capacitor is given by the formula \( U = \frac{1}{2} C V^2 \). - Capacitance (C) measures the ability to store charge per unit voltage. - The work done to charge a capacitor is derived from integrating the voltage as charge is added. - The energy stored increases with the square of the voltage, not linearly or inversely. This understanding is crucial for solving problems related to capacitors in circuits, energy storage, and electrical engineering applications.