The correct option for the formula calculating the energy (U) stored in a capacitor with capacitance (C) and voltage (V) across it is **B. \( U = \frac{1}{2} C \cdot V^2 \)**. ### 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) across it. The relationship is given by the formula: \[ C = \frac{Q}{V} \] - Rearranging this gives us: \[ Q = C \cdot V \] 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 (or energy stored) is the integral of the voltage with respect to 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: \[ U = \frac{1}{C} \cdot \frac{Q^2}{2} \bigg|_0^Q = \frac{Q^2}{2C} \] 3. **Substituting for Q**: - Now, substituting \( Q = C \cdot V \) into the energy formula: \[ U = \frac{(C \cdot V)^2}{2C} = \frac{C \cdot V^2}{2} \] - This leads us to the final formula: \[ U = \frac{1}{2} C \cdot V^2 \] ### Why Other Options Are Incorrect - **Option A: \( U = C \cdot V^2 \)**: - This option is incorrect because it does not account for the fact that the energy stored in a capacitor is not simply the product of capacitance and the square of the voltage. The factor of \( \frac{1}{2} \) is crucial as it reflects the average voltage during the charging process. - **Option C: \( U = C \cdot V \)**: - This option is also incorrect. It suggests that the energy stored is directly proportional to the capacitance and voltage, which is not the case. This formula does not represent energy; rather, it represents charge (Q) when rearranged. - **Option D: \( U = \frac{1}{2} V \cdot C^2 \)**: - This option is incorrect because it incorrectly places the capacitance squared in the formula. The energy stored in a capacitor depends on the capacitance and the square of the voltage, not the square of the capacitance. ### Summary of Key Points - The energy stored in a capacitor is given by the formula \( U = \frac{1}{2} C \cdot V^2 \). - This formula arises from the work done to charge the capacitor, integrating the voltage with respect to charge. - The factor of \( \frac{1}{2} \) is essential, as it accounts for the average voltage during the charging process. - Understanding the relationships between charge, voltage, and capacitance is crucial for deriving the energy formula correctly. By grasping these concepts, you will have a solid understanding of how energy is stored in capacitors and the significance of the formula.