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 = 1/2 CV²** ### 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 is derived from the work done to charge it. When charging a capacitor, the voltage across it increases from 0 to V. The work done (or energy stored) can be calculated by integrating the charge over the voltage: \[ U = \int_0^Q V \, dQ \] - 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: \[ U = \frac{1}{2} CV^2 \] ### Why Other Options Are Incorrect - **A. U = CV**: - This option suggests that the energy stored is simply the product of capacitance and voltage. However, this is incorrect because it does not account for the fact that the voltage across the capacitor increases as it charges. The energy stored is not linear with respect to voltage; it is quadratic, as shown in the correct formula. - **C. U = V²/R**: - This formula is related to the power dissipated in a resistor (Ohm's law) and is not applicable to capacitors. It does not represent the energy stored in a capacitor. Instead, it describes the energy per unit time (power) in a resistive circuit, which is not relevant to the energy stored in a capacitor. - **D. U = C/V**: - This option incorrectly suggests that the energy stored is inversely proportional to voltage. This is not correct; energy stored in a capacitor increases with the square of the voltage, not inversely. This formula does not represent any physical quantity related to energy storage in capacitors. ### Summary of Key Points - The energy stored in a capacitor is given by the formula \( U = \frac{1}{2} CV^2 \). - Capacitance (C) is the ability of a capacitor to store charge per unit voltage. - The energy stored increases with the square of the voltage applied across the capacitor. - Understanding the relationship between charge, voltage, and energy is crucial for solving problems related to capacitors. This thorough understanding of the energy stored in capacitors will help you tackle related problems in your physics studies effectively.