Faraday’s Law

Three simulations of classic electromagnetic induction situations. (1) A rectangular loop enters and leaves a region with a magnetic field, pulled by a motor or dragged with the mouse. (2) A coil and a cylindrical magnet approach and pass through each other. (3) The current in coil 1, controlled by a switch, a rheostat and a DC or AC power supply, induces current in coil 2. In all of them, observe the magnetic flux, the induced electromotive force, the direction of the current (Lenz’s law), the magnetic force that brakes the motion and the energy balance.

Quasi-static regime (dimensions much smaller than the wavelength), in which Faraday’s law, $\varepsilon = -\dfrac{d\Phi}{dt}$, holds exactly. At each time step, the circuit ($L\,\dfrac{di}{dt} + R\,i = \varepsilon$) and the motion ($m\,\dfrac{dv}{dt} = F_{ext} + F_{mag}$, with $F_{mag} = i\oint d\vec\ell\times\vec B$) are solved together by the implicit midpoint method, which preserves the energy balance exactly; Lenz’s law and energy conservation come out of this, with no ready-made curve.

Loop in a field: the flux is the integral of $B$ over the area of the loop; the self-inductance comes from Grover’s formula for a rectangle. The field can vary in time (an electromagnet that follows the chosen value with a time constant of 0.5 s): the emf then also has the term $-N A\,\dfrac{dB}{dt}$, and the work done by the field’s source appears in the energy balance.

Magnet and coil: the uniformly magnetized magnet is equivalent to a solenoid with surface current $K = M = B_r/\mu_0$. The flux through the coil is the sum, by numerical quadrature, of the mutual inductances between coaxial loops, given by complete elliptic integrals; the field lines are traced by integrating the exact field of the cylinder (Derby and Olbert).

Two coils: $L_1$ from Nagaoka’s formula, $L_2$ from Maxwell’s formula for a short coil and $M$ from the same quadrature of elliptic integrals. The equations $V = R_1 i_1 + \dfrac{d\lambda_1}{dt}$ and $0 = R_2 i_2 + \dfrac{d\lambda_2}{dt}$, with $\lambda_1 = L_1 i_1 + M i_2$ and $\lambda_2 = M i_1 + L_2 i_2$, are integrated exactly at each step (matrix exponential). With the switch open, circuit 1 is modeled by a 10 kΩ resistance (the spark). Time in this simulation runs in slow motion, since the time constants $L/R$ are in milliseconds.

LED: bicolor and with no polarity (two LEDs in antiparallel, red and green), with a resistor in series. Each junction follows the Shockley equation, $i = I_s\,(e^{u/nV_T} - 1)$, plus an internal resistance; the forward voltage is about 1.8 V (red) and 2.0 V (green), and the light emitted is proportional to the current. The circuit becomes nonlinear: in simulations 1 and 2, the current is obtained at each step by solving the diode equation (the inductance is neglected, since $L/R$ is much shorter than the duration of the pulses); in simulation 3, the coil equations are integrated by the TR-BDF2 method.

Marco P. M. de Souza