Force & Stored Energy
The plates pull together; the energy lives in the field.
Lesson at a glanceShow the key points
- F = Q²/2Aε₀ (31.7), from one plate's field σ/2ε₀ acting on the other.
- U = Q²/2C = ½CV² = ½QV (31.8–31.10).
- Energy density of any electric field: u = ½ε₀E².
The plates of a charged capacitor pull on each other. Work done against that pull is stored as energy, and the energy turns out to live in the electric field itself.
Force between the plates
Give a parallel-plate capacitor +Q and −Q. The positive plate on its own makes a field E₊ = σ/2ε₀ = Q/2Aε₀ (a large charged sheet). The negative plate sits in that field and feels F = QE₊:
Plates of area 0.020 m² carry ±2.0 µC. Find the force between them.
Energy stored in a capacitor
Start with the plates touching, give them ±Q, then slowly pull one plate away to a distance d. The force is constant (it doesn’t depend on the gap), so you do work W = Fd = Q²d/2Aε₀ = Q²/2C, where C is the final capacitance. That work is stored as electrostatic potential energy:
A flash unit charges a 220 µF capacitor to 300 V. How much energy is stored?
The energy is in the field
Pull the plates from d₁ to d₂ with Q fixed. You do extra work Q²(d₂ − d₁)/2Aε₀. Meanwhile a new region of field appears, of volume A(d₂ − d₁). Dividing, the energy per unit volume of the field is
Air breaks down at about 3.0 × 10⁶ V/m. What is the energy density of a field that strong?
- 1Plates attract with F = Q²/2Aε₀ (31.7), using one plate's field σ/2ε₀.
- 2Stored energy U = Q²/2C = ½CV² = ½QV (31.8–31.10).
- 3The energy is stored in the field, with density u = ½ε₀E².
Check yourself
Keep exploring
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