Electromagnetic Induction

Magnetic flux measures how much magnetic field passes through a surface. For a uniform field $\Phi_B = \vec{B} \cdot \vec{A} = BA\cos\theta$, and in general $\Phi_B = \int \vec{B} \cdot d\vec{A}$. The area vector is perpendicular to the surface, so flux is largest when the field passes straight through and zero when it runs along the surface.

Faraday’s law says a changing magnetic flux induces an emf, $\mathcal{E} = -N\frac{d\Phi_B}{dt}$ for a coil of N turns, whether the field changes, the area changes or the loop turns. Lenz’s law gives the direction: the induced current makes its own magnetic field that opposes the change in flux. Faraday’s law is one of Maxwell’s equations, which together also predict electromagnetic waves.

Once a current is induced in a loop, the magnetic field pushes on the parts of the loop inside the field, $\vec{F} = I\vec{\ell} \times \vec{B}$. That force opposes the motion that caused it, so a bar sliding on rails or a loop leaving a field slows down, and Newton’s second law then gives an equation for its speed. The force grows with the induced current, so it is bigger for a faster loop and smaller for a loop with more resistance.

An inductor, usually a coil like a solenoid, pushes back whenever the current through it changes: it makes an emf $\mathcal{E} = -L\frac{dI}{dt}$, where L is its inductance. A solenoid’s inductance depends on its number of turns, length, area and core, $L = \frac{\mu N^2 A}{\ell}$, and an inductor stores energy in its magnetic field, $U_L = \frac{1}{2}LI^2$.

In an LR circuit the inductor fights changes in current, so the current rises or falls exponentially instead of jumping, with time constant $\tau = \frac{L}{R}$. The loop rule gives $\mathcal{E} = IR + L\frac{dI}{dt}$, so after a switch closes $I(t) = \frac{\mathcal{E}}{R}\left(1 - e^{-t/\tau}\right)$. Right after the switch closes an inductor with no current acts like a break in the circuit, and after a long time it acts like a plain wire.

In an LC circuit, energy sloshes back and forth between the capacitor’s electric field and the inductor’s magnetic field. The charge obeys $\frac{d^2q}{dt^2} = -\frac{1}{LC}q$, the same equation as a mass on a spring, so it oscillates with angular frequency $\omega = \frac{1}{\sqrt{LC}}$. Energy conservation, $\frac{Q_{max}^2}{2C} = \frac{1}{2}LI_{max}^2$, gives the maximum current.