The correct option for the formula for the energy (U) stored in a capacitor with capacitance (C) and voltage (V) across its plates is **B. U = 1/2 CV^2**. ### 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 created by the charge on the plates. - To charge a capacitor, you start from zero charge and increase the charge to \( Q \). The voltage across the capacitor increases as you add more charge. The voltage at any point while charging can be expressed as: \[ V = \frac{Q}{C} \] - The work done (or energy stored) in charging the capacitor can be calculated by integrating the voltage over the charge: \[ U = \int_0^Q V \, dQ \] - Substituting \( V = \frac{Q}{C} \) into the integral gives: \[ U = \int_0^Q \frac{Q}{C} \, dQ = \frac{1}{C} \int_0^Q Q \, dQ \] - The integral of \( Q \) from 0 to \( Q \) is: \[ \int_0^Q Q \, dQ = \frac{Q^2}{2} \] - Therefore, substituting back, we have: \[ U = \frac{1}{C} \cdot \frac{Q^2}{2} = \frac{Q^2}{2C} \] - Now, using the relationship \( Q = CV \), we can substitute \( Q \) back into the equation: \[ U = \frac{(CV)^2}{2C} = \frac{C V^2}{2} \] - This simplifies to: \[ U = \frac{1}{2} C V^2 \] 3. **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 gives the charge stored in the capacitor, not the energy. The energy is related to the square of the voltage, not just the product of capacitance and voltage. - **C. U = C/V**: This option is incorrect because it implies that energy is inversely proportional to voltage, which is not true. The energy stored in a capacitor increases with the square of the voltage, not decreases. - **D. U = V^2/C**: This option is also incorrect. While it correctly relates voltage and capacitance, it does not account for the factor of 1/2 that arises from the integration process. The energy stored is not simply the voltage squared divided by capacitance. ### Summary - The energy stored in a capacitor is given by the formula \( U = \frac{1}{2} C V^2 \). - This formula shows that energy increases with the square of the voltage and is directly proportional to capacitance. - The derivation involves understanding the relationship between charge, voltage, and capacitance, and integrating the work done to charge the capacitor. - Remember that the energy stored is not simply the product of capacitance and voltage, but rather half of that product when considering the voltage's effect on energy storage. ### Revision Summary - The correct formula for energy stored in a capacitor is \( U = \frac{1}{2} C V^2 \). - Capacitance (C) is the charge per unit voltage. - Energy increases with the square of the voltage across the capacitor. - Incorrect options fail to account for the integration of voltage over charge or misrepresent the relationship between voltage, capacitance, and energy.