The correct option for the formula for the energy (U) stored in a capacitor with capacitance (C) and voltage (V) across it is: **B. U = 0.5 CV²** ### 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 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 a capacitor is charged, work is done against the electric field to move charge from one plate to another. 3. **Derivation of the Energy Formula**: - To derive the energy stored, consider that when charging a capacitor, the voltage across it increases as more charge is added. The work done (or energy stored) can be calculated by integrating the voltage over the charge: \[ U = \int_0^Q V \, dQ \] - Since \( V = \frac{Q}{C} \), we can substitute this into the integral: \[ U = \int_0^Q \frac{Q'}{C} \, dQ' \] - This integral evaluates to: \[ U = \frac{1}{C} \int_0^Q Q' \, dQ' = \frac{1}{C} \left[ \frac{(Q')^2}{2} \right]_0^Q = \frac{Q^2}{2C} \] - Now, substituting \( Q = CV \) into the equation gives: \[ U = \frac{(CV)^2}{2C} = \frac{C V^2}{2} \] - Thus, the formula for the energy stored in a capacitor is: \[ U = 0.5 CV^2 \] ### Why Other Options Are Incorrect - **A. U = CV**: - This option suggests that the energy stored is directly proportional to the capacitance and voltage. However, this formula only represents the charge stored in the capacitor, not the energy. The energy is related to the square of the voltage, not just the voltage itself. - **C. U = 2CV**: - This option incorrectly implies that the energy stored is double the product of capacitance and voltage. The factor of 2 is not justified in the context of energy storage in capacitors. The correct relationship involves a factor of 0.5, as derived above. - **D. U = C/V**: - This option suggests that the energy stored is inversely proportional to the voltage, which is incorrect. This formula does not represent any physical quantity related to energy stored in a capacitor. Instead, it incorrectly combines capacitance and voltage in a way that does not reflect the principles of energy storage. ### Summary of Key Points - The energy stored in a capacitor is given by the formula \( U = 0.5 CV^2 \). - This formula is derived from the work done to charge the capacitor, taking into account the increasing voltage as charge is added. - The other options (A, C, D) do not accurately represent the relationship between capacitance, voltage, and energy stored in a capacitor. - Remember that energy is proportional to the square of the voltage, not just the voltage itself. This understanding is crucial for solving problems related to capacitors in physics, especially in circuits and energy storage applications.