Electromagnetic induction

 1. Electromagnetic Induction

The phenomenon of generation of induced emf and induced current due to change in magnetic field lines associated with a closed circuit is called electromagnetic induction.

2. Magnetic Flux

Magnetic flux through a surface of area A placed in a uniform magnetic field is φm=B⃗ .A⃗ =BAcosθ, θ being angle between B⃗  and normal to A⃗ . If magnetic field is not uniform, then φm=∫AB⃗ .dA⃗ , where integral extends for whole area A.

The SI unit of magnetic flux is weber. Magnetic flux is a scalar quantity; because of being scalar product of two vectors B⃗ andA⃗ .

3. Faraday’s Laws of Electromagnetic Induction

(i) Whenever there is a change in magnetic flux linked with a coil, an emf is induced in the coil. The induced emf is proportional to the rate of change of magnetic flux linked with the coil.

i.e., ε∝∆φ∆t

(ii) emf induced in the coil opposes the change in flux, i.e.,

ε∝–∆φ∆t⇒ε=–k∆φ∆t

where k is a constant of proportionality.

Negative sign represents opposition to change in flux.

In SI system φ is in weber, t in second, ε in volt, when k=1,ε=–∆φ∆t

If the coil has N-turns, then ε=–N∆φ∆t

4. Induced Current and Induced Charge

If a coil is closed and has resistance R, then current induced in the coil,

i=εR=–NR∆φ∆t

Induced charge, q=I∆t=–N∆φR=TotalfluxlinkageResistance

5. Lenz’s Law

It states that the direction of induced emf is such that it tends to produce a current which opposes the change in magnetic flux producing it.

6. EMF Induced in a Moving Conducting Rod

EMF induced in a conducting rod of length l moving with velocity v in a magnetic field of induction B, such that B, l and v are mutually perpendicular, is given by

ε = Bvl

force required to keep the rod in constant motion is F=BIL=B2l2vr

7. Self Induction

When the current in a coil is changed, an induced emf is produced in the same coil. This phenomenon is called self-induction. If L is self-inductance of coil, then

Nφ∝IorNφ=LI⇒L=NφI

L is also called coefficient of self induction.

The graph between effective magnetic flux (Nφ) and current I is straight line of slope self inductance L.

Also induced emf ε=–L∆I∆t

The unit of self inductance is henry (H). The self induction acts as inertia in electrical circuits; so it is also called electrical inertia.

The self inductance of a solenoid consisting core of relative permeability µr is L= µr µ0 n2Al

where n=Nl is the number of turns per metre length.

8. Mutual Induction

When two coils are placed nearby and the current in one coil (often called primary coil) is changed, the magnetic flux linked with the neighbouring coil (often called secondary coil) changes; due to which an emf is induced in the neighbouring coil. This effect is called the mutual induction. If M is mutual inductance of two coils, then φ2 I1 or φ2 =MI1

Definition of mutual inductance: M=φ2I1.

The mutual inductance of two coils is defined as the magnetic flux linked with the secondary coil when the current in primary coil is 1 ampere.

Also induced emf in secondary coil ε2=–M∆I1∆t⇒M=ε2∆I1/∆t.

The mutual inductance of two coils is defined as the emf induced in the secondary coil when the rate of change of current in the primary coil is 1 A /s.

The SI unit of mutual inductance is also henry (H). The mutual inductance of two coils does not depend on the fact which coil carries the current and in which coil emf is induced i.e., M12 =M21 = M

This is also called reciprocity theorem of mutual inductance.

If L1 and L2 are self-inductances of two coils with 100% flux linkage between them, then

M=L1L2−−−−−√, otherwise M=kL1L2−−−−−√, where k is coefficient of flux linkage between the coils.

Mutual Inductance of solenoid-coil system

M=μ0N1N2Al

where A is area of coil, l is length of solenoid, N1 is number of turns in solenoid and N2 is number of turns in coil.

9. Eddy Currents

When a thick piece of a conductor is placed in a varying magnetic field the magnetic flux linked with the conductor changes, so induced currents are induced in the body of conductor, which causes heating of conductor. The currents induced in the conductor are called the eddy currents. In varying magnetic field, the free electrons of conductor experience Lorentz force and traverse closed paths; which are equivalent to small current loops. These currents are the eddy currents; they cause heating effect and sometimes the conductor becomes red-hot.

Eddy current losses may be reduced by using laminated soft iron cores in galvanometers, transformers, etc. and making holes in the core. Few of the application of eddy currents is in induction, furnace, induction motor and many more.