Electromagnetic induction

Electromagnetic induction includes an induced electromotive force associated with a changing magnetic field or motion of a conductor in a magnetic field. Faraday's law of induction and motional electromotive force specify the relevant contributions.

For a conducting disc of radius $a$ rotating in a uniform axial magnetic field $B$, the shaft-to-rim motional electromotive force is ${\int }_0^a\omega rBdr=\omega B{a}^2/2$. A current $I$ between shaft and rim gives the opposing torque $IB{a}^2/2$, making electromagnetic conversion power $\mathcal{E}I$ equal to mechanical torque times angular speed.

Braking caused by induced currents whose magnetic force opposes motion. energy removed from mechanical motion can become resistive heat; the force depends on magnetic-field gradients or boundary crossing, resistance and inductance.

A rectangular loop entering a uniform magnetic field while only one horizontal segment is immersed has motional electromotive force $Bav$. Neglecting self-inductance, its electric current is $Bav/R$ and its upward magnetic force is $(Ba{)}^{2}v/R$. The equation $m\dot{v}=mg-(Ba{)}^{2}v/R$ gives exponential relaxation toward $mgR/(Ba{)}^2$. This entry model ends when both horizontal segments enter the field; then their net magnetic force and the rate of change of external magnetic flux vanish.

A thin square loop rotating about a central in-plane axis has $\mathcal{I}=m{a}^2/6$. With ohmic current and decay slow compared with a rotation, the averaged torque is $-B^2{a}^4\omega /(2R)$. Its angular-speed decay time is $mR/(3B^2{a}^2)$; its kinetic-energy decay time is half that value.

A square wire loop partly in a uniform field has linear drag $B^2{a}^{2}v/R$, giving velocity-relaxation time ${\tau }_t=mR/(B^2{a}^2)$. For uniform wire of mass density ${\rho }_m$ and resistivity ${\rho }_e$, ${\tau }_t=16{\rho }_m{\rho }_e/B^2$, independent of wire dimensions.

The instantaneous-current approximation uses $I=\mathcal{E}/R$ and requires the electrical relaxation time $L/R$ to be negligible relative to the relevant mechanical changes. Otherwise circuit self-inductance changes the braking transient.

A square resistive loop of side $a$, moving normally across a sharp field edge at speed $v$, has induced current magnitude $Bav/R$ and drag magnitude $B^2{a}^{2}v/R$ while partially overlapping the field. Completely enclosed in a uniform static field it has no changing flux and, with neglected self-inductance, no current.

For a complete entry and exit of a small square loop, integrating the drag over two distances $a$ gives impulse magnitude $2B^2{a}^3/R$. This is independent of speed in the ideal sharp-edge, zero-inductance model, though dissipated energy is not.

Changing the magnetic flux through a long solenoid produces an exterior azimuthal electric field $E_{\varphi }(r,t)=-\dot{\Phi }(t)/(2\pi r)$ by Faraday's law of induction. A charge $q$ whose displacement is negligible during the switching receives mechanical momentumThe result assumes the usual quasistatic field approximation and neglects the particle's displacement while the pulse acts. It describes an electromagnetic induction effect, distinct from the static Aharonov-Bohm effect.

Mutual induction is electromagnetic induction in one electrical circuit associated with a changing electric current in another. In a fixed linear arrangement, mutual inductance quantifies the coupling.