The correct option for the formula for the energy (U) stored in a capacitor with capacitance (C) charged to a voltage (V) is: **B. U = 0.5CV²** ### 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 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**: - The energy stored in a capacitor can be calculated by integrating the work done to move charge \( Q \) from 0 to \( Q \) at a voltage \( V \): \[ U = \int_0^Q V \, dQ \] - 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, using the relationship \( Q = CV \), we can substitute \( Q \) back into the equation: \[ U = \frac{(CV)^2}{2C} = \frac{C V^2}{2} \] - Thus, we arrive at the final formula: \[ U = 0.5CV^2 \] ### Why Other Options Are Incorrect - **A. U = CV**: - This option suggests that the energy stored is equal to the product of capacitance and voltage. However, this is incorrect because it does not account for the fact that energy is related to the square of the voltage. The energy stored in a capacitor increases with the square of the voltage, not linearly. - **C. U = V/C**: - This option implies that the energy stored is inversely proportional to capacitance, which is not correct. This formula does not represent energy at all; rather, it rearranges the capacitance formula incorrectly. - **D. U = C/V**: - Similar to option C, this option suggests an incorrect relationship. It implies that energy decreases with increasing voltage, which contradicts the fundamental principles of how capacitors store energy. ### Common Pitfalls - **Confusing Charge and Energy**: Students often confuse the formulas for charge and energy. Remember that charge \( Q \) is related to capacitance and voltage, while energy \( U \) involves the square of the voltage. - **Forgetting the Factor of 0.5**: The factor of 0.5 is crucial in the energy formula. It arises from the integration process and is often overlooked. - **Units**: Ensure that you are consistent with units when calculating energy. Capacitance is in farads, voltage in volts, and energy will be in joules. ### Revision Summary - The energy stored in a capacitor is given by the formula \( U = 0.5CV^2 \). - Capacitance relates charge and voltage: \( C = \frac{Q}{V} \). - Energy is derived from the work done to charge the capacitor, leading to the quadratic relationship with voltage. - Be cautious of common mistakes, such as confusing charge with energy and neglecting the factor of 0.5 in the energy formula.