Electromagnetic Induction: NEET notes.

01 · NCERT §6.1, §6.2

The experiments of Faraday and Henry

• Oersted and Ampere showed that currents make magnetic fields. Around 1830 Faraday in England and Henry in the USA showed the reverse link: a changing magnetic field drives a current in a closed coil. Producing current this way is called electromagnetic induction.
• Experiment 6.1: push the N pole of a bar magnet towards a coil joined to a galvanometer and the pointer kicks. It deflects only while the magnet moves; a magnet held still gives nothing.
• Pulling the magnet away reverses the deflection. Using the S pole instead reverses both results. Moving faster gives a bigger deflection.
• Holding the magnet still and moving the coil gives the same effects: what matters is relative motion between magnet and coil.
• Experiment 6.2: a second coil carrying a steady current from a battery can replace the magnet. Moving either coil relative to the other again gives a current while the motion lasts.
• Experiment 6.3: with both coils held still, pressing the key in the battery circuit gives a brief kick in the galvanometer; holding it pressed gives none; releasing it gives a brief kick the other way. An iron rod along the common axis makes the kicks much larger.
• So relative motion is not essential. What the three experiments share is a magnetic field through the coil that is changing with time.
• Example 6.1: to get a bigger deflection, put a soft-iron rod in the current-carrying coil, use a stronger battery, or move it faster. Without a galvanometer, a small torch bulb in the circuit glows to show the induced current.

02 · NCERT §6.3

Magnetic flux

• For a flat area A in a uniform field B, the magnetic flux is ΦB = B·A = BA cos θ, where θ is the angle between B and the area vector A (the normal to the surface).
• For a curved surface or a field that varies from place to place, split the surface into small elements dA and add B·dA over all of them.
• Flux is a scalar. Its SI unit is the weber (Wb), equal to T m².
• Flux is largest, BA, when the field is along the normal (θ = 0°), zero when the field lies in the plane of the surface (θ = 90°), and −BA when the surface is flipped (θ = 180°).
• Flux can therefore be changed in three ways: change B, change the area A, or change the angle θ.

03 · NCERT §6.4

Faraday's law of induction

• Faraday's law: an emf is induced in a circuit whenever the magnetic flux through it changes, and the size of the emf equals the rate of change of flux: ε = −dΦB/dt.
• The minus sign fixes the direction of the emf (Lenz's law, next section).
• For a closely wound coil of N turns, each turn links the same flux, so ε = −N dΦB/dt. More turns give more emf.
• Experiment 6.3 explained: pressing the key makes the current, and so the flux through the other coil, rise quickly; a steady current means steady flux and no emf; releasing the key makes the flux fall, giving an emf the other way.
• Example 6.2: a 10 cm square loop of resistance 0.5 Ω has a 0.10 T field at 45° to its normal. The field falls steadily to zero in 0.70 s. Initial flux = 0.1 × 10⁻² × cos 45° = 10⁻³/√2 Wb, so ε = 1.0 mV and I = 2 mA.
• The earth's field also threads such a loop, but it is steady over the experiment, so it induces nothing.
• Example 6.3: a 500-turn coil of radius 10 cm and resistance 2 Ω faces the earth's horizontal field of 3.0 × 10⁻⁵ T and is turned through 180° in 0.25 s. The flux per turn goes from +3π × 10⁻⁷ to −3π × 10⁻⁷ Wb, so the average emf is about 3.8 × 10⁻³ V and the current 1.9 × 10⁻³ A.
• Switching a large electromagnet on or off can induce emfs large enough to damage sensitive instruments close by.

04 · NCERT §6.5

Lenz's law and conservation of energy

• Lenz's law (1834): the induced emf has the polarity that would drive a current opposing the change in flux that produced it. This is the minus sign in Faraday's law.
• An N pole pushed towards a coil raises the flux, so the induced current makes the near face of the coil an N pole, which repels the magnet. Seen from the magnet's side the current runs anticlockwise.
• Pulling the N pole away lowers the flux, so the current reverses (clockwise from the magnet's side) and the near face becomes an S pole, which pulls back on the retreating magnet.
• With an open circuit no current flows, but an emf still appears across the open ends, and Lenz's law still gives its polarity.
• If the induced current aided the change, a small push would make the magnet speed up by itself, giving energy for nothing. Lenz's law is what energy conservation demands.
• In the real case you must push against the repulsion; the work you do appears as Joule heat from the induced current.
• Example 6.4: a loop entering a field region gains flux and one leaving it loses flux, so the currents circulate in opposite senses. A loop wholly inside or wholly outside a uniform field region has no induced current.
• Example 6.5: a still loop between fixed magnets, however strong, has no current, since the flux is not changing. A loop moving through a steady electric field gets no current either. A rectangular loop leaving a field region at steady speed has a constant emf; a circular one does not, as its area inside the field changes at a varying rate.

05 · NCERT §6.6

Motional emf

• A rod of length l slides at speed v along two rails in a uniform field B at right angles to the plane of the circuit. The enclosed area changes at the rate lv, so the flux Blx changes at Blv and the induced emf is ε = Blv. This is called motional emf.
• Lorentz-force view: every free charge q in the rod moves with it at speed v, so it feels a force qvB along the rod. Carrying a charge along the rod's length l takes work qvBl, and emf is work per unit charge, so ε = Blv again.
• Here the flux changes because the conductor moves through a steady field. When the conductor is at rest and the field changes, v = 0 and the only possible force is qE, so a changing magnetic field must create an electric field.
• The electric field made by a changing magnetic field behaves differently from the field of static charges. Moving charges exert forces on magnets, and moving magnets exert forces on charges: electricity and magnetism are linked.
• With the rails closed through a resistance R, the current is I = Blv/R. The field then pushes on the current-carrying rod with a force BIl that opposes its motion, as Lenz's law requires; keeping v steady needs an equal pull.

06 · NCERT §6.6

Rotating rods and wheels

• A rod of length R turning at angular speed ω about one end, in a uniform field B along the axis, has different parts moving at different speeds v = ωr.
• Adding the emf B v dr from each small piece gives ε = ½BωR² between the pivot and the free end.
• The same result follows from Faraday's law: the rod sweeps a sector of area ½R²θ, and B times the rate of change of that area is ½BR²ω.
• Example 6.6: a 1 m rod turning at 50 rev/s in a 1 T field gives ε = ½ × 1.0 × 2π × 50 × 1² = 157 V between the centre and the rim.
• Example 6.7: a wheel with 10 spokes of 0.5 m, turning at 120 rev/min in the earth's horizontal field of 0.4 G, gives ε = ½ × 4π × 0.4 × 10⁻⁴ × 0.5² = 6.28 × 10⁻⁵ V between axle and rim.
• The number of spokes does not matter: the spokes are emfs in parallel, each giving the same value.

07 · NCERT §6.7, §6.7.1

Mutual inductance

• The flux a current sets up through a coil grows in step with that current. For N closely wound turns, the flux linkage NΦB divided by I is a fixed number for the coil, its inductance.
• Inductance depends only on geometry and on the material inside, much as capacitance depends on plate area, gap and dielectric. It is a scalar with dimensions [M L² T⁻² A⁻²]; its SI unit is the henry (H).
• Two long coaxial solenoids of length l: inner S₁ with radius r₁ and n₁ turns per metre, outer S₂ with n₂. A current I₂ in S₂ gives field μ₀n₂I₂, and the flux linkage with S₁ is N₁Φ₁ = M₁₂I₂ with M₁₂ = μ₀n₁n₂πr₁²l.
• Working the other way, the flux of S₁ lies only inside S₁, and M₂₁ comes out the same. In general M₁₂ = M₂₁ = M, which helps when one direction is hard to calculate.
• With a core of relative permeability μr, M = μrμ₀n₁n₂πr₁²l. M also depends on how far apart the coils are and how they are oriented.
• Example 6.8: a small loop of radius r₁ at the centre of a large coaxial loop of radius r₂ (r₁ ≪ r₂) sits in a nearly uniform field μ₀I₂/2r₂, so M = μ₀πr₁²/2r₂.
• A changing current in one coil induces an emf in the other: ε₁ = −M dI₂/dt. This explains the kicks in Experiment 6.3.

08 · NCERT §6.7.2

Self-inductance and magnetic energy

• A changing current in a coil changes the coil's own flux, inducing an emf in the same coil: self-induction. The flux linkage is NΦB = LI, where L is the self-inductance.
• The self-induced emf is ε = −L dI/dt. It opposes any change in the current, rise or fall, and is called the back emf.
• For a long solenoid of cross-section A, length l and n turns per metre, L = μ₀n²Al. With a core of relative permeability μr, such as soft iron, L = μrμ₀n²Al.
• L behaves like inertia, the electrical analogue of mass: it resists both the growth and the decay of current.
• Work done against the back emf while the current builds from 0 to I is stored as magnetic energy: W = ½LI², just as a mass stores ½mv².
• With two coils carrying currents, the emf in coil 1 is ε₁ = −L₁ dI₁/dt − M₁₂ dI₂/dt.
• Example 6.9: for a solenoid the stored energy is (B²/2μ₀)Al, so the energy per unit volume is B²/2μ₀. This matches the electric energy density ½ε₀E²: both grow as the square of the field, and both hold for any region of space.

09 · NCERT §6.8

AC generator

• An ac generator turns mechanical energy into electrical energy by rotating a coil in a magnetic field, which changes the angle θ between B and the coil's area vector.
• Its parts: a coil (the armature) on a rotor shaft whose axis is at right angles to B, turned by some outside means. The coil's ends connect to the outside circuit through slip rings and brushes.
• Turning at steady angular speed ω with θ = ωt, the flux through N turns is NBA cos ωt, so ε = NBAω sin ωt = ε₀ sin ωt, with peak ε₀ = NBAω.
• The emf is largest when θ = 90° or 270°, where the coil's plane lies along B and the flux, though zero, is changing fastest. It is zero when the plane faces the field and the flux is largest.
• The polarity reverses every half turn, so the current is alternating (ac). With ω = 2πν, ε = ε₀ sin 2πνt, where ν is the rotation frequency.
• Commercial generators are turned by falling water (hydro-electric), by steam raised with coal or other fuel (thermal), or by steam from nuclear fuel. A typical large machine gives about 100 MW, and some give 500 MW. Usually the coils are fixed and electromagnets rotate.
• The supply frequency is 50 Hz in India and 60 Hz in some countries such as the USA.
• Example 6.10: a 100-turn coil of area 0.10 m² turned at 0.5 rev/s in 0.01 T gives ε₀ = 100 × 0.01 × 0.1 × 2π × 0.5 = 0.314 V.

Revise

Must-know facts

1. 01An emf is induced only while the magnetic flux through a circuit is changing.
2. 02ΦB = B·A = BA cos θ, θ between B and the normal; unit weber, 1 Wb = 1 T m².
3. 03Faraday's law: ε = −N dΦB/dt for a coil of N turns.
4. 04Flux changes if B, the area or the angle between B and the normal changes.
5. 05Lenz's law: the induced current opposes the change in flux that causes it; this follows from energy conservation.
6. 06An N pole approaching a coil makes the near face an N pole (repulsion); receding, an S pole (attraction).
7. 07Motional emf across a rod moving at right angles to B: ε = Blv.
8. 08Rod of length R rotating about one end with angular speed ω: ε = ½BωR².
9. 09A time-varying magnetic field produces an electric field.
10. 10Flux linkage NΦ = LI (self) or N₁Φ₁ = MI₂ (mutual); unit henry, dimensions [M L² T⁻² A⁻²].
11. 11Long coaxial solenoids: M = μ₀n₁n₂πr₁²l; M₁₂ = M₂₁.
12. 12Long solenoid: L = μ₀n²Al, or μrμ₀n²Al with a core.
13. 13Self-induced emf ε = −L dI/dt opposes both rise and fall of current.
14. 14Energy stored in an inductor: ½LI²; magnetic energy density B²/2μ₀.
15. 15AC generator: ε = NBAω sin ωt, peak ε₀ = NBAω.
16. 16Generator emf is maximum when the coil's plane is parallel to B and zero when it faces B.
17. 17Supply frequency: 50 Hz in India, 60 Hz in the USA.

Where marks go

Common traps

Reference

Formulas

Magnetic flux

ΦB = B·A = BA cos θ

θ between B and the normal; unit Wb = T m².

Faraday's law

ε = −N dΦB/dt

Minus sign is Lenz's law.

Motional emf

ε = Blv

Rod, velocity and field mutually perpendicular.

Rotating rod

ε = ½BωR²

Between pivot and free end; field along the axis.

Mutual inductance, coaxial solenoids

M = μ₀n₁n₂πr₁²l

r₁ is the inner radius; multiply by μr for a core.

Mutually induced emf

ε₁ = −M dI₂/dt

M₁₂ = M₂₁.

Self-inductance of a solenoid

L = μrμ₀n²Al

μr = 1 for air.

Self-induced emf

ε = −L dI/dt

Back emf.

Energy in an inductor

U = ½LI², u = B²/2μ₀

u is energy per unit volume.

AC generator emf

ε = NBAω sin ωt, ε₀ = NBAω

ω = 2πν; θ = 0 at t = 0.

Glossary

Key terms

Electromagnetic induction

Producing an emf, and a current in a closed circuit, by changing the magnetic flux through it.

Magnetic flux (ΦB)

B·A summed over a surface; a scalar measured in webers.

Weber (Wb)

SI unit of magnetic flux, 1 T m².

Faraday's law

Induced emf equals minus the rate of change of flux linkage.

Lenz's law

The induced current flows so as to oppose the change in flux that produced it.

Motional emf

The emf Blv across a conductor moving through a magnetic field.

Flux linkage

NΦB for a closely wound coil of N turns.

Inductance

Flux linkage per unit current; depends on geometry and core material.

Henry (H)

SI unit of inductance, 1 Wb/A.

Mutual inductance (M)

Flux linkage in one coil per unit current in another; the same either way round.

Self-inductance (L)

A coil's own flux linkage per unit current in it; electrical inertia.

Back emf

The self-induced emf −L dI/dt that opposes changes in current.

Magnetic energy density

Energy stored per unit volume in a magnetic field, B²/2μ₀.

AC generator

A machine that turns a coil (or magnets) to make an alternating emf NBAω sin ωt.

Armature

The rotating coil of a generator.

Slip rings and brushes

Sliding contacts that carry the generator coil's current to the outside circuit.