EM.4 · Polarisation, magnetisation and material fields

Putting a dielectric into a capacitor changes the field differently depending on whether its battery remains connected.

Prerequisites: 13, 15.

• Relate free and bound charge to D, E and polarisation
• Compare fixed-charge and fixed-voltage dielectric changes
• Use magnetic constitutive response and free-current boundary conditions

Electric polarisation 电极化强度 P is electric dipole moment per unit volume, measured in C/m². Define D=ε₀E+P so that ∇·D=ρ_free; the total charge, including bound charge, still appears in ∇·E=ρ_total/ε₀. Bound volume charge is ρ_b=−∇·P and bound surface charge is σ_b=P·n, where n points outward from the material. Uniform P therefore gives no bound charge in the bulk but can give surface charges of opposite sign. A dielectric is not an ideal metal: polarisation need not make its internal electric field zero.

For a linear isotropic dielectric, P=ε₀χ_e E and D=εE with ε=ε₀(1+χ_e)=ε₀ε_r. These simple scalar relations assume the response is linear and ignore anisotropy, strong dispersion and nonlinear effects. In a fully filled, large parallel-plate capacitor with negligible edge effects, D normal to the plates equals the free surface-charge density. If σ_free stays fixed, increasing ε reduces E=σ_free/ε. The bound charges oppose the applied field; their magnitude is not automatically equal to the free plate charge.

For plate area A and spacing d, C=εA/d. Disconnecting the battery fixes free charge Q: V=Q/C falls and stored energy U=Q²/(2C) falls when a dielectric increases C. Leaving an ideal voltage source connected fixes V: Q=CV and stored energy U=CV²/2 both rise. The source supplies energy and moving the dielectric can involve mechanical work; compare the stated electrical energy quantity rather than assuming the capacitor alone is an isolated system. At a material interface n·(D₂−D₁)=σ_free, whereas static tangential E is continuous. Normal E generally changes when permittivity changes.

Magnetisation M is magnetic dipole moment per unit volume, measured in A/m. In SI, B=μ₀(H+M); for a linear isotropic response M=χ_m H and B=μ₀(1+χ_m)H. B and H have different units and roles. A long uniform solenoid has H≈nI when end and demagnetising effects are negligible; material response then changes B. Free surface current sets n×(H₂−H₁)=K_free, and normal B remains continuous. Ferromagnetic hysteresis and saturation cannot be represented by one constant χ_m over every field.

A capacitor is disconnected before inserting a linear dielectric with relative permittivity 4. Free charge stays fixed; capacitance 电容 becomes four times its original value.

If an ideal battery stays connected instead, V is fixed: The source exchanges energy. Do not compare isolated charge and fixed voltage as the same experiment.

Use free charge in the D equation and total charge in the E equation. Do not confuse fixed Q with fixed V, or identify B numerically with H.

electric polarisation: Electric dipole moment per unit volume, P, contributing bound charge.

magnetisation: Magnetic dipole moment per unit volume, M, entering B=μ₀(H+M).