The correct option is **B. U = ½ 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 the plates increases as more charge is added. The work done to move a small charge \(dq\) against the voltage \(V\) is given by: \[ dU = V \, dq \] - However, the voltage is not constant; it increases as the charge increases. The average voltage during the charging process can be expressed as: \[ V_{\text{avg}} = \frac{V}{2} \] - This is because the voltage starts at 0 (when the capacitor is uncharged) and goes up to \(V\) (when fully charged). 3. **Integrating to Find Total Energy**: - To find the total energy stored in the capacitor, we integrate the work done from 0 to the final charge \(Q\): \[ U = \int_0^Q V \, dq \] - Substituting \(V\) with \(\frac{Q}{C}\) (from the capacitance formula), we have: \[ U = \int_0^Q \frac{q}{C} \, dq \] - This integral evaluates to: \[ U = \frac{1}{C} \cdot \frac{Q^2}{2} = \frac{1}{2C} \cdot (CV)^2 = \frac{1}{2} CV^2 \] - Thus, the formula for the energy stored in a capacitor is: \[ U = \frac{1}{2} CV^2 \] ### Why Other Options Are Incorrect: - **Option A: U = CV** - This option suggests that the energy stored is directly proportional to the capacitance and voltage. However, this formula does not account for the fact that the voltage increases as the capacitor charges. The energy stored is not simply the product of capacitance and voltage; it must include the factor of \(1/2\) to reflect the average voltage during charging. - **Option C: U = C/V** - This option implies that the energy stored decreases with increasing voltage, which is incorrect. The energy stored in a capacitor is not inversely related to voltage. Instead, it is directly related to the square of the voltage, as shown in the correct formula. - **Option D: U = V/C** - This option suggests that energy is proportional to voltage divided by capacitance, which does not represent the relationship between energy, capacitance, and voltage. This formula does not have the correct dimensions for energy and is not derived from the principles of capacitor charging. ### Summary of Key Points: - The energy stored in a capacitor is given by the formula \(U = \frac{1}{2} CV^2\). - This formula accounts for the increasing voltage as the capacitor charges, reflecting the average voltage during the charging process. - The other options (A, C, D) do not correctly represent the relationship between energy, capacitance, and voltage. - Understanding the derivation of this formula is crucial for solving problems related to capacitors in physics.