Magnetic Flux and Electromagnetic Induction

Electromagnetic induction is the process by which a changing magnetic field produces (induces) an e.m.f. in a conductor. It was discovered by Michael Faraday in 1831, and it powers the entire modern world: every generator, every transformer, every wireless charger, every induction cooker, every metal detector, every RFID reader, every electric guitar pick-up, every eddy-current brake — all are applications of the two laws we meet in this lesson, Faraday's law and Lenz's law. This lesson is the anchor for OCR's PAG 9 practical activity on electromagnetic induction.

Spec mapping (OCR H556 Module 6.3 — electromagnetism). This lesson covers the definition of magnetic flux $\Phi = BA\cos\theta$Φ=BAcosθ with $\theta$θ between $\vec{B}$B and the normal to the surface; flux linkage $N\Phi$NΦ for a coil of $N$N turns; Faraday's law in the form $\varepsilon = -N\,\mathrm{d}\Phi/\mathrm{d}t$ε=−NdΦ/dt; Lenz's law and its derivation from conservation of energy; the four ways of changing flux linkage (change $B$B, change $A$A, change $\theta$θ, or move the coil); and qualitative understanding of rotating-coil generation of alternating e.m.f.

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Magnetic Flux

Before we can state Faraday's law we need one more quantity: magnetic flux, symbol $\Phi$Φ (Greek phi). Intuitively, magnetic flux is a measure of the total "amount" of magnetic field passing through a surface.

For a flat surface of area $A$A in a uniform field $B$B making an angle $\theta$θ between the field and the normal to the surface,

$\Phi = BA\cos\theta.$Φ=BAcosθ.

The unit is the weber (Wb): $1$1 Wb $= 1$=1 T m$^2$2.

Two special cases to commit to memory:

• Field perpendicular to the surface ($\theta = 0^{\circ}$θ=0∘): $\Phi = BA$Φ=BA, the maximum.
• Field parallel to the surface ($\theta = 90^{\circ}$θ=90∘): $\Phi = 0$Φ=0, no flux through the loop.

Watch the angle! $\theta$θ is the angle between the field and the normal to the surface, not between the field and the surface itself. If the field lies in the plane of the loop, $\theta = 90^{\circ}$θ=90∘ and $\Phi = 0$Φ=0. Mixing this up is the most common flux error at A-Level.

Flux Linkage

For a coil of $N$N turns, each turn is linked by the same flux $\Phi$Φ. The total flux linkage is

$N\Phi = NBA\cos\theta.$NΦ=NBAcosθ.

Flux linkage also has units of weber (some textbooks write "weber-turns" for clarity). It is the quantity that appears in Faraday's law for a coil — not the single-turn flux $\Phi$Φ.

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Worked Example 1 — Calculating Flux

A circular coil of $N = 50$N=50 turns has radius $8.0$8.0 cm. It is placed in a uniform magnetic field of $B = 0.12$B=0.12 T. Calculate the flux linkage when (a) the field is perpendicular to the plane of the coil, (b) the field is parallel to the plane of the coil, (c) the field is at $30^{\circ}$30∘ to the plane of the coil.

(a) Field perpendicular to coil. This means the field is parallel to the normal, so $\theta = 0^{\circ}$θ=0∘ and $\cos\theta = 1$cosθ=1.

$A = \pi r^2 = \pi(0.080)^2 \approx 2.011 \times 10^{-2}\ \text{m}^2$A=πr2=π(0.080)2≈2.011×10−2 m2

$\Phi = BA = (0.12)(2.011 \times 10^{-2}) \approx 2.413 \times 10^{-3}\ \text{Wb}$Φ=BA=(0.12)(2.011×10−2)≈2.413×10−3 Wb

$N\Phi = (50)(2.413 \times 10^{-3}) \approx 0.121\ \text{Wb}.$NΦ=(50)(2.413×10−3)≈0.121 Wb.

(b) Field parallel to coil. Field in the plane of the coil means the angle between field and normal is $90^{\circ}$90∘. $\cos 90^{\circ} = 0$cos90∘=0, so $N\Phi = 0$NΦ=0.

(c) Field at $30^{\circ}$30∘ to the plane of the coil. Careful: "$30^{\circ}$30∘ to the plane" means $60^{\circ}$60∘ to the normal, so $\theta = 60^{\circ}$θ=60∘ in the formula.

$N\Phi = NBA\cos 60^{\circ} = (50)(0.12)(2.011 \times 10^{-2})(0.5) \approx 0.060\ \text{Wb}.$NΦ=NBAcos60∘=(50)(0.12)(2.011×10−2)(0.5)≈0.060 Wb.

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Faraday's Law

Faraday observed that whenever the flux linking a coil changes, an e.m.f. is induced in the coil. Quantitatively, the induced e.m.f. is equal to the rate of change of flux linkage:

$\varepsilon = -\frac{\mathrm{d}(N\Phi)}{\mathrm{d}t}.$ε=−dtd(NΦ)​.

The minus sign is Lenz's law (see next section). The magnitude is the rate of change of flux linkage in webers per second ($=$= volts).

Any way of changing the flux linkage will induce an e.m.f.:

1. Changing the field $B$B (e.g. switching a nearby magnet on or off, or pushing/pulling a magnet near the coil).
2. Changing the area $A$A of the circuit (e.g. sliding a conductor along a pair of rails in a field).
3. Changing the angle $\theta$θ — rotating the coil in the field. Generators work this way.
4. Moving the coil into or out of a field — a special case of changing $B$B at the coil's location.

Faraday's law is the fundamental principle behind all generators and transformers.

Why "flux linkage" not just "flux"?

For a single turn, the induced e.m.f. is $-\mathrm{d}\Phi/\mathrm{d}t$−dΦ/dt. For a coil of $N$N turns, each turn has the same $\mathrm{d}\Phi/\mathrm{d}t$dΦ/dt, and the e.m.f.s of the turns add in series, giving $\varepsilon = -N\,\mathrm{d}\Phi/\mathrm{d}t = -\mathrm{d}(N\Phi)/\mathrm{d}t$ε=−NdΦ/dt=−d(NΦ)/dt. The product $N\Phi$NΦ is the natural quantity. Dropping $N$N is the single most common error in induction calculations.

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Lenz's Law

Lenz's law fixes the direction of the induced current: the induced current flows in the direction that opposes the change in flux which caused it.

Lenz's law is a direct consequence of conservation of energy. If the induced current flowed in the same direction as the change, then the change would accelerate and we would get something for nothing — perpetual motion. Instead, the induced current creates a field that opposes the change, and an external agent must do work to maintain the change, keeping energy conservation intact.

Examples of Lenz's Law

1. Bar magnet approaching a coil (N pole first). The flux through the coil increases. The induced current flows in the direction that makes the near face of the coil a N pole, repelling the incoming magnet. The external agent must do work against this repulsion, and that work becomes the electrical energy of the induced current.

2. Bar magnet withdrawn from a coil (N pole first). The flux decreases. The induced current now flows the other way, making the near face a S pole, attracting the magnet and opposing its withdrawal. Again, the person pulling the magnet does work.

3. Magnet dropped through a vertical copper tube. The changing flux induces "eddy currents" in the tube walls, and these currents produce a field that decelerates the magnet. The magnet falls much more slowly than it would in free fall — a dramatic visual demonstration of Lenz's law. (OCR does not require detailed analysis of eddy currents.)

Mnemonic for Lenz's Law

"The induced current always tries to keep things the same." If flux is increasing, the induced current produces flux that opposes the increase. If flux is decreasing, the induced current produces flux that opposes the decrease. Either way, the system "resists change", and the agent producing the change must do work.

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Worked Example 2 — Bar Magnet Through a Coil

A bar magnet is pushed into a $200$200-turn coil. The flux linking each turn increases from $0$0 to $0.004$0.004 Wb in $0.10$0.10 s. Calculate the average e.m.f. induced in the coil.

$\varepsilon = -N\,\frac{\Delta \Phi}{\Delta t} = -200 \times \frac{0.004 - 0}{0.10} = -8.0\ \text{V}.$ε=−NΔtΔΦ​=−200×0.100.004−0​=−8.0 V.

The magnitude is $8.0$8.0 V. The minus sign indicates that the e.m.f. opposes the change (so the induced current would try to push the magnet back out). For numerical calculations you usually only quote the magnitude, but the sign is the formal statement of Lenz's law and examiners may want to see it.

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Worked Example 3 — Moving Rod on Rails (Motional E.M.F.)

A straight rod of length $L = 20$L=20 cm slides at $v = 2.0$v=2.0 m s$^{-1}$−1 along a pair of horizontal conducting rails, in a vertical magnetic field of $0.30$0.30 T. Find the induced e.m.f.

Derivation of ε = BLv. Moving the rod to the right sweeps out new area at a rate $\mathrm{d}A/\mathrm{d}t = Lv$dA/dt=Lv, increasing the flux through the circuit at a rate

$\frac{\mathrm{d}\Phi}{\mathrm{d}t} = B \times \frac{\mathrm{d}A}{\mathrm{d}t} = BLv.$dtdΦ​=B×dtdA​=BLv.

The induced e.m.f. is therefore

$\varepsilon = BLv = (0.30)(0.20)(2.0) = 0.12\ \text{V}.$ε=BLv=(0.30)(0.20)(2.0)=0.12 V.

The formula $\varepsilon = BLv$ε=BLv is a direct consequence of Faraday's law applied to a rigid rod sweeping out area in a uniform field. It turns up in many OCR practical contexts. Direction (by Lenz's law) is such that the induced current produces a force opposing the rod's motion — so an external agent must do work against this force to maintain the constant velocity.

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Worked Example 4 — Rotating Coil (Qualitative Only)

A rectangular coil is rotated in a uniform field. Qualitatively explain how the flux linkage varies with time and when the e.m.f. is a maximum.

As the coil rotates, the angle $\theta$θ between the field and the normal to the coil changes continuously: $\theta = \omega t$θ=ωt. Flux linkage varies as

$N\Phi = NBA\cos(\omega t).$NΦ=NBAcos(ωt).

When plotted against time, this is a cosine curve. The e.m.f. is the rate of change:

$\varepsilon = -\frac{\mathrm{d}(N\Phi)}{\mathrm{d}t} = NBA\omega\sin(\omega t).$ε=−dtd(NΦ)​=NBAωsin(ωt).

The e.m.f. is maximum when the curve is changing fastest — at the zero crossings of flux, when the plane of the coil is parallel to the field ($\theta = 90^{\circ}$θ=90∘). The e.m.f. is zero when the flux is at a maximum or minimum, i.e. when the coil is perpendicular to the field.

OCR-specific. OCR A-Level Physics A does not require you to derive the rotating-coil e.m.f. formula $\varepsilon = NBA\omega\sin(\omega t)$ε=NBAωsin(ωt) algebraically, nor to work with RMS values. A qualitative understanding that a rotating coil generates an alternating e.m.f., maximum at $\theta = 90^{\circ}$θ=90∘ and zero at $\theta = 0^{\circ}$θ=0∘ or $180^{\circ}$180∘, is sufficient.

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PAG 9 — The Practical Anchor

This lesson is the home of OCR's PAG 9 (Practical Activity Group) on electromagnetic induction. Typical experiments:

• Drop a bar magnet through a coil and measure the induced e.m.f. with a data-logger or oscilloscope; verify $\varepsilon \propto \mathrm{d}\Phi/\mathrm{d}t$ε∝dΦ/dt qualitatively.
• Search-coil flux measurement. A small flat coil is moved through a varying field; the induced e.m.f. is integrated over time to give the total change in flux.
• Solenoid-and-search-coil. A pulse of current through a solenoid produces a flux change; a search coil placed inside or outside reads the induced e.m.f.

In PAG 9 you are expected to demonstrate (a) that flux change produces an e.m.f., (b) that the direction of the e.m.f. obeys Lenz's law, and (c) that the magnitude of the e.m.f. scales with the rate of change of flux. The mathematical relation $\varepsilon = -N\,\mathrm{d}\Phi/\mathrm{d}t$ε=−NdΦ/dt is the analytic backbone.

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