Key Points - Electromagnetic Induction

- Magnetic Flux:
Magnetic flux through a plane of area dA placed in a uniform magnetic field B
φ = ∫ B.d A
If the surface is closed, then
φ = ∫ B.d A
This is because magnetic lines of force are closed lines and free magnetic poles do not exist.

- Faraday’s Law: a) First Law: whenever there is a change in the magnetic flux linked with a circuit with time, an induced emf is produced in the circuit which lasts as long as the change in magnetic flux continues. b) Second Law: According to this law,
? dφ ?
Induced emf, E ∝ ?
?
? dt ?

- Lenz’s Law:
The direction of the induced emf or current in the circuit is such that it opposes the cause due to which it is produced, so that,
? dφ ?
E = −N ?
?
? dt ?
Where N is the number of turns in coil
Lenz’s law is based on energy conservation.

- Induced EMF and Induced Current: a) Induced EMF, dφ
E = −N dt
N (φ2 − φ1 )
=− t b) Induced current,
E
N ? dφ ?
I = =− ?
?
R
R ? dt ?
N (φ2 − φ1 )
R t
Charge depends only on net change in flux does not depends on time.
Induced Emf due to Linear Motion of a Conducting Rod in a Uniform Magnetic Field
The induced emf,
=−

- 
E = −l.(vxB )
If e, v and B are perpendicular to each other, then
E = Bvl

- Induced EMF due to Rotation of a Conducting Rod in a Uniform Magnetic Field:
The induced emf,
1
E = Bω l 2 = Bπ nl 2 = BAn
2
Where n is the frequency of rotation of the conducting rod.

- Induced EMF due to Rotation of a Metallic Disc in a Uniform Magnetic Field:
1
EOA = Bω R 2 = Bπ R 2 n = BAn
2

- Induced EMF, Current and Energy Conservation in a Rectangular Loop Moving in a Non – Uniform Magnetic Field with a Constant Velocity: a) The net increase in flux crossing through the coil in time Δt is,
?φ = ( B2 − B1 )lv?t b) Induced emf in the coil is,
E = ( B1 − B2 )lv c) If the resistance of the coil is R, then the induced current in the coil is,
E ( B1 − B2 )
= lv
R
R d) Resultant force acting on the coil is
F = Il ( B1 − B2 )( towards left)
I= e) The work done against the resultant force l 2v 2
?t joule
R
Energy supplied in this process appears in the form of heat energy in the circuit. f) Energy supplied due to flow of current I in time Δt is,
H = I 2 R ?t
W = ( B1 − B2 ) 2
Or H = ( B1 − B2 ) 2 l 2v 2
?t joule
R
Or H = W

- Rotation of Rectangular Coil in a Uniform Magnetic Field: a) Magnetic flux linked with coil
φ = BAN cosθ
=BAN cosω t b) Induced emf in the coil dφ
E=
= BAN ω sin ωt = E0 sin ωt dt c) Induced current in the coil.
I=
E BAN ω
= sin ωt
R
R
E0 sin ωt
R d) Both Emf and current induced in the coil are alternating

- Self-Induction and Self Inductance: a) The phenomenon in which an induced emf is produced by changing the current in a coil is called self in induction.
=
φ ∝ I or φ = LI
φ or L=
I dI dt
E
L=
−(dI / dt )
E = −L where L is a constant, called self inductance or coefficient of self – induction. b) Self inductance of a circular coil
L=
µ0 N 2π R
=
µ0 N 2 A
2
2. c) Self inductance of a solenoid
L=
µ0N 2 A l d) Two coils of self – inductances L1 and L2, placed far away (i.e., without coupling) from each other. i)
For series combination:
L = L1 + L2 . .... Ln ii)
For parallel combination:
1.1 1
1
= + + .... +
L L1 L2
Ln

- Mutual Induction and Mutual Inductance: a) On changing the current in one coil, if the magnetic flux linked with a second coil changes and induced emf is produced in that coil, then this phenomenon is called mutual induction.
φ2 ∝ I1 or φ2 = MI1
Or M =
E2 = −
φ2
I1 d φ2 dI
= −M 1 dt dt
M=
E2
−( dI1 / dt )
Therefore, M12 = M21 = M b) Mutual inductance two coaxial solenoids
µNN A
M= 0 1 2 l c) If two coils of self- inductance L1 and L2 are wound over each other, the mutual inductance is,
M = K L1 L2
Where K is called coupling constant. d) Mutual inductance for two coils wound in same direction and connected in series
L = L1 + L2 + 2 M e) Mutual inductance for two coils wound in opposite direction and connected in series
L = L1 + L2 − 2 M f) Mutual inductance for two coils in parallel
L=
L1 L2 − M 2
L1 + L2 ± 2 M
Energy Stored in an Inductor:
1
U B = LI 2 max
2
Magnetic Energy Density:
UB =
B2
2.µ0

- Eddy Current:
When a conductor is moved in a magnetic field, induced currents are generated in the whole volume of the conductor. These currents are called eddy currents.

- Transformer: a) It is a device which changes the magnitude of alternating voltage or current.
Es ns
=
=K
Ep np b) For ideal transformer:
I p ns
=
Is np c) In an ideal transformer:
E p I p = Es I s d) In step – up transformer: ns > n p or K > 1
E s > E p and I s < I p
e) In step – down transformer: ns < n p or K < 1
E s < E p and I s > I p f) Efficiency
EI
η = s s x100%
Ep I p

- Generator or Dynamo:
It is a device by which mechanical energy is converted into electrical energy. It is based on the principle of electromagnetic induction.

- Different Types of Generator: a) AC Generator
It consists of field magnet, armature, slip rings and brushes. b) DC Generator
It consists of field magnet, armature, commutator and brushes.

- Motor: It is a device which converts electrical energy into mechanical energy.
Back emf e ∝ ω
Current flowing in the coil,
E − eb ia =
R
E = eb + ia R
Where R is the resistance of the coil.
Out put Power = ia eb
Efficiency,
η= eb
×100%
E