**ELECTROMAGNETIC INDUCTION**

Electromagnetic induction is a very important phenomenon of electromagnetism. It is a phenomenon in which EMF is produced in the coil, whenever magnetic flux linked with the coil changes. The produced EMF is called induced EMF and current so produced due to induced EMF is called induced current.

**MAGNETIC FLUX (Φ)**

*.It is the total no. of magnetic field lines passing perpendicularly through the surface , when it is placed in the magnetic field(B).

Φ = B.A = BAcosθ

B is the magnetic field , A is the surface area and θ is the angle between the direction of the magnetic field and the normal vector of the surface area.

*. The SI units of magnetic flux(φ) is weber (Wb).

*. Any one can change the magnetic flux by-

– changing the intensity of magnetic field.

– changing the coil orientation wrt to magnetic field.

– changing the surface area.

**FARADAY’S LAWS OF ELECTROMAGNETIC INDUCTION**

**FARADAY’S FIRST LAW**– whenever magnetic flux linked with the coil or loop of wire or any closed circuit changes, then there will always produce an EMF called induced EMF. This induced EMF will remains untill and unless magnetic flux keeps changing.

**FARADAY’S SECOND LAWS –**Second laws gives the magnitude of induced EMF. The induced EMF is directly proportional to the rate of change of magnetic flux. Mathematically it is given as-

Induced EMF = ε = -(dφ/dt) = -(φ2 – φ1)/t

**LENZ’S LAW**

*. According to lenz’s law the direction of the flow of induced current in the circuit is such a way that it opposes the change in magnetic flux that produce it.

*. It is used to find the direction of current.

*. It follows the rule of energy conservation.

**APPLICATIONS OF LENZ’S LAW**

*. When south pole of a bar magnet is used to move towards the coil , then the current induced in the coil is in clockwise direction.

*. When south pole of a bar magnet is used to move away from the coil then the direction of the induced current in the coil is anticlockwise.

*. When any current carrying coil is moved towards a stationary coil then the direction of the induced current is as follows:

*. When any current carrying coil is moved away from a stationary coil then the direction of the induced current is as follows:

*. When any current flowing through the straight wire changes then then the induced current in the nearby coil is in anticlockwise direction , as shown below:

**FLEMING RIGHT HAND RULE**

It is a rule which helps us to find the direction of induced current and EMF in a conductor. Which are in the effect of changing magnetic field.

**HOW TO APPLY THIS RULE**: Stretch the fingers of your right hand in such a way that the fore finger , central finger and the thumb are in the position of mutually perpendicular direction ,and the thumb are allign in the direction of the magnetic force or in the direction of the motion of the conductor.

Fore finger are in the direction of magnetic field , and the central finger are in the direction of induced current or EMF.

**MOTIONAL EMF**

*. If a conducting rod of length l is moving with velocity v on the two conducting parallel rails in uniform magnetic field (B) , as shown above.

x is the distance travelled by the rod of length l in time t.

Then magnetic flux is given as Φ = B.A

Where area A = xl= vtl ,. Φ = B.vlt

*. Induced EMF is given as , |ε|= (dφ/dt) = Blv

*. Induced current is I = ε/R = Blv/R

*. Magnetic force is F = lIB = l.(Blv/R).B=B²l²v/R

*. Power used in moving the conductor =

P = dW/dt = F×v = (B²l²v/R)×v = B²l²v²/R

*. Electrical power used =

P(thermal) =dH/dt =l²R =(Blv/R)^2.R = B²l²v²/R

*. If the rod is moving in vertical plane – then it achieve a maximum constant velocity v and it’s magnetic force F = mg.

F = mg =B²l²v/R , v = mgR/B²l²

*. If conducting rod performing rotational motion then motional EMF induced across the end is –

ε = (1/2)Bωl² =Bl²πυ = Bl²π/T

**EDDY CURRENTS**

*. It is the current induced in the bulk pieces of conductors , when magnetic flux linked with it changes is called eddy currents.

*. Eddy currents so produced is always perpendicular to the direction of the magnetic field.

*. It shows both heating and magnetic effect.

*. Magnitude of eddy currents is i = e/R.

*. It’s direction can be given by the lenz’s law or Fleming right hand rule.

**APPLICATION OF EDDY CURRENT**

*. It is used in electromagnetic damping.

*. Induction furnace

*. Electric power metre

*. Magnetic braking in electronic trains.

**INDUCTANCE**

*. It is a sacalr quantity and play the same role in electrical circuit as inertia played in mechanics.

*. It is the measure of the ratio of the total magnetic flux linkage to the current.ie.

L = N(φB)/I. Where φB is magnetic flux.

*. It’s dimensional formula is ML^2T^-2A^-2 and it’s unit is Henry denoted as H.

**SELF INDUCTION**

*. Self induction is the property of a coil by virtue of which it opposes any change in the strength of the current by inducing a EMF called back EMF.

*. When the current in the coil is switched ON , then the self induction opposes the growth of current. And when current in the coil is switched OFF , then self induction opposes the decrease in the current.

*. It is also called inertia of electricity.

*. The self induced EMF in the coil is e = -L(dI/dt)

*. The coefficient of self induction or self induction L is = N(φB)/I = NBA/I

*. Self induction of the long solenoid having cross sectional area A , length l and no. of turns N is = L = μ0N²A/l.

*. Self induction of the long solenoid having core of any other magnetic material μ is =

L = μ0μrN²A/l = μ0μrn²Al = μ0μrn²V

*. Self induction of the planar coil of radius R is =

μ0N²πR/2

**MUTUAL INDUCTION**

*. It is phenomenon of inducing a opposing EMF in secondary coil by changing the current or magnetic flux linked with the primary coil.

M₁₂ = N₁φ₁/I₂ , M₂₁ = N₂φ₂/I₁

*. The mutual induction of the two coil depends upon –

– geometry of the two coil ie. shape, size , numbers of turns , nature of materials on which two coils are bound.

– distance between the two coils.

– relative placement of the two coils ie. Orientation of the two coils.

*. Mutual induction of the two long coaxial solenoid is = M₁₂ = μ0n₁n₂Al,..M₂₁= μ0n₁n₂Al

*. Mutual induction can be given as:

M₁₂=M₂₁=M = μ0N₁N₂A/l

*. Mutual induction of two concentric and coplanar coil M(c1c2) = N₂B₁A₂/I₁

= μ0N₁N₂πr₂²/2r₁

*. Magnetic energy per unit volume can be given as: = uB = UB/V = B²/2μ0,. [UB is magnetic energy]

***. COMBINATION OF INDUCTANCE**

SERIES COMBINATION:

* Ls = L₁ + L₂ , if M = 0 , if M is not equal to 0 then,

Ls = L₁ + L₂ +-2M , + sign occurs when bindings are in same sense and – sign are used when bindings are in opposite sense.

*. PARALLEL COMBINATION

1/Lp = 1/L₁ + 1/L₂ if M = 0 , 1/Lp =L₁L₂ /L₁ + L₂

if M is not equal to 0 then,

1/Lp =L₁L₂- M^2/L₁ + L₂+-M

**ENERGY STORED IN THE INDUCTOR**

*.Magnetic energy stored in the inductor having I current is U = (1/2)LI^2.

*. Power consumed by the inductor = iL(di/dt)

*. Energy consumed in dt time = iL(di/dt)dt

**GROWTH OF CURRENT IN SERIES L-R CIRCUIT**

*. Equation of EMF in the circuit is=

e = lR + L(dI/dt)

*. When key is closed the current in the circuit increases exponentially with respect to the time.

The current in the circuit at any instant of time t is I = e(1-e^[-Rt/L])/R

*. The quantity L/R is called the time constant of the circuit and it is denoted as τ.

*. At one time constant , the current in the circuit is 63% of the final current.

*. If there is any change in the circuit containing inductor then there is no instantaneous effect in the flux of the inductor.

L1I1 = L2I2

**CURRENT DECAY IN L-R CIRCUIT**

*. EMF equation is given as :

lR + L(dI/dt) = 0

*. Current at any instant in the circuit is =

I = I0e^[-Rt/L]

*. Current in the circuit after one time constant

I = I0e^-1 = 0.37% of initial current.

**AC GENERATOR**

*. It is a device which is used to obtain the AC current or EMF by rotating the coil in the magnetic field with some external force.

*. Due to rotation of the coil flux linkage with coil continuously gets changing.

*. Due to change in magnetic flux EMF generated is |e| = dφ/dt = d(NBAcosωt)/dt= NBAωsinωt

*. e = NBAωsinωt = e0sinωt , e0 is the peak voltage.

*. Alternating current generated is :

I = e/R = (NBAω/R)sinωt= I0sinωt