5 Electromagnetic Induction

Magnetic Flux and Surface Orientation

Magnetic flux describes how much magnetic field passes through a chosen surface. For a uniform field and a flat surface, calculate it with

Here, $B$ is magnetic field strength, $A$ is area, and $\theta$ is the angle between the field and the surface’s normal, the direction perpendicular to the surface. When the field is perpendicular to the surface, $\theta = 0$ and flux is greatest. When the field is parallel to the surface, $\theta = 90^\circ$ and flux is zero. The SI unit is the weber, $\text{Wb} = \text{T}\cdot\text{m}^2$.

Flux is a useful starting point because induction depends not simply on the presence of a magnetic field, but on whether the flux through a loop changes.

Takeaway: Field strength, surface area, and orientation together determine magnetic flux.

Faraday’s Law and Lenz’s Law

A changing magnetic flux through a conducting loop induces an electromotive force, or emf. Faraday’s law expresses this relationship for a coil of $N$ turns:

The flux can change if the magnetic field changes, the loop’s area changes, or the loop rotates. The negative sign describes Lenz’s law: the induced current creates a magnetic field that opposes the change in flux, not necessarily the original field itself.

For instance, if a magnetic field directed into the page through a loop is increasing, the induced field points out of the page. The current must flow counterclockwise to create that opposing field. This opposition is consistent with conservation of energy.

Example: A coil has $200$ turns and area $0.010\,\text{m}^2$. It is perpendicular to a field that increases from $0$ to $0.30\,\text{T}$ in $0.20\,\text{s}$. The magnitude of its average induced emf is

If the coil is part of a closed circuit with resistance $R$, the induced current’s magnitude is $I = \frac{|\mathcal{E}|}{R}$.

Takeaway: Faraday’s law gives the emf magnitude from the rate of flux change; Lenz’s law determines the direction.

Self-Induction and Inductance

A current in a coil creates a magnetic field and magnetic flux through the coil. If the current changes, the resulting change in flux induces an emf in the same coil. This process is called self-induction.

The coil’s self-inductance, $L$, relates its flux linkage to current and describes the emf produced when current changes:

The induced emf opposes changes in current, so an inductor resists a rapid increase or decrease in current. Inductance is measured in henries, $\text{H} = \text{V}\cdot\text{s}/\text{A}$. The energy stored in an inductor’s magnetic field is

An ideal inductor stores energy rather than dissipating it. Real circuits may also lose energy because of resistance.

Takeaway: Inductance links current changes to induced emf, while the inductor’s magnetic field stores energy.

Energy Oscillations in an LC Circuit

An ideal LC circuit contains a capacitor of capacitance $C$ and an inductor of inductance $L$, with negligible resistance. When a charged capacitor is connected to the inductor, the capacitor discharges and current builds. The inductor’s magnetic field then sustains the current, which charges the capacitor with the opposite polarity. Energy continually shifts between the capacitor’s electric field and the inductor’s magnetic field.

For an ideal circuit, the charge varies sinusoidally:

Here, $q_0$ is the maximum charge and $\varphi$ sets the initial phase. Current is the rate of change of charge, $i(t) = \frac{dq}{dt}$. The oscillation period and frequency are

For example, when $L = 0.10\,\text{H}$ and $C = 10\,\mu\text{F}$,

In a real circuit, resistance dissipates energy, so the oscillations gradually weaken.

Takeaway: In an ideal LC circuit, energy repeatedly transfers between electric and magnetic fields, with the oscillation rate set by $L$ and $C$.