Electromagnetic Induction — Notes

Faraday’s experiments: a current is induced in a closed coil only while the magnetic flux through it changes — by moving a magnet or the coil, or by changing the current in a nearby coil. Faster change gives a larger current.

Magnetic flux: $\Phi_B = \vec B\cdot\vec A = BA\cos\theta$ (θ between $\vec B$ and the normal), unit weber (Wb = T m²). For a non-uniform field $\Phi_B = \int \vec B\cdot d\vec A$. Flux linkage of $N$ turns: $N\Phi_B$.

Faraday’s law: $\varepsilon = -N\dfrac{d\Phi_B}{dt}$; induced current $I = \varepsilon/R$. Flux can be changed by changing $B$, $A$ or θ. A steady flux induces nothing.

The induced current flows in such a direction that its magnetic field opposes the change in flux producing it (the minus sign in Faraday’s law).

Method: (1) direction of the flux; (2) increasing or decreasing; (3) induced field opposite to the change; (4) right-hand rule for the current.

• N pole approaching a coil → near face of the coil becomes N (repels); N pole receding → near face S (attracts).
• Energy conservation: because the induced current opposes the motion, work must be done to keep the flux changing; this work becomes electrical energy and heat. If it helped the change, energy would appear from nothing.

Sliding rod (length $l$, speed $v$, perpendicular to uniform $B$): flux $\Phi = Blx$, so $\varepsilon = \dfrac{d\Phi}{dt} = Blv$. Equivalently each free charge feels $q\vec v \times \vec B$ along the rod.

With circuit resistance $R$: $I = \dfrac{Blv}{R}$; magnetic force on the rod $F = IlB = \dfrac{B^2l^2v}{R}$, opposing the motion; power needed $Fv = \dfrac{B^2l^2v^2}{R} = I^2R$.

Rotating rod (length $l$, angular speed ω, about one end, perpendicular to $B$): $\varepsilon = \dfrac{1}{2}B\omega l^2$.

A changing current in a coil changes its own flux and induces a back emf opposing the change: $N\Phi = LI$, $\varepsilon = -L\dfrac{dI}{dt}$. Unit: henry (H = V s A⁻¹ = Wb A⁻¹).

Long solenoid: $L = \dfrac{\mu_0 N^2A}{l} = \mu_0 n^2 A l$; with a core $\mu_r\mu_0 n^2 A l$.

Factors: number of turns (∝ N² at fixed length), area, length, and the core’s permeability — not the current. Inductance plays the role of inertia for current. (Energy stored: $\tfrac12 LI^2$.)

A changing current in coil 1 induces an emf in coil 2: $N_2\Phi_2 = MI_1$, $\varepsilon_2 = -M\dfrac{dI_1}{dt}$; $M_{12} = M_{21}$.

Coaxial solenoids (inner radius $r_1$, turns per metre $n_1$, $n_2$, common length $l$): $M = \mu_0 n_1 n_2 \pi r_1^2 l$.

Factors: numbers of turns, areas, separation, relative orientation (coaxial: large; axes perpendicular: very small), and a common core ($\times\mu_r$). Used in transformers and induction chargers.