Electromagnetic Induction

1. Check this before you revise anything

The "Additional Exercises" section has been removed from this chapter, as from all 14 chapters of the current Class 12 Physics book. The questions run contiguously from 6.1 to 6.8 with no gaps.

Eddy currents have been removed. Older editions carried a full section on them, with magnetic braking in trains, the induction furnace, dead-beat galvanometers and electric power meters as applications. Searching this chapter for "eddy" returns zero hits, and so do "magnetic braking", "induction furnace" and "dead beat".

The section titled "Energy Consideration: A Quantitative Study" is also gone. The energy argument survives, but only qualitatively, inside section 6.5 on Lenz's law.

A cross-chapter gap worth knowing about. Example 6.3, Example 6.7 and Exercise 6.6 all use the horizontal component of the Earth's magnetic field as a given quantity. That component used to be defined in Chapter 5, but as noted on that chapter's page, the entire section on the Earth's magnetism has been removed — the word "Earth" no longer appears anywhere in Chapter 5.

So this chapter uses a quantity the book no longer teaches. Treat $H_E$ simply as "the component of the field parallel to the ground", which is all these questions require.

2. The Experiments, and Magnetic Flux (Textbook 6.2 to 6.3)

Faraday and Henry independently found that electricity can be produced from magnetism, but only under one condition.

The three experiments, and what each isolates:

• A bar magnet moved towards or away from a closed coil deflects a galvanometer. Holding the magnet still, however close, gives nothing.
• A coil carrying a steady current, moved relative to a second coil, induces a current in the second. Again, no relative motion means no effect.
• With both coils stationary, merely switching the current on or off in the first still induces a current in the second.

The third experiment is decisive. Nothing moves, so motion cannot be the essential ingredient. What all three share is a changing magnetic flux through the circuit.

Magnetic flux measures how much field threads a surface:

${\Phi }_B=\vec{B}\cdot \vec{A}=BA\cos \theta$

where $\theta$ is the angle between $\vec{B}$ and the normal to the area, not the plane. The unit is the weber, with 1 Wb = 1 T m${}^2$.

Three ways to change the flux, and every induction question uses at least one:

• Change $B$, as when a magnet approaches or a current is switched.
• Change $A$, as when a loop is pulled out of a field region.
• Change $\theta$, as when a coil rotates — which is what a generator does.

3. Faraday's Law and Lenz's Law (Textbook 6.4 to 6.5)

Faraday's law. The induced emf equals the negative rate of change of flux:

$\epsilon =-\frac{d{\Phi }_B}{dt}$

and for a coil of $N$ turns, where the flux links each turn:

$\epsilon =-N\frac{d{\Phi }_B}{dt}$

The emf lasts only while the flux is changing. In Exercise 6.3, current in a solenoid is switched off over a finite time, and the induced emf exists only during that interval. A steady current, however large, induces nothing.

Lenz's law is what the minus sign means: the induced current flows in the direction that opposes the change producing it.

Note the precise wording. The induced effect opposes the change, not the field itself. An approaching north pole is met by an induced north pole that repels it; a receding north pole is met by an induced south pole that attracts it, trying to hold it back. Both oppose the change, in opposite senses.

Why the sign must be negative. Suppose the induced current aided the change instead. It would strengthen the flux, which would drive more current, which would strengthen the flux further — energy from nothing. Lenz's law is conservation of energy written into the sign.

The mechanical counterpart: work must be done against the opposing force to keep a magnet moving, and that work is exactly the electrical energy dissipated in the circuit.

Applying it in practice. Exercises 6.1 and 6.2 are pure direction questions, over six and two configurations respectively. The reliable procedure is: decide whether flux is increasing or decreasing, decide which way an induced field must point to oppose that, then use the right-hand rule to get the current direction.

Exercise 6.1(f) is the instructive case — the field line lies in the plane of the loop, so the flux through it is zero and never changes, and no current is induced at all.

4. Motional emf (Textbook 6.6)

When a conducting rod of length $l$ moves with velocity $v$ perpendicular to a field $B$:

$\epsilon =Blv$

Two derivations, and both are examinable. From flux: as the rod slides, the circuit area changes at rate $lv$, so $d\Phi /dt=Blv$. From the Lorentz force: free charges in the moving rod feel $qvB$, which drives them to one end until the resulting electric field balances the magnetic force.

The second route explains which end is positive, which the flux argument alone does not. This is the whole content of Exercise 6.6(b) and (c).

The rotating rod. A rod rotating about one end sweeps out area at a rate that grows along its length, so $v=\omega r$ is not constant and the field must be integrated:

$\epsilon ={\int }_0^{l}B\omega rdr=\frac{1}{2}B\omega l^2$

The factor of $\frac{1}{2}$ is the single most-forgotten result in the chapter. Exercise 6.5 gives $\frac{1}{2}\times 0.5\times 400\times 1^2=100$ V.

5. Inductance (Textbook 6.7)

Inductance measures a circuit's opposition to a change in current, and both varieties are defined through flux linkage.

Mutual inductance (6.7.1). The flux linking coil 2 due to current in coil 1:

$N_2{\Phi }_2=M{I}_1,{\epsilon }_2=-M\frac{d{I}_1}{dt}$

$M$ depends only on the geometry of the pair and the medium between them, never on the currents. It is symmetric: $M_{12}=M_{21}$.

Exercise 6.8 asks only for the change in flux linkage, $M\Delta I=1.5\times 20=30$ Wb. The time given is not needed for that part, though it would give the emf.

Self-inductance (6.7.2). A changing current in a coil changes its own flux, inducing an emf that opposes the change:

$N\Phi =LI,\epsilon =-L\frac{dI}{dt}$

This is why Exercise 6.7 works: the average emf and the rate of current change give $L=200\times 0.1/5.0=4$ H.

For a long solenoid, computing the flux linkage gives:

$L={\mu }_0{n}^{2}Al$

with $n$ the turns per unit length. The dependence on $n^2$ — not $n$ — is worth noting: doubling the winding density quadruples the inductance.

Energy stored. Work done against the back emf while establishing the current is stored in the field:

$U=\frac{1}{2}L{I}^2$

Compare $\frac{1}{2}C{V}^2$ for a capacitor: the inductor stores energy in a magnetic field and does so through current, where the capacitor stores it electrically through voltage.

6. The AC Generator (Textbook 6.8)

A generator is the third route to changing flux, by changing $\theta$: a coil is rotated at constant angular speed in a fixed magnetic field.

With the flux $\Phi =BA\cos \omega t$, Faraday's law gives:

$\epsilon =NBA\omega \sin \omega t={\epsilon }_0\sin \omega t,{\epsilon }_0=NBA\omega$

Where the peaks fall is not where intuition puts them. The emf is maximum when the coil's plane is parallel to the field, where the flux is momentarily zero but changing fastest. It is zero when the plane is perpendicular, where the flux is maximum but momentarily stationary.

Peak emf grows with the number of turns, the area, the field and the rotation rate. Slip rings keep the connection to the external circuit while the coil turns, which is what makes the output alternating rather than direct.

This sinusoidal emf is exactly the supply that Chapter 7 analyses.

Summary

• Induction requires a changing flux; a steady field or steady current induces nothing, however large.
• ${\Phi }_B=BA\cos \theta$, with $\theta$ measured from the normal; the unit is the weber, 1 Wb = 1 T m${}^2$.
• Flux can be changed through $B$, through $A$, or through $\theta$ — the last is how a generator works.
• Faraday's law: $\epsilon =-d{\Phi }_B/dt$, and $\epsilon =-Nd{\Phi }_B/dt$ for $N$ turns.
• Lenz's law: the induced current opposes the change in flux, not the flux itself.
• An approaching pole is repelled and a receding pole attracted — both oppose the change.
• The minus sign is conservation of energy; the opposite sign would create energy from nothing.
• Motional emf $\epsilon =Blv$; the Lorentz-force derivation also tells you which end goes positive.
• A rod rotating about one end gives $\epsilon =\frac{1}{2}B\omega l^2$ — the factor of one half is easily lost.
• Mutual inductance: $N_2{\Phi }_2=M{I}_1$ and ${\epsilon }_2=-Md{I}_1/dt$, with $M$ set by geometry alone and symmetric between the coils.
• Self-inductance: $N\Phi =LI$ and $\epsilon =-LdI/dt$.
• Solenoid: $L={\mu }_0{n}^{2}Al$, so inductance goes as the square of the turns per unit length.
• Energy stored in an inductor is $U=\frac{1}{2}L{I}^2$, the magnetic counterpart of $\frac{1}{2}C{V}^2$.
• AC generator: $\epsilon =NBA\omega \sin \omega t$, peaking when the coil plane is parallel to the field.
• Eddy currents and the quantitative energy section have been removed from this chapter.
• The Earth's horizontal field component is used in Examples 6.3 and 6.7 and Exercise 6.6, although Chapter 5 no longer defines it.
• The Additional Exercises block has been removed, leaving Exercises 6.1 to 6.8.