Magnetic braking

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.