偏光、磁化および物質場
導入済み| English | 意味 |
|---|---|
| electric polarisation/ɪˈlektrɪk ˌpəʊləraɪˈzeɪʃn/ | 分極強度 |
| magnetisation/ˌmæɡnɪtaɪˈzeɪʃn/ | 磁化強度 |
A decision before an answer
- Putting a dielectric into a capacitor changes the field differently depending on whether its battery remains connected.
- Your goal: Relate free and bound charge to D, E and polarisation.
Separate free and bound charge
- Electric polarisation 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.
An isolated ideal parallel-plate capacitor is fully filled with a linear dielectric of ε_r=3. Which pair gives voltage and stored-energy ratios?
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
Free Q stays fixed and C triples; both V=Q/C and U=Q²/(2C) become one third.
Apply the material relation
- 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.
Uniform electric polarisation in a finite dielectric necessarily implies:
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
ρ_b=−∇·P=0 for uniform P, but σ_b=P·n can be nonzero.
Hold charge or voltage fixed
- 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.
Take a fully filled capacitor with ε_r=4 and fixed free plate density σ. Gauss’s law for D gives D=σ, then E=σ/(4ε₀) and P=ε₀(4−1)E=3σ/4. C rises fourfold, so fixed-Q voltage and stored energy become one quarter. With the battery connected instead, V and E=V/d stay fixed while Q and U become four times larger. Separately, H=400 A/m and χ_m=0.5 give M=200 A/m and B=μ₀(600 A/m)≈0.754 mT.
In a linear magnetic material with H=500 A/m and χ_m=0.2, M=____ A/m.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
M=χ_mH=0.2·500=100.
Distinguish magnetic fields
- 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.
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.
Which answer fits this case?
Relate free and bound charge to D, E and polarisation
Normal electric field must be continuous across every dielectric interface even if the permittivities differ.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
With no free surface charge, normal D is continuous; E=D/ε can differ.
Keep the distinctions
- electric polarisation 电极化强度 — Electric dipole moment per unit volume, P, contributing bound charge.
- magnetisation 磁化强度 — Magnetic dipole moment per unit volume, M, entering B=μ₀(H+M).
- 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.
Match each term with its precise meaning in this lesson.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
難易度 3/5 · 200 コイン + 今日初回完了時に小ボーナス付与
The order follows the stated process; check each stage before the next.