A parallel-plate capacitor is charged to voltage V and then disconnected from the battery. The plates are pulled apart, doubling the separation d. What happens to the stored energy?
AIt stays the same because the charge Q on the plates is conserved
BIt doubles, because U = Q²/(2C) and C = ε₀A/d, so halving C doubles U
CIt halves, because the electric field E = Q/(ε₀A) weakens as the plates move further apart
DIt stays the same because U = ½CV² and V doesn't change after disconnection
When disconnected, Q is fixed (no path for charge to flow). The right formula is U = Q²/(2C). Pulling the plates apart doubles d, which halves C = ε₀A/d. With Q fixed and C halved, U = Q²/(2C) doubles. Note: option A is wrong because conserved charge does not mean conserved energy — work is done by the electric field as the plates separate. Option D is wrong because V = Q/C does change when C changes and Q is fixed — disconnecting fixes Q, not V.
Question 2 Multiple Choice
Where is the energy of a charged capacitor physically stored?
AIn the chemical potential of the battery that originally charged it
BIn the electric field occupying the space between the plates, as energy per unit volume u = ½ε₀E²
CIn the surface charge distribution on the capacitor plates
DIn the connecting wires as kinetic energy of electrons
This is the conceptual leap at the heart of field theory. When you charge a capacitor, you do work against the repulsion of existing charges. That work is stored in the electric field itself — every cubic meter of space with field strength E contains ½ε₀E² joules of energy. This is not a metaphor or accounting convenience. The field-energy view is confirmed by the fact that electromagnetic waves (with no charges present) carry energy through empty space, described by the same field-energy density expression.
Question 3 True / False
Any region of space that contains an electric field contains real, physically stored energy proportional to the square of the field strength.
TTrue
FFalse
Answer: True
This is the meaning of u = ½ε₀E². The energy is in the field, not in the charges. This perspective generalizes beyond capacitors: the electric field between any charged objects, the field around a point charge, and even the oscillating fields of electromagnetic waves all carry energy described by this density. The field is a real physical entity, not just a mathematical tool for calculating forces.
Question 4 True / False
The three formulas U = ½CV², U = ½QV, and U = Q²/(2C) give different values for the same capacitor and the user should choose the most accurate one.
TTrue
FFalse
Answer: False
All three formulas are exactly equivalent for a given capacitor — they describe the same quantity using different pairs of variables. The choice between them is about which variables are known or held constant, not accuracy. Use ½CV² when voltage V is given (e.g., charged from a fixed battery). Use Q²/(2C) when charge Q is held fixed (e.g., after disconnecting from the battery). Use ½QV as a bridge form. They will always give the same numerical answer for the same physical situation.
Question 5 Short Answer
Why is it significant that energy is stored in the electric field rather than in the charge configuration? What does this perspective enable?
Think about your answer, then reveal below.
Model answer: If energy were only a property of charge configurations, it would be impossible to account for energy transfer through empty space — there are no charges in the space between a distant transmitter and receiver. The field-energy perspective reveals that fields are real physical entities that carry energy independently. This generalizes to magnetic fields (u = ½μ₀⁻¹B²), to electromagnetic waves (which are propagating field energy described by the Poynting vector), and ultimately to quantum field theory, where all particles are excitations of fields. Starting from the capacitor, the field-energy view is the conceptual seed that grows into all of classical and quantum electromagnetism.
The factor ½ε₀E² is directly analogous to ½kx² for a spring — both represent the work done in building up the field (or displacement) against a restoring force. This analogy shows that field energy is not exotic; it follows the same pattern as mechanical potential energy stored in elastic systems. The leap is recognizing that space itself can be the 'spring.'