Electromagnetic Induction (EMI)

Electromagnetic induction is a fundamental principle in physics that describes generating an electromotive force (emf) or voltage in a conductor when exposed to a varying magnetic field. Michael Faraday first discovered this phenomenon in the 1830s, and it has shaped many electrical devices and technologies. Electromagnetic induction is a cornerstone of modern electrical engineering and physics, forming the basis of many devices and technologies that are integral to daily life and industrial processes.

1.0Magnetic Flux

• The magnetic flux through a surface placed in a magnetic field is defined as the total number of magnetic field lines passing through that surface.
•  Magnetic flux, $\varphi =\vec{B}\cdot \vec{A}=\mathrm{BA}\cos \theta$
• If a coil has more than one turn,  the flux through the whole coil is the sum of the flux through the individual turn. If the magnetic field is uniform, the flux through one turn is, $\varphi =\vec{B}\cdot \vec{A}=\mathrm{BA}\cos \theta$
• for N turns, the total flux linkage, $\varphi =\mathrm{NBA}$

• Magnetic flux is a scalar quantity
• S I unit-Weber(Wb)
• CGS Unit-Maxwell(Mx)
• Dimensional Formula-[ML²T^-2A^-1]
• 1 Wb = 10⁸ Mx
• For a  Non- Uniform Magnetic Field Magnetic Flux is given by, $\varphi =\int \vec{B}\cdot \overrightarrow{dA}$
• Gauss Law in Magnetism,Net magnetic flux through closed surfaces is always zero, $\varphi =\int \vec{B}\cdot \overrightarrow{dA}=0$
• Since incoming field lines = Outgoing Field Lines
• Net magnetic flux through a closed surface is zero
• Magnetic monopoles do not exist
• The Rate of change of flux is given by

1. Instantaneous Rate $=\frac{d\varphi }{dt}$
2. Average Rate $=\frac{\Delta \varphi }{\Delta t}=\frac{{\varphi }_{\text{Final }}-{\varphi }_{\text{Initial }}}{\Delta t}$

2.0Faraday’s Law of Induction and Lenz Law Experiment

• When the magnet is held stationary anywhere near or inside the coil, the galvanometer does not show any deflection.

• When the North pole of a powerful bar magnet approaches the coil, the galvanometer indicates a deflection to the right of the zero mark.

•  When the N-pole of a strong bar magnet is moved away from the coil, the galvanometer shows a deflection left to the zero mark.

• If the same experiments are conducted by approaching or moving away the South pole of the magnet towards the coil, the direction of the current in the coil is opposite to that observed in the case of the North pole.

• The deflection in the galvanometer is more when the magnet moves faster and less when the magnet moves slower.

The Conclusion of  the Experiment

• Whenever magnetic flux changes through the coil over  time an EMF is induced in it.
• An electromotive force (EMF) is generated whenever there is relative motion between the magnetic source (magnet) and the coil. When the magnet and coil approach each other, the magnetic flux linked with the coil increases, inducing an EMF. Conversely, when they move apart, the magnetic flux linked with the coil decreases, again inducing an EMF. This induced EMF persists as long as there is a change in the magnetic flux. 
• Due to this emf, an electric current starts to flow, and the galvanometer shows deflection.
• The deflection in the galvanometer lasts as long as the relative motion between the magnet and coil continues.

3.0Faraday’s Law

Faraday’s First Law-”Whenever a conductor is placed in a varying magnetic field,an electromotive force is induced.

Faraday’s Second Law-The induced emf in a coil is equal to the rate of change of flux linkage.

$|e|=\frac{d\varphi }{dt}$

Induced EMF $\propto$ Relative Velocity

4.0Direction of Induced Current 

Lenz Law-The induced EMF's polarity is such that it generates a current opposing the change in magnetic flux responsible for its creation.

$\mathrm{e}=-\frac{d\varphi }{dt}$ (Negative sign denotes opposition)

• If $\varphi$ - t curve is given as the negative of its slope it will give us induced EMF. 

e = -(Slope of $\varphi$ - t curve)

• Lenz’s Law follows the Conservation of Energy.

5.0Induced Parameters

1. Induced emf (e)
2. Induced current (I)
3. Induced charge (q)                                                           
4. Induced heat (H)
5.  Induced electric field (E_in)

• Instantaneous induced emf ($\mathrm{e}$)$={\lim }_{\Delta t\to 0}\left(\frac{-\Delta \varphi }{\Delta t}\right)=\frac{-d\varphi }{dt}$
• Induced Current at Instant, $\mathrm{I}=\frac{e}{R}=\frac{-1}{R}\left(\frac{d\varphi }{dt}\right)$
• In small Time intervals dt, induced charge, $\mathrm{dq}=-\frac{d\varphi }{R}$
• Induced Heat, $\mathrm{H}={\int }_0^t{I}^{2}Rdt={\int }_0^t\frac{e^2}{R}dt$

6.0Lenz Law and Conservation of Energy

1. When the North Pole of the Bar magnet comes towards the coil, it experiences a repulsive force due to which its speed will decrease.

2. To move the magnet towards the coil with constant speed, some part of mechanical work has to be done to overcome the force of repulsion.

3. This mechanical work is converted into electrical energy.

4. This electrical energy is converted into heat energy due to Joule’s Effect.

Here, kinetic energy gets converted into electrical energy which further gets, converted into thermal energy and given by: $\frac{1}{2}\mathrm{M}{v}^2=i^{2}R\Delta t=ms\Delta \theta$

7.0Types of EMI

1. Static EMI-If (A, $\theta$) $\to$ Constant and $\frac{dB}{dt}\to \frac{d\varphi }{dt}$

$\Rightarrow \frac{dI}{dt}\to \frac{dB}{dt}\to \frac{d\varphi }{dt}$

2.  Dynamic EMI- If (B, $\theta$) $\to$ Constant and $\frac{dA}{dt}\to \frac{d\varphi }{dt}$

Periodic EMI- If (A,B)Constant and $\frac{d\theta }{dt}\to \frac{d\varphi }{dt}$

8.0Self Induction(L)

When current through the coil changes, with respect to time then magnetic flux linked with the coil also changes with respect to time. Due to this an emf and a current is induced in the coil. According to Lenz law, induced current opposes the change in magnetic flux. This phenomenon is called self-induction and a factor by virtue of which the coil shows opposition to change in magnetic flux called self-inductance of the coil. 

Case 1. Current through  the coil is constant: $\mathrm{L}=\frac{N\varphi }{I}=\frac{NBA}{I}=\frac{{\varphi }_{\text{Total }}}{I}$

Case 2. Induced EMF in Self Induction: $e_s=\frac{-LdI}{dt}$

• Self Inductance is a scalar quantity
• SI Unit- Henry
• Dimensional Formula- [ML²T^-2A^-2]
• L depends on : Geometry of inductor and Medium($\mu ={\mu }_0{\mu }_r$)

Self Inductance of Circular Coil

$L=\frac{N\varphi }{I}=\frac{{\mu }_0{N}^2\pi R}{2}$

Self Inductance of Solenoid

$\mathrm{L}={\mu }_0\left(\frac{N^2}{l^2}\right)(lA)={\mu }_0{n}^{2}Al={\mu }_0{n}^{2}V$

9.0Power in an Inductor

P = VI = EI

$\mathrm{P}=\mathrm{LI}\frac{dI}{dt}+I^{2}R$

10.0L-R Circuit Analysis

Case 1- Current Growth

• $\mathrm{I}=I_0\left(1-e^{-\frac{t}{\lambda }}\right)$ at t = 0, where $I_0=\frac{E}{R}$ and $\lambda =\frac{L}{R}(\lambda -\text{ Time Constant })$
• Just after closing the key t = 0, inductor behaves like open circuit so, I = 0
• After some time, the inductor behaves like a simple connecting wire (short circuit) and current in the circuit is constant. $I_0=\frac{E}{R}$
• Time Constant of circuit $(\lambda )$ - It is a time in which current increases upto 63% or 0.63 times the peak value.Its S I unit is second. $\lambda =\frac{L}{R}\sec$
• Half-Life (T)- It is a time upto which current increases upto 50% or 0.50 times of peak current value. $\mathrm{T}=0.693\lambda =0.693\frac{L}{R}$
• Rate of Growth of Current at any instant. $\frac{dI}{dt}=\frac{E}{L}\left(e^{-\frac{t}{\lambda }}\right)$
• For $\mathrm{t}=0\Rightarrow {\left(\frac{dI}{dt}\right)}_{\max }=\frac{E}{L}$
• For $\mathrm{t}=\infty \Rightarrow {\left(\frac{dI}{dt}\right)}_{\text{min }}\to 0$
• The Maximum or initial value of rate of growth of current does not depend upon the resistance of the coil.

Case-2 Current Decay

• $\mathrm{I}=I_0{e}^{-\frac{Rt}{L}}=I_0{e}^{-\frac{t}{\lambda }}$
• For $\mathrm{t}=0\Rightarrow \mathrm{I}=I_0=\frac{E}{R}$
• For t $\to \infty \Rightarrow \mid \to 0$

Time Constant ($\lambda$) - It is a time in which the current decreases upto 37% or 0.37 times of the peak current value. $\lambda =\frac{L}{R}\sec$

Rate of Decay of current at any instant- $\frac{-dI}{dt}=\frac{E}{L}{e}^{-\frac{t}{\lambda }}$

• For $\mathrm{t}=0\Rightarrow {\left(\frac{-dI}{dt}\right)}_{\max }=\frac{E}{L}$
• For $t\to \infty \Rightarrow {\left(\frac{-dI}{dt}\right)}_{\min }\to 0$

11.0Combination of Inductors

Series Combination

$L_{eq}=L_1+L_2$

Parallel Combination

$\frac{1}{L_{eq}}=\frac{1}{L_1}+\frac{1}{L_2}$

Note-If an inductor is cut into two parts ,its time constant remains the same.

12.0Mutual Induction(M)

Whenever current is passing through the primary coil or circuit, changes with respect to time then magnetic flux in neighboring secondary coil or circuit will also change with respect to time. According to Lenz Law for opposition of flux change an emf and a current induced in the neighboring coil or circuit. This phenomenon is called 'Mutual induction'.

Note-Units and Dimensions of M is same as L ( self Induction)

• M is a scalar quantity
• M does not depend on current through the primary or flux through the secondary coil
• M is the combined property of the primary and secondary coil and is the same for both coils

Mutual inductance of two coaxials Solenoids.

$M_{S_1{S}_2}=\left(\frac{{\mu }_0{N}_1{N}_{2}A}{l}\right)$

Mutual Inductance of Two Concentric and Coplanar Coils

$M_{C_1{C}_2}=\left(\frac{{\mu }_0{N}_1{N}_2\pi r_2^2}{2r_1}\right)$

Mutual inductance in terms of self Inductance $\mathrm{M}=K\sqrt{L_1{L}_2}$ , K is the coupling factor

Induced EMF in Mutual Induction

• $e_2=-\mathrm{M}\left(\frac{d{I}_1}{dt}\right)$
• e₂ = induced emf in the secondary coil
• $\left(\frac{d{I}_1}{dt}\right)$= rate of change of current in the primary coil.
  

Dynamic EMI( Motional EMF)

• A conductor PQ is placed in a uniform magnetic field B, directed normal to the plane of paper outwards. PQ is moved with a velocity v, the free electrons of PQ also move with the same velocity. The electrons experience a magnetic Lorentz force, $F_m=-e(\vec{v}\times \vec{B})$
• Force exerted by electric field on free electrons

${\vec{F}}_e=-\mathrm{eE}$

$\Rightarrow {\mathrm{F}}_{\mathrm{m}}=-\overrightarrow{F_e}$

$\Rightarrow \vec{E}=-(\vec{v}\times \vec{B})$

• The potential difference between the ends P and Q is 

$\mathrm{V}=-\vec{E}\cdot \vec{l}=(\vec{v}\times \vec{B})\cdot \vec{l}$

• Since this EMF is generated by the movement of a conductor, it is referred to as motional EMF.

Dynamic Induction due to Rotation of Rod

$\epsilon =\frac{1}{2}B\omega l^2$

Dynamic induction due to Rotation of cycle Wheel Spokes

${\mathrm{e}}_{\mathrm{net}}=\frac{B\omega l^2}{2}$

Dynamic Induction due to Rotation of Disc

• $\mathrm{e}=\frac{B\omega r^2}{2}$
• P.D. between center and circumference, $\mathrm{e}=\frac{B\omega r^2}{2}$
• P.D. between two point on  circumference, e = 0

13.0Eddy Currents

• Eddy currents are the current induced in the body of a conductor when the magnetic flux linked with the conductor changes.
• It is a group of induced currents which are produced when metal bodies placed in a time varying magnetic field or they move in an external magnetic field in such a way that flux through them changes with respect to time.
• Some of the applications of eddy currents are : Electromagnetic damping, induction furnace, 
• Eddy currents have several adverse effects: they resist motion relative to the magnetic field, result in energy loss as heat, and contribute to a reduction in the lifespan of electrical devices.
• To minimize eddy currents, laminated cores are used in a transformer

14.0Induced Electric Field

• An electric field can apply force on charge at rest, a time varying magnetic field will produce an induced electric field, and this induced electric field will exert force on free electrons at rest and current is produced.
• When magnetic field changes with time in region then an electric field induces within and outside the region
• Unlike electro-static fields, these induced electric field lines always form closed loops and are non-conservative in nature.
• For Induced Electric Field: $\oint \overrightarrow{E_{in}}\cdot \vec{d}l=\frac{-d\varphi }{dt}$
• Induced electric field

1. Inside point (r<R)

$\mathrm{E}=\frac{rdB}{2dt}\Rightarrow \left(E_{\text{inside }}\propto \mathrm{r}\right)$

2. Outside Point (r>R)

$\mathrm{E}=\frac{R^{2}dB}{2rdt}\Rightarrow E_{\text{out }}\propto \frac{1}{r}$

3. Surface point (r=R)

$E=\frac{RdB}{2dt}$

15.0Variation of induced Electric Field with Distance

16.0 Periodic EMI

• When a coil, which is placed in a uniform magnetic field, rotates with constant angular frequency about the shown axis then magnetic flux through the coil changes periodically with respect to time so an emf of periodic nature induced in the coil. This phenomenon is known as periodic emi.

• $\varphi$ = NBA Cos $\omega$t, magnetic flux through rotating coil at any instant t
• e = NBA$\omega$ Sin $\omega$t, induced emf in rotating coil
• I = I₀ Sin $\omega$t, induced current in load circuit

17.0Sample Questions on EMI

Q-1 A rectangular loop of area 0.06 m² is placed in a magnetic field 1.2 T with its plane inclined 30° to the field direction. Find the flux linked with the plane of the loop.

Solution:    

Area of loop A = 0.06 m², B = 1.2 T and = 90° – 30° = 60°

So, the flux linked with the loop is

$\varphi =\mathrm{BA}\mathrm{Cos}\theta =1.2\times 0.06\times cos60°=1.2\times 0.06\times \frac{1}{2}=0.036\mathrm{Wb}$

Q-2. A conducting circular loop is placed in a uniform magnetic field 0.04 T with its plane perpendicular to the magnetic field. The radius of the loop starts shrinking at 2 mm/s. Find the induced emf in the loop when the radius is 2 cm.

Solution:

$ϵ=\frac{d\varphi }{dt}=\frac{d(B)\pi r^2}{dt}=2\pi rB\left(\frac{dr}{dt}\right)$

$=(2\pi )\left(2\times 10^{-2}\right)\left(2\times 10^{-3}\right)(0.04)=3.2\pi \mu \mathrm{V}$

Q-3 A bar magnet is moving towards a circular coil with a kinetic energy of 1180 J.If mass of the silver ring is 1 Kg and its specific heat is 236 J/kg ^oC then find rise in temperature of the ring.

Solution: 

$\text{ K.E }=ms\Delta \theta \Rightarrow 1180=(1)(236)\Delta \theta$

$\Delta \theta =5$, So, the temperature rise will be 5°C.

Q-4 Self inductance of two coils are 2H & 8H. If 50% flux of primary coil is linked with secondary coil, then find the coefficient of mutual inductance.

Solution:

$\mathrm{M}=K\sqrt{L_1{L}_2}$

$M=\frac{1}{2}\sqrt{(2)(8)}$

M = 2 H

Q-5. How does the mutual inductance of a pair of coils change when :

• the distance between the coils is increased?
• the number of turns in each coil is decreased?
• a thin Iron rod is placed between the two coils, other factors remaining the  same? Justify your answer in each case.

Solution:

(i) The mutual inductance of two coils decreases when the distance between them is increased. This is because the flux passing from one coil to another decreases.

(ii) Mutual inductance $M=\left(\frac{{\mu }_0{N}_1{N}_{2}A}{l}\right)\Rightarrow M\propto N_1{N}_2$      

Clearly, when the number of turns N₁ and N₂ in the two coils is decreased, the mutual inductance decreases.

(iii) When an iron rod is placed between the two coils the mutual inductance increases, because $M\propto permeability(\mu ).$