The correct option is **A. \( U = \frac{1}{2} C V^2 \)**. ### Detailed Explanation To understand why option A is correct, we need to delve into the concept of capacitors and how energy is stored in them. 1. **Capacitance Definition**: - 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. Mathematically, this is expressed as: \[ C = \frac{Q}{V} \] - Rearranging this gives us: \[ Q = C V \] 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 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, as the charge increases, the voltage also increases. The voltage at any point while charging is given by \( V = \frac{Q}{C} \). Therefore, we can express \( dU \) in terms of \( Q \): \[ dU = \frac{Q}{C} \, dq \] 3. **Integrating to Find Total Energy**: - To find the total energy stored in the capacitor, we need to integrate from 0 to \( Q \): \[ U = \int_0^Q \frac{q}{C} \, dq \] - This integral evaluates to: \[ U = \frac{1}{C} \cdot \frac{Q^2}{2} = \frac{Q^2}{2C} \] - Now, substituting \( Q = C V \) into the equation gives: \[ U = \frac{(C V)^2}{2C} = \frac{C V^2}{2} \] - Thus, we arrive at the formula: \[ U = \frac{1}{2} C V^2 \] ### Why Other Options Are Incorrect - **Option B: \( U = C V \)**: - This option suggests that the energy stored is directly proportional to the product of capacitance and voltage. However, this 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 \( \frac{1}{2} \) due to the integration of the varying voltage. - **Option C: \( U = C V^2 \)**: - This option incorrectly implies that the energy stored is proportional to the square of the voltage multiplied by capacitance without the \( \frac{1}{2} \) factor. This would suggest that energy increases too rapidly with voltage, which is not consistent with the physics of capacitors. - **Option D: \( U = \frac{1}{2} V^2 C \)**: - While this option is mathematically equivalent to option A (since multiplication is commutative), it is presented in a different order. However, it is still correct in terms of the formula. The key point is that it is essentially the same as option A, but it is not the standard form typically presented in textbooks. ### Summary of Key Points - The energy stored in a capacitor is given by the formula \( U = \frac{1}{2} C V^2 \). - This formula arises from the integration of the work done to charge the capacitor as the voltage increases. - The other options either misrepresent the relationship between energy, capacitance, and voltage or lack the necessary factor of \( \frac{1}{2} \). - Understanding the derivation of this formula is crucial for grasping how capacitors function in electrical circuits. By mastering this concept, you will have a solid foundation for further studies in electricity and circuits.