What is Magnetic flux through a circuit? — Physics

Definition 29.1 University Physics — Year 1 · Chapter 29 — Electromagnetic Induction and Applications

Orient a closed circuit $\Gamma$ (a sense of travel); its surface gets the normal $\vect n$ given by the right-hand rule. The magnetic flux through the circuit is $\Phi = \sum\vect B\cdot\vect n\,\dd S$ over any surface bounded by $\Gamma$ (weber, $1\,\mathrm{Wb} = 1\,\mathrm{T}\,\mathrm{m}^{2}$ ) — “any”, because the flux of $\vect B$ through a closed surface is zero. For a uniform field and a plane coil of $N$ turns and area $S$ , $\Phi = NBS\cos\alpha$ , $\alpha$ the angle between $\vect B$ and $\vect n$ .

Orientation and Lenz’s law. Left: the circuit’s orientation fixes n and the sign of . Middle: a magnet approaching a ring increases the flux; the induced current creates a field opposing the magnet’s and the ring repels it. Right: a bar sliding on rails in a field pointing toward the reader; the flux of the circuit grows, the induced current flows so that the Laplace force on the bar opposes its motion. — Orientation and Lenz’s law. Left: the circuit’s orientation fixes $\vect n$ and the sign of $\Phi$ . Middle: a magnet approaching a ring increases the flux; the induced current creates a field opposing the magnet’s and the ring repels it. Right: a bar sliding on rails in a field pointing toward the reader; the flux of the circuit grows, the induced current flows so that the Laplace force on the bar opposes its motion.

Examples

Example 29.4 (Orders of magnitude)

A coil of $100$ turns and $10\,\mathrm{cm}^{2}$ in the Earth’s field, flipped in $0.1\,\mathrm{s}$ : $\Delta\Phi = 2 \times 100 \times 10^{-3} \times 5 \times 10^{-5} = 1 \times 10^{-5}\,\mathrm{Wb}$ , $e \sim 0.1\,\mathrm{mV}$ — measurable with a sensitive galvanometer, the way the Earth’s field was once mapped. The same coil in the $1\,\mathrm{T}$ gap of a magnet, pulled out in $0.1\,\mathrm{s}$ : $1\,\mathrm{V}$ . A transformer coil of $1000$ turns with $1\,\mathrm{T}$ at $50\,\mathrm{Hz}$ over $10\,\mathrm{cm}^{2}$ : $e_{\max} = 1000 \times 10^{-3} \times 314 = 314\,\mathrm{V}$ — the mains.

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