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 = 0.5CV²**. ### 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 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 can be evaluated as follows: \[ U = \frac{1}{C} \int_0^Q Q \, dQ = \frac{1}{C} \left[ \frac{Q^2}{2} \right]_0^Q = \frac{1}{C} \cdot \frac{Q^2}{2} = \frac{Q^2}{2C} \] 3. **Substituting for Q**: - Now, we can substitute \( Q = CV \) back into the equation: \[ U = \frac{(CV)^2}{2C} = \frac{C^2V^2}{2C} = \frac{1}{2}CV^2 \] - Thus, we arrive at the formula: \[ U = 0.5CV^2 \] ### Why Other Options Are Incorrect - **Option A: U = CV²** - This option suggests that the energy stored is directly proportional to the capacitance and the square of the voltage. However, this does not account for the fact that the energy is accumulated as the capacitor charges, which involves integrating the voltage over the charge. Therefore, this option is incorrect. - **Option C: U = C/V** - This option implies that the energy stored is inversely proportional to the voltage, which is not consistent with the physics of capacitors. The energy stored increases with the square of the voltage, not decreases. Thus, this option is incorrect. - **Option D: U = 0.5V²/C** - While this option has the correct form of a fraction, it incorrectly places voltage in the numerator and capacitance in the denominator. The energy stored in a capacitor is not defined this way. The correct relationship shows that energy is proportional to capacitance and the square of voltage, not inversely related to capacitance. Therefore, this option is also incorrect. ### Summary of Key Points - The energy stored in a capacitor is given by the formula \( U = 0.5CV^2 \). - This formula is derived from the work done to charge the capacitor, integrating the voltage with respect to charge. - The energy stored increases with the square of the voltage and is directly proportional to the capacitance. - Understanding the relationship between charge, voltage, and capacitance is crucial for deriving the energy formula correctly. This thorough understanding will help you tackle questions related to capacitors and energy storage in your physics exams.