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electromagnetism

Electrostatics & Gauss's law, potential, conductors & capacitance, magnetostatics, relativity of fields, charged-particle motion, induction, Maxwell's equations, EM waves, fields in matter.

electrostatics · potential · conductors · magnetostatics · relativity of fields · motion of charges · induction · Maxwell & waves · dielectrics · magnetic matter

Electrostatics & Gauss's law

Charge. Two signs; locally conserved (charge in a region changes only by current through its boundary); quantized in units of \(e=1.602\times10^{-19}\,\mathrm{C}\); a frame-independent scalar (see relativity of fields).

Coulomb & superposition. \(\vv F=\dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1q_2}{r^2}\,\uvec r\), \(\uvec r\) from the other charge toward the one acted on — like signs repel. \(\varepsilon_0=8.854\times10^{-12}\,\mathrm{F\,m^{-1}}\), \(k_e=(4\pi\varepsilon_0)^{-1}=8.99\times10^9\,\mathrm{N\,m^2\,C^{-2}}\). Forces and fields from many charges add vectorially.

Field. \(\vv E=\vv F/q\) on a test charge; point charge \(\vv E=Q\,\uvec r/4\pi\varepsilon_0 r^2\). Continuous distributions (line \(\lambda\), surface \(\sigma\), volume \(\rho\)): integrate \(\dd\vv E\) over the distribution. Field lines start on \(+\), end on \(-\); line density \(\propto E\).

Energy of a system. Work to assemble from infinity \(U=\tfrac12\sum_{j\ne k}\dfrac{q_jq_k}{4\pi\varepsilon_0 r_{jk}}\) (the \(\tfrac12\) undoes double counting). Ionic lattice (NaCl type): the alternating sum gives \(U\approx-1.747\,e^2/4\pi\varepsilon_0 a\) per ion pair, \(a\) the nearest-neighbour spacing.

Flux & Gauss's law. \(\Phi=\int_S\vv E\cdot\dd\vv a\); over any closed surface \(\oint\vv E\cdot\dd\vv a=Q_{\mathrm{enc}}/\varepsilon_0\) — equivalent to Coulomb + superposition. Computes \(E\) only when symmetry (spherical, cylindrical, planar) fixes it on the Gaussian surface.

Standard fields.

Sheets of charge. Crossing a surface charge \(\sigma\): \(\Delta E_\perp=\sigma/\varepsilon_0\), \(E_\parallel\) continuous. Force per area on a charge layer \(=\sigma\times\)(average of the fields on the two sides); on a conductor surface this is an outward electrostatic pressure \(\sigma^2/2\varepsilon_0=\varepsilon_0E^2/2\).

Field energy. Energy density \(u=\tfrac12\varepsilon_0E^2\); \(U=\tfrac{\varepsilon_0}{2}\int E^2\,\dd v\) over all space equals the assembly work. Energies of superposed fields don't add: \(U\) contains the interaction term \(\varepsilon_0\int\vv E_1\cdot\vv E_2\,\dd v\). Point-charge self-energy diverges — for point charges use the pair sum.

Potential

Line integral & definition. For electrostatic fields \(\int\vv E\cdot\dd\vv s\) is path-independent, so \(\phi_2-\phi_1=-\int_1^2\vv E\cdot\dd\vv s\) defines the potential (volts). Potential energy \(U=q\phi\); work by the field \(W=q(\phi_i-\phi_f)\).

Canonical potentials (reference at \(\infty\)): point charge \(\phi=Q/4\pi\varepsilon_0r\); discrete/continuous \(\phi=\sum q_i/4\pi\varepsilon_0r_i=\int\rho\,\dd v/4\pi\varepsilon_0 r\); spherical shell: outside as point charge, inside constant \(Q/4\pi\varepsilon_0R\); uniformly charged disk, on axis: \(\phi=(\sigma/2\varepsilon_0)\bigl(\sqrt{R^2+z^2}-|z|\bigr)\). Charge extending to infinity needs a finite reference: infinite line \(\phi=-(\lambda/2\pi\varepsilon_0)\ln(r/r_0)\). Assembly energy \(U=\tfrac12\int\rho\phi\,\dd v\).

Gradient. \(\vv E=-\nabla\phi\); \(\nabla\phi\) points up the steepest ascent, so field lines run downhill in \(\phi\) and cross equipotential surfaces at right angles. Uniform field between plates: \(E=V/d\).

Dipole. \(\vv p=q\boldsymbol\ell\) (from \(-\) to \(+\)); generally \(\vv p=\int\vv r\,\rho\,\dd v\), origin-independent for neutral distributions. Far field: \(\phi=\dfrac{p\cos\theta}{4\pi\varepsilon_0r^2}\), \(\vv E=\dfrac{p}{4\pi\varepsilon_0r^3}\bigl(2\cos\theta\,\uvec r+\sin\theta\,\hat{\boldsymbol\theta}\bigr)\). Multipole expansion of any bounded distribution: \(\phi=\dfrac{1}{4\pi\varepsilon_0}\Bigl(\dfrac{Q}{r}+\dfrac{\vv p\cdot\uvec r}{r^2}+\dfrac{\text{quadrupole}}{r^3}+\cdots\Bigr)\) — the leading nonvanishing moment dominates far away.

Divergence. \(\nabla\cdot\vv F=\partial_xF_x+\partial_yF_y+\partial_zF_z\) = outward flux per unit volume. Divergence theorem: \(\oint_S\vv F\cdot\dd\vv a=\int_V\nabla\cdot\vv F\,\dd v\). With Gauss's law: \(\nabla\cdot\vv E=\rho/\varepsilon_0\).

Curl. \((\nabla\times\vv F)_z=\partial_xF_y-\partial_yF_x\) etc. = circulation per unit area. Stokes: \(\oint_C\vv F\cdot\dd\vv s=\int_S(\nabla\times\vv F)\cdot\dd\vv a\). Electrostatic fields: \(\nabla\times\vv E=0\) ⇔ path independence ⇔ \(\vv E=-\nabla\phi\).

Poisson & Laplace. \(\nabla^2\phi=-\rho/\varepsilon_0\); in empty space \(\nabla^2\phi=0\). Harmonic functions: the value at a point equals the average over any sphere around it; no interior maxima or minima. Earnshaw: no charge can rest in stable equilibrium in an electrostatic field.

Conductors & capacitance

Electrostatic equilibrium. Inside the material \(\vv E=0\); all net charge sits on surfaces; the conductor is one equipotential; just outside, \(\vv E=(\sigma/\varepsilon_0)\,\uvec n\), normal to the surface. \(\sigma\) concentrates at sharp features — points break down first.

Cavities & screening. An empty cavity in a conductor is field-free whatever happens outside (Faraday cage). A charge \(q\) inside a cavity induces \(-q\) on the cavity wall; the compensating \(+q\) spreads over the outer surface exactly as if the interior were empty.

Uniqueness. Given \(\rho\) in a region and the potential (or total charge) on each boundary conductor, \(\phi\) is unique — a solution found by any means, including guessing, is the solution.

Image charges. Point charge \(q\) at height \(h\) above a grounded infinite plane: the field above the plane is that of \(q\) plus an image \(-q\) at depth \(h\). Induced density \(\sigma(s)=-qh/2\pi(h^2+s^2)^{3/2}\) (\(s\) along the plane), integrating to \(-q\); attraction toward the plane \(F=q^2/4\pi\varepsilon_0(2h)^2\).

Capacitance. \(C=Q/\phi\) (isolated conductor) or \(Q/V\) (pair); farads. Isolated sphere \(C=4\pi\varepsilon_0a\); parallel plates \(C=\varepsilon_0A/s\), \(\times\kappa\) when filled with dielectric (dielectrics). Combination rules and RC transients: circuits. Several conductors: \(Q_i=\sum_jC_{ij}\phi_j\) with symmetric coefficients \(C_{ij}=C_{ji}\).

Capacitor energy. \(U=\tfrac12Q\phi=\tfrac12C\phi^2=Q^2/2C\) — for parallel plates exactly the field energy \(\tfrac12\varepsilon_0E^2\times\)volume.

Magnetostatics

Lorentz force. \(\vv F=q(\vv E+\vv v\times\vv B)\); \(B\) in tesla (\(1\,\mathrm{T}=10^4\,\mathrm{gauss}\)). The magnetic term does no work.

Force on currents. \(\dd\vv F=I\,\dd\vv l\times\vv B\); straight wire \(F=BIl\sin\theta\). Parallel wires: \(F/L=\mu_0I_1I_2/2\pi d\), same direction attract. Torque on a planar coil \(\boldsymbol\tau=\vv m\times\vv B\), \(\tau=mB\sin\theta\), magnetic moment \(\vv m=NI\vv A\).

Ampère & no monopoles. \(\oint\vv B\cdot\dd\vv s=\mu_0I_{\mathrm{enc}}\), i.e. \(\nabla\times\vv B=\mu_0\vv J\) (steady currents); \(\nabla\cdot\vv B=0\) always — no magnetic charge, field lines close on themselves. \(\mu_0=4\pi\times10^{-7}\,\mathrm{T\,m\,A^{-1}}\); \(\mu_0\varepsilon_0=1/c^2\).

Biot–Savart. \(\dd\vv B=\dfrac{\mu_0I}{4\pi}\dfrac{\dd\vv l\times\uvec r}{r^2}\) for steady currents; superpose over the circuit.

Vector potential. \(\vv B=\nabla\times\vv A\) (makes \(\nabla\cdot\vv B=0\) automatic); \(\vv A=\dfrac{\mu_0}{4\pi}\int\dfrac{\vv J\,\dd v}{r}\), thin wire \(\dfrac{\mu_0I}{4\pi}\oint\dfrac{\dd\vv l}{r}\); \(\vv A\) is fixed only up to addition of a gradient.

Standard fields.

Discontinuity. Across a current sheet, \(B_\perp\) is continuous and \(B_\parallel\) jumps by \(\mu_0\mathcal J\) (direction \(\perp\) the current).

Relativity & field transformations

Charge invariance. Total charge (measured by Gauss flux) is the same in every frame — unlike energy or mass. A moving charge distribution keeps its total charge.

Boosted point charge. For sources at rest in the primed frame: \(E_\parallel\) unchanged, \(E_\perp\) larger by \(\gamma\) in the frame where they move. Charge with constant velocity \(\beta c\): field is radial from the present position with \(E=\dfrac{Q}{4\pi\varepsilon_0r^2}\dfrac{1-\beta^2}{(1-\beta^2\sin^2\theta)^{3/2}}\) — weakened fore-and-aft, strengthened transverse (pancaked). If the charge starts or stops, the news propagates outward at \(c\); the shell stitching old to new field carries a transverse pulse — radiation.

Force transformation. Between a particle's rest frame and the lab: \(F_\parallel\) equal; \(F_\perp^{\mathrm{lab}}=F_\perp^{\mathrm{rest}}/\gamma\).

Origin of magnetism. A current-carrying wire is neutral in the lab; in the frame of a charge drifting alongside, positive lattice and negative carriers length-contract unequally, the wire acquires net charge, and the resulting radial electric force is exactly what the lab calls \(q\vv v\times\vv B\). Magnetism = electrostatics + relativity; any moving charged object makes a magnetic field (Rowland). Kinematics: relativity.

Transformations. Frame \(F'\) moving at \(\vv v\) relative to \(F\):

Special cases. If the sources are all static in a frame moving at \(\vv u\) relative to you (pure \(E\) there): \(\vv B=(\vv u\times\vv E)/c^2\). If \(E\) vanishes in a frame moving at \(\vv u\) relative to you (neutral wire): \(\vv E=\vv B\times\vv u\).

Invariants. \(\vv E\cdot\vv B\) and \(E^2-c^2B^2\) are the same in every frame — a field that is pure \(E\) somewhere can never look like pure \(B\) elsewhere, and a light-like field (\(E\perp B\), \(E=cB\)) is light-like in all frames.

Motion of charges

In electric fields. \(a=qE/m\); between plates projectile-like. Energy from potential: \(q\Delta V=\Delta K\); from rest nonrelativistic \(v=\sqrt{2qV/m}\). Electron accelerated through \(V\): \(K=eV\), \(\lambda=h/\sqrt{2meV}\) nonrel.

Circular motion. If \(\vv v\perp\vv B\): \(qvB=mv^2/r\), \(r=mv/(qB)=p/(qB)\). Angular frequency \(\omega=qB/m\), period \(T=2\pi m/(qB)\), cyclotron \(f=qB/(2\pi m)\). If component \(v_\parallel\) present: helix pitch \(p_{\rm helix}=v_\parallel T\).

Velocity selector. \(qE=qvB\Rightarrow v=E/B\).

Mass spectrometer. After accelerating voltage: \(r=\dfrac1B\sqrt{2mV/q}\).

Hall effect. Carriers pile up on one side until the transverse field balances: \(qv_dB=qE_H\), \(\vv E_H=-(\vv J\times\vv B)/nq\); \(V_H=E_Hw\); with thickness \(t\), \(V_H=BI/(nqt)\). Sign of \(V_H\) reveals the sign of the actual carriers — the one place a current of \(-\) charges one way differs from \(+\) charges the other way.

Induction & inductance

Flux/Faraday/Lenz. Magnetic flux \(\Phi=\int\vv B\cdot\dd\vv a\) (\(=BA\cos\theta\) uniform); flux linkage \(N\Phi\). Induced emf \(\mathcal E=-N\dfrac{\dd\Phi}{\dd t}\) — universal, whether the loop moves, the source moves, or \(B\) simply changes; sign opposes the change causing it (Lenz). Differential form: \(\nabla\times\vv E=-\partial\vv B/\partial t\) — induced \(E\) is non-conservative, \(\oint\vv E\cdot\dd\vv s=-\dd\Phi/\dd t\).

Motional emf. \(\mathcal E=Blv\) for rod length \(l\) moving perpendicular to \(B\) — the \(q\vv v\times\vv B\) force on the rod's charges. Rod on rails: current \(I=Blv/R\); magnetic drag \(F=BIl=B^2l^2v/R\); mechanical power \(Fv=I^2R\).

Rotating coil (AC generator). \(\Phi=NBA\cos\omega t\), \(\mathcal E=NBA\omega\sin\omega t\), peak \(\mathcal E_0=NBA\omega\). AC analysis (rms, impedance, power): circuits.

Mutual inductance. \(\Phi_{21}=M I_1\), \(\mathcal E_2=-M\,\dd I_1/\dd t\); reciprocity \(M_{12}=M_{21}\); henry \(=\mathrm{Wb\,A^{-1}}=\mathrm{V\,s\,A^{-1}}\).

Self-induction. \(L=\Phi/I\), \(\mathcal E_L=-L\,\dd I/\dd t\); long solenoid \(L=\mu_0n^2\times\text{volume}=\mu_0N^2A/l\). An inductor's current cannot jump — it changes continuously.

RL transients. Growth \(I=I_0(1-e^{-tR/L})\), decay \(I=I_0e^{-tR/L}\), time constant \(\tau=L/R\).

Magnetic energy. \(U_L=\tfrac12LI^2\); field energy density \(B^2/2\mu_0\), \(U=\dfrac{1}{2\mu_0}\int B^2\,\dd v\) — the magnetic twin of \(\tfrac12\varepsilon_0E^2\).

Transformers. Ideal (coupled coils on a shared flux): \(\dfrac{V_s}{V_p}=\dfrac{N_s}{N_p}\), \(\dfrac{I_s}{I_p}=\dfrac{N_p}{N_s}\), \(V_pI_p=V_sI_s\). Step-up: \(N_s\gt N_p\); transmission loss \(P_{\rm loss}=I^2R\) reduced by high \(V\), low \(I\). Eddy currents: induced loops oppose flux change; heating \(P=I^2R\) — laminate the core.

Maxwell's equations & EM waves

Displacement current. Steady-state Ampère (\(\nabla\times\vv B=\mu_0\vv J\)) forces \(\nabla\cdot\vv J=0\), contradicting continuity \(\nabla\cdot\vv J=-\partial\rho/\partial t\) when charge builds up (a charging capacitor). Fix: add \(\varepsilon_0\,\partial\vv E/\partial t\) — the displacement current density — to \(\vv J\); the growing \(E\) in a capacitor gap sustains \(B\) exactly as the wire's current does.

Maxwell's equations.

Gauss's law; no monopoles; Faraday; Ampère–Maxwell. Integral forms: flux of \(E\) = charge\(/\varepsilon_0\); flux of \(B\) = 0; circulation of \(E\) = \(-\dd\Phi_B/\dd t\); circulation of \(B\) = \(\mu_0(I+\varepsilon_0\,\dd\Phi_E/\dd t)\). In matter, the free-charge forms use \(\vv D\) and \(\vv H\) (dielectrics, magnetic matter).

Plane waves. Source-free Maxwell gives \(\nabla^2\vv E=\mu_0\varepsilon_0\,\partial^2\vv E/\partial t^2\): any waveform \(f(\uvec k\cdot\vv r-ct)\) travels at \(c=1/\sqrt{\mu_0\varepsilon_0}=2.998\times10^8\,\mathrm{m\,s^{-1}}\), with \(\vv E\perp\vv B\perp\uvec k\), \(E=cB\), and \(\vv E\times\vv B\) along \(\uvec k\). Superposition holds; two equal counter-propagating waves make a standing wave — \(E\) and \(B\) nodes offset by \(\lambda/4\), oscillations offset by \(T/4\).

Energy transport. \(u=\tfrac12\varepsilon_0E^2+B^2/2\mu_0\), split equally in a travelling wave. Poynting vector \(\vv S=\vv E\times\vv B/\mu_0\) = power per area anywhere in any EM field. Sinusoidal wave: \(\bar S=\tfrac12\varepsilon_0cE_0^2=E_0^2/2Z_0\), \(Z_0=\sqrt{\mu_0/\varepsilon_0}\approx377\,\Omega\). Waves carry momentum density \(S/c^2\) — hence radiation pressure \(\bar S/c\) on an absorber.

Other frames. Both invariants vanish for a plane wave (\(\vv E\cdot\vv B=0\), \(E=cB\)), so every observer sees a light-like wave at speed \(c\); only frequency and direction change (Doppler, aberration — relativity).

Electric fields in matter

Dielectric constant. Filling a capacitor with insulator multiplies \(C\) by \(\kappa\ge1\): molecular polarization builds surface charge next to the plates that partially cancels the free charge. Air \(\approx1.0006\); typical solids/liquids 2–10; water \(\approx80\). At fixed \(Q\) the stored energy drops by \(\kappa\) — a slab is pulled into the gap.

Dipole in a field. Torque \(\vv N=\vv p\times\vv E\); orientation energy \(U=-\vv p\cdot\vv E\); nonuniform field pulls with \(F_x=\vv p\cdot\nabla E_x\) (etc.). Dipole potential/field: potential.

Molecular moments. Induced: \(\vv p=\alpha\vv E\); \(\alpha/4\pi\varepsilon_0\) has dimensions of volume, of order the atomic volume (\(\sim10^{-30}\,\mathrm{m^3}\)). Polar molecules (water, HCl) carry permanent \(p\sim10^{-30}\text{–}10^{-29}\,\mathrm{C\,m}\), far exceeding typical induced moments; an applied field partially aligns them against thermal agitation.

Polarization & bound charge. \(\vv P\) = dipole moment per volume \(=N\vv p\). Uniform \(\vv P\) ⇔ bound surface charge \(\sigma_b=\vv P\cdot\uvec n=P\cos\theta\); nonuniform ⇔ volume charge \(\rho_b=-\nabla\cdot\vv P\). Macroscopic \(\vv E\) in matter = spatial average of the microscopic field; uniformly polarized slab: interior average \(-\vv P/\varepsilon_0\); uniformly polarized sphere: interior uniform \(\vv E=-\vv P/3\varepsilon_0\), exterior an exact dipole field with \(\vv p=\tfrac43\pi R^3\vv P\).

Dielectric sphere in uniform \(E_0\). Polarizes uniformly: \(\vv P=3\varepsilon_0\dfrac{\kappa-1}{\kappa+2}\vv E_0\), interior field \(\vv E=\dfrac{3}{\kappa+2}\vv E_0\) — reduced but uniform.

Displacement \(\vv D\). \(\vv D\equiv\varepsilon_0\vv E+\vv P\); \(\nabla\cdot\vv D=\rho_{\mathrm{free}}\) (Gauss with only free charge). Linear dielectric: \(\vv P=\chi_e\varepsilon_0\vv E\), \(\vv D=\varepsilon\vv E\), \(\varepsilon=\kappa\varepsilon_0\), \(\kappa=1+\chi_e\). Point charge buried in infinite dielectric: \(E=q/4\pi\kappa\varepsilon_0r^2\).

Microscopic view. Dilute (gases): each molecule feels \(\approx\) the macroscopic field, \(\chi_e=N\alpha/\varepsilon_0\). Dense matter: the local field at a molecule exceeds the average \(E\) (neighbouring dipoles help), so \(\chi_e\) grows faster than \(N\alpha/\varepsilon_0\) (Clausius–Mossotti direction). Polar substances: partial alignment gives \(\chi_e\approx Np^2/\varepsilon_0kT\) — falls as \(1/T\).

Changing fields. Electronic (stretch) polarization follows up to optical frequencies; orientational polarization of whole molecules cannot, dying out around microwave frequencies — water: \(\kappa\approx80\) static but \(n^2\approx1.77\) optical. Moving bound charge is real current: \(\vv J_{\mathrm{bound}}=\partial\vv P/\partial t\), giving Ampère–Maxwell in matter \(\nabla\times\vv B=\mu_0\bigl(\vv J_{\mathrm{free}}+\partial\vv D/\partial t\bigr)\).

Waves in a dielectric. Speed \(v=1/\sqrt{\mu_0\varepsilon}=c/\sqrt\kappa\); refractive index \(n=\sqrt\kappa\) (at that frequency); still \(\vv E\perp\vv B\perp\uvec k\) with \(E=vB\).

Magnetic fields in matter

Three responses. Diamagnetic: weak moment antiparallel to \(\vv B\), pushed toward weaker field (all matter; orbital origin). Paramagnetic: weak moment parallel, pulled toward stronger field (unpaired spins). Ferromagnetic: strong spontaneous alignment surviving without applied field (spins + quantum exchange interaction).

Magnetic dipoles. No magnetic charge (\(\nabla\cdot\vv B=0\)): matter's magnetism comes from currents and spins. Current loop: \(\vv m=I\vv a\); \(\vv A=\dfrac{\mu_0}{4\pi}\dfrac{\vv m\times\uvec r}{r^2}\); far field \(B_r=\dfrac{\mu_0m}{2\pi r^3}\cos\theta\), \(B_\theta=\dfrac{\mu_0m}{4\pi r^3}\sin\theta\) — same shape as the electric dipole field. Torque \(\vv N=\vv m\times\vv B\), energy \(-\vv m\cdot\vv B\), force in a nonuniform field \(\vv F=\nabla(\vv m\cdot\vv B)\).

Atomic moments. Orbital: \(\vv m=-\dfrac{e}{2m_e}\vv L\). An applied \(B\) perturbs the orbit, inducing \(\Delta\vv m=-\dfrac{e^2\langle r^2\rangle}{4m_e}\vv B\) — always opposing: diamagnetism. Spin moment \(e\hbar/2m_e=9.27\times10^{-24}\,\mathrm{J\,T^{-1}}\) (Bohr magneton); partial thermal alignment of spins gives paramagnetism.

Magnetization & bound currents. \(\vv M\) = moment per volume. Uniform \(\vv M\) ⇔ surface current density \(\mathcal J=M\) (\(\vv M\times\uvec n\)) — a uniformly magnetized cylinder is equivalent to a solenoid; nonuniform ⇔ \(\vv J_{\mathrm{bound}}=\nabla\times\vv M\).

The field \(\vv H\). \(\vv H\equiv\vv B/\mu_0-\vv M\); \(\nabla\times\vv H=\vv J_{\mathrm{free}}\), \(\oint\vv H\cdot\dd\vv l=I_{\mathrm{free}}\) — what coil windings control. Linear media: \(\vv M=\chi_m\vv H\), \(\vv B=\mu\vv H\), \(\mu=\mu_0(1+\chi_m)\). \(\chi_m\sim-10^{-5}\) diamagnets, \(+10^{-5}\)–\(10^{-3}\) paramagnets; paramagnetic \(\chi_m\approx\mu_0Nm^2/kT\) (Curie \(1/T\) law).

Permanent magnets. Outside: dipole-like, \(\vv m=\vv MV\). Inside: \(\vv B\) (continuous loops, follows the magnetization) and \(\mu_0\vv H\) (runs pole-to-pole, opposing \(\vv M\)) differ — a magnet is a solenoid of bound current, not a bar of charge.

Ferromagnetism. Exchange-aligned domains; applied field grows favourable domains and rotates \(\vv M\); saturation when all spins align. Irreversible domain-wall motion gives hysteresis: magnetization remains at zero applied field, and a reverse field is needed to erase it — the multi-valued B–H loop. Soft iron (narrow loop) for cores, effective \(\mu/\mu_0\) in the thousands; hard materials (wide loop) for permanent magnets. Above the Curie point (iron \(770\,^\circ\mathrm{C}\)) thermal agitation wins and the material turns paramagnetic.