1. Concept Overview
A capacitor stores energy in the electric field created between its plates. When a voltage is applied, positive and negative charges collect on opposite plates, building up an electric field. Work must be done to move charges onto the plates, and this work is stored as electrical energy inside the capacitor.
2. How a Capacitor Stores Energy
As charge accumulates on the plates, the electric field grows stronger. To add more charge, additional work must be done against this increasing electric field. This work becomes the stored energy of the capacitor.
3. Derivation of Energy Formula
The small amount of work needed to move a small charge \(dq\) to a capacitor at potential \(V\) is:
\( dW = V \, dq \).
Since \( V = \dfrac{q}{C} \),
\( dW = \dfrac{q}{C} dq \).
3.1. Integrating the Work
Total work (energy) required to charge the capacitor from 0 to \(Q\):
\( W = \int_0^Q \dfrac{q}{C} dq = \dfrac{1}{2} \dfrac{Q^2}{C} \).
4. Standard Energy Formulas
The energy stored in a charged capacitor can be written in three equivalent ways:
- \( U = \dfrac{1}{2} C V^2 \)
- \( U = \dfrac{1}{2} Q V \)
- \( U = \dfrac{Q^2}{2C} \)
4.1. Why Three Forms?
Each form is useful depending on which quantities (Q, V, C) are known. All three represent the same stored energy.
5. Energy Density in a Capacitor
The energy stored is actually in the electric field between the plates. The energy per unit volume (energy density) is:
\( u = \dfrac{1}{2} \varepsilon_0 E^2 \).
This formula applies to any electric field, not just capacitors.
6. Effect of Dielectric on Stored Energy
Placing a dielectric between the plates changes the stored energy depending on whether charge or voltage is kept constant.
6.1. Case 1: Capacitor Connected to a Battery (Voltage Constant)
Capacitance increases, but voltage stays the same. Since \( U = \dfrac{1}{2} C V^2 \), energy increases. The battery supplies extra charge to maintain the voltage.
6.2. Case 2: Capacitor Isolated (Charge Constant)
Capacitance increases, but charge is fixed. Using \( U = \dfrac{Q^2}{2C} \), energy decreases. The electric field weakens inside the dielectric.
7. Recharge and Discharge
During charging, the capacitor absorbs energy. During discharge, the stored energy is released to the circuit. This property makes capacitors useful for energy storage, filtering, signal smoothing and power management.
8. Worked Examples
8.1. Example 1: Using U = ½CV²
A capacitor with \( C = 10 \, \mu F \) is charged to \( V = 100 \, V \).
\( U = \dfrac{1}{2} C V^2 = \dfrac{1}{2} \times 10^{-5} \times 100^2 \)
\( U = 0.05 \, J \).
8.2. Example 2: Using U = Q² / (2C)
A capacitor stores charge \( Q = 40 \, \mu C \) and has capacitance \( C = 5 \, \mu F \).
\( U = \dfrac{Q^2}{2C} = \dfrac{(40 \times 10^{-6})^2}{2 \times 5 \times 10^{-6}} \)
\( U = 0.16 \, J \).
9. Physical Interpretation
You can think of a capacitor like a spring for electric charge. Just as compressing a spring stores mechanical energy, separating charges stores electrical energy. When the capacitor is discharged, the electric field collapses and the stored energy flows out into the circuit.