Electromagnetic Induction

 Introduction

Electromagnetic induction

Electromagnetic induction in the phenomenon of generating an e.m.f by changing the number of magnetic lines of force associated with a circuit. The e.m.f so generated is called induced e.m.f and the corresponding current is called induced current.

Magnetic flux ($\phi$)

The number of magnetic lines of force crossing a surface normally is called magnetic flux linked with the surface.

$\phi = \overrightarrow{B}. \overrightarrow{A} = BA \cos \theta$

Where ‘B’ is strength of magnetic field, ‘A’ is the area of the surface and is ‘θ’ the angle which normal to the surface area makes with the direction of the magnetic field.

The SI unit of $\phi$ is Weber. The cgs unit is Maxwell.

1 Weber= $10^8$ Maxwell. Also, Weber =1 Tesla meter$^2$

If a coil has N turns, then

$\phi =NBA \cos θ$

Case I: If the magnetic field $\overrightarrow{B}$ is normal to the plane of coil then,

$θ=0^{\circ}, \phi = NBA \cos 0^{\circ} = NBA$ (Maximum)

Case II: If $\overrightarrow{B}$ s parallel to the plane of the coil the $θ= 90^{\circ}$

$θ=90^{\circ}$

∴ $\phi$ =0 (Minimum)

The magnetic flux can be +ve, -ve or zero depending on the angle θ. Theflux is +ve for $0^{\circ}< θ>90^{\circ} and -ve for$180^{\circ} > θ>90^{\circ} $

The magnetic flux is taken as -ve if field lines enter the area and +ve if field lines leave the area.

The magnetic flux through a closed surface is always zero i.e.

$\oint \overrightarrow{B}. \overrightarrow{dA} = 0$

This is called Gauss’s theorem for magnetostatics.

Faraday’s law of Electromagnetic induction

First law

Whenever there is change in the magnetic flux linked with a coil, an induced e.m.f is produced in the coil. There is an induced current only when coil circuit is complete (i.e closed). The induced e.m.f lasts so long as the change in flux takes places.

Second law The magnitude of e.m.f induced in the circuit is directly proportional to the rate of change of magnetic flux linked in the circuit i.e

$E \propto \dfrac{d \phi}{dt} = \dfrac{\phi_2 - \phi_1}{t} = - \dfrac{d\phi}{dt}$

$\phi = NBA \cos \theta = NBA \cos \omega t$

$E= - \dfrac{d}{dt} (NBA \cos \omega t) =$

Cases

If area A alone is changing, then

E=-NB cosωt(dA/dt)

If the magnetic field B alone is changing, then

E=-NA cosωt(dB/dt)

If θ alone is changing

∴E=-NBA cosωt ((dcosωt))/dt =-NBAωsinωt

Thus, induced e.m.f depends on

i. Number of turns (N)

ii. Area of coil (A)

iii. Angular velocity (ω) of the coil

Emf induced in the coil if B, or A or θ is changing.

Emf induced in the coil doesn’t depend on resistance of coil.

Emf induced in the coil depends on relative motion of coil and electromagnet system i.e. it the coil or magnet is brought with the high speed towards than more emf will be induced in the coil and vice-versa.

Emf induced $\propto$speed $\propto$ 1/time

Current Induced in the coil (i)

Current induced $i =\dfrac{E}{R}$

∴$i = \dfrac{d \phi}{dt} \dfrac{1}{R}$

Current induced in the coil depends on the resistance of the coil and relative motion of coil and electromagnetic system.

∴current induced(i)∝$\dfrac{1}{R} \propto$ speed $\propto$ 1/(time taken)

Charge induced (q)

$q= idt = \dfrac{d \phi}{dt} \dfrac{1}{R} dt$

∴$i = \dfrac{d \phi}{R}$

Charge induced in the coil depends on resistance of coil and in independent of relative motion of coil and electromagnet system i.e., even a coil or magnet brought with high speed or less speed, amount of charge flow in the coil will be constant.

Len’s Law

The direction of induced e.m.f. is given by Lenz’s law.

According to this law, the direction of induced e.m.f. in a circuit is always such as to oppose the change in magnetic flux responsible to it.

Len’s law is in accordance with the principle of conversation of energy. In fact, work done in moving the magnet w.r.t, the coil change into electric energy producing induced current.

The direction of induced current

This is given by Fleming’s Right-Hand Rule. According to this rule, if we stretch the first figure, the central finger and thumb of our right hand is mutually perpendicular direction, such that first finger points along the magnetic field and thumb points along the direction of motion of conductor, then central figure would give the direction of induced current +ve terminal of battery.

Right Hand Palm Law

If four finger point the direction of $\overrightarrow{B}$ and thumb points the direction of velocity then palm gives the +ve terminal of battery/direction of induced current. [Fleming’s Left Hand Law gives the direction of force ]

Conducting rod moving in uniform magnetic field

When a conducting rod of length ‘l’ moves with the velocity ‘v’ in uniform magnetic field of induction ‘B’ such that ‘l’ makes angle θ with v, then the magnitude of the average induced emf is given by

E=vBlsinθ=B ($\overrightarrow{l} \times \overrightarrow{v}$)

Emf induced in the rod will be zero if out of B, l and v any one parallel to other

To move rod with constant speed a= 0

∴$F_{ext}-F_m=m \times 0= 0$

∴$F_{ext}=F_m$

$F_{ext}=BIl=F_m =B(\dfrac{E}{R})l=B(\dfrac{Bvl}{R})l$

∴$F_{ext}=(B^2 l^2 v)/R$

Mechanical power applied to pull the rod is given by P=(work done )/(time )

= F*V

∴P= $(B^2 l^2 v^2)/R$

Conducting rod rotating with angular velocity ω in a uniform magnetic field.

When a rod of length ‘l’ rotates with angular velocity ω in a uniform magnetic field B, the induced emf across the ends of rotating rod is:

$E=1/2 Bωl^2=1/2 B2πfl^2=Bπfl^2=BAf$

∴E=(Bωl^2)/2=BAf

Where $A=πl^2$) is area swept by the rod in the rotation and f is the frequency of rotation.

EMF induced in a close loop moving in a magnetic field

Emf induced in the rod Blv can be used only for straight conductor.

If a zig-zig wire or rod or loop is moving then, $E= Bvl_{\text{displacement}}$

If a closed loop of any shapr and size moving inside magnetic field then emf induced is closed loop will be zero provided that no load is applied across the closed-loop. (Since $l_\text{displacement}$ of closed loop is zero)

Once load is applied, cell will be in parallel, thus total emf induced will be 2rBv.

Eddy Current

Eddy currents are the currents induced in the body of a conductor when the amount of magnetic flux linked with the conductor changes.

For example, when we move a metal plate out of a magnetic field, the relative motion of the field and the conductor again induces a current in the conductor. The magnitude of eddy current is

i=(induced emf)/resistance=E/R=-(d∅/dt)/R

The direction of eddy current is given by Lenz’s law or Fleming’s right-hand rule.

Eddy current are produced when a metal is kept in a varying magnetic field.

Eddy current are basically the current induced in the body of a conductor due to change in magnetic flux induced with the conductor.

Application of Eddy currents

i. Electromagnetic damping is designing dead beat galvanometers.

ii. In induction furnace to melt the substance.

iii. As electromagnetic brakes in controlling the speed of electric trains.

iv. In induction AC motors.

v. In speedometers of automobiles and energy meters.

vi. Eddy current are also used in deep heat treatment of the human body.

Undesirable Effects

i. They oppose the relative motion.

ii. They involve loss of energy in the form of heat.

iii. The excessive heating may break the insulation in the appliances and reduce their life.

Self-Induction

In the property of a coil by virtue of v\ which the coil opposes any chance in the strength of current flowing through it by inducing an e.m.f in itself. ∴ ∅=LI

Where L is a constant called coefficient of self-induction or self-inductance of the coil.

For N turns of coil,∅=NLI

The emf induced in the coil is

E=-(d∅)/dt=(-d)/dt (LI)=-L dI/dt

∴=-L dI/dt

where E is the back e.m.f

The S.I unit of L is Henry or Weber/Ampere.

Energy Stored in Inductor

Energy density=(energy )/volume=$1/2 B^2/μ_0$

∴Energy stored by magnetic field

=$B^2/(2μ_0 )$.Volume

An inductor stores energy in the form of magnetic energy given by $E=1/2 LI^2$ For Solenoid, $B=μ_0 nI$

$E=(μ_0 nI)^2/(2μ_0 )* volume =1/2 (μ_0 n^2.V)I$