Eddy Currents and Inductance: Quick Revision

A brief note on Eddy Currents and Inductance. Read it before a test. Try each question at the end before you open its answer.

Eddy currents

• What they are: currents that swirl inside a solid conductor when the flux through it changes. By Lenz's law they oppose the change.
• A moving conductor is braked, and the work done against the drag becomes heat. Magnetic brakes and damping use this.
• A disc of radius $a$ and thickness $t$ in a field changing at $\dfrac{dB}{dt}$ loses $P = \dfrac{\pi t a^4}{8\rho}\left(\dfrac{dB}{dt}\right)^2$.
• Thin insulated sheets and slots break up the loops. In a sheet, $\dfrac{P}{V} = \dfrac{\pi^2 f^2 B_m^2 d^2}{6\rho}$, and the loss goes as $d^2$.
• Other uses: the induction furnace and cooker, and the metal detector.

Self-inductance

• Definition: $N\phi = LI$. The unit is the henry, $1\ \text{H} = 1\ \text{V s/A}$.
• Emf: $\varepsilon = -L\dfrac{dI}{dt}$. It opposes the change of the current, not the current.
• Long solenoid: $L = \dfrac{\mu_r\mu_0 N^2 A}{l}$. It goes as $N^2$.
• Breaking a circuit quickly gives a large emf, and a spark.
• Series: $L_1 + L_2$. Parallel: $\dfrac{1}{L} = \dfrac{1}{L_1} + \dfrac{1}{L_2}$.

Mutual inductance and energy

• Mutual inductance: $N_2\phi_2 = MI_1$ and $\varepsilon_2 = -M\dfrac{dI_1}{dt}$.
• Coil inside a long solenoid: $M = \mu_0 nN_2\pi r_2^2$. A wider coil gets only the flux of the solenoid.
• Coupling: $M = k\sqrt{L_1L_2}$ with $0 \leq k \leq 1$. In series, $L = L_1 + L_2 \pm 2M$.
• Energy stored: $U = \dfrac{1}{2}LI^2$. In the field: $u = \dfrac{B^2}{2\mu_0}$ in each cubic metre.

A worked example

A coil of $0.15$ H carries $4.0$ A. The current falls to zero steadily in $0.025$ s. Find the energy stored and the emf.

1. The energy. $U = \dfrac{1}{2}LI^2 = \dfrac{1}{2} \times 0.15\ \text{H} \times (4.0\ \text{A})^2 = 1.2\ \text{J}$.
2. The emf. $|\varepsilon| = L\dfrac{\Delta I}{\Delta t} = 0.15\ \text{H} \times \dfrac{4.0\ \text{A}}{0.025\ \text{s}} = 24\ \text{V}$.

Practise with new numbers

Check yourself

1. Sheets of a core are made half as thick. By what factor does the eddy-current loss change?

2. A coil has a steady current. What is the self-induced emf?

3. The current in an inductor is made three times larger. By what factor does the stored energy change?