Electromagnetic Induction Class 12 Notes CBSE Physics Chapter 6 (Free PDF Download)

Section-A (1 Mark Questions)

1. A metallic wire coil is stationary in a uniform magnetic field. What is the emf induced in the coil

Ans. No emf is induced in the coil as there is no change in the magnetic flux linked with the coil.

2. Why does metallic piece become very hot when it is surrounded by a coil carrying high frequency (H.F) alternating current?

Ans. When a metallic piece is surrounded by a coil carrying high frequency (H.F) alternating current, it becomes hot because eddy currents are produced which in turn produces joule’s heating effect.

3. The induced emf is also called back emf. Why?

Ans. It is because induced emf produced in a circuit always opposes the cause which produces it.

4. The electric current flowing in a wire in the direction from B to A Find out the direction of the induced current in the metallic loop kept the wire as shown in the figure.

Ans. According to Lenz’s law, the direction of induced current will oppose the cause of its production. So, the current in loop will induce in such a way that it will support the current flowing in the wire i.e., in the same direction. So, the direction of current in the loop will be clockwise.

5. Two spherical bobs, one metallic and the other of glass, of the same size are allowed to fall freely from the same height above the ground. Which of the two would reach earlier and why?

Ans. The glass bob will reach earlier on ground as acceleration due to gravity is independent of mass of the falling bodies. Being insulator, no induced current is developed in it due to the earth’s magnetic field.

6. The closed loop PQRS of wire is moved into a uniform magnetic field at right angles to the plane of the paper as shown in the figure. Predict the direction of the induced current in the loop.

Ans. Since, magnetic flux increasing when the loop moves into uniform magnetic field. So, the induced current should oppose this increase. Thus, flow will be from QPSRQ. i.e. anti-clockwise.

7. State the Faraday’s law of electromagnetic induction.

Ans. First law:  Whenever a conductor is placed in a varying magnetic field, EMF induces and this emf is called an induced emf and if the conductor is a closed circuit than the induced current flows through it.

Second law: The magnitude of the induced EMF is equal to the rate of change of flux linkages.

8. Write S.I. unit of magnetic flux. Is it a scalar or a vector?

Ans. SI unit of Magnetic flux is Weber. It is a Scalar quantity.

9. On what factors does the self-inductance of a solenoid depends?

Ans. Factors on which self-inductance depends:

(a) No of turns in the solenoid 

(b) Length of the solenoid 

(c) Core inside the solenoid

(d) Area of the cross-section.

10. Define self–induction of a coil. Give one example.

Ans. The phenomenon of electric current in a coil due to growth or decay of current in the coil.

Example: Choke Coil. 

Section-B (2 Marks Questions)

11. A bar magnet falls from a height ‘h’ through a metal ring. Will its acceleration be equal to g? Give reason for your answer.

Ans. Acceleration will be less than ‘g’ because of induction of current in the coil as stated by Lenz’s Law which opposes the falling of magnet.

12. If the change in flux through the loop in time △t is △Φ and the resistance of loop be ‘r’. Find out the amount of charge flowing through the conductor in time △t.

Ans. $\varepsilon =-\dfrac{\Delta \theta }{\Delta t}$

current(I)= $\dfrac{\left | \varepsilon  \right |}{r}=\dfrac{1}{r}\dfrac{\Delta \theta }{\Delta t}$

or, $\dfrac{\Delta Q}{\Delta T}=\dfrac{1}{r}\dfrac{\Delta \phi }{\Delta t}$

or, $\Delta Q=\dfrac{\Delta \phi }{r}$ .

13. Mention one advantage of eddy current. 

Ans. Magnetic braking in trains: Strong electromagnets are situated above the rails in some electrically powered trains. When the electromagnets are activated, the eddy currents induced in the rails oppose the motion of the train. As there are no mechanical linkages, the braking effect is smooth.

14. Name two scientists who concluded electric currents were induced in closed coils when subjected to changing magnetic fields by performing experiments. 

Ans. Michael Faraday in England and Joseph Henry in USA. 

15. State Lenz’s Law. On which fundamental principle is the law based upon. 

Ans. The polarity of induced emf is such that it tends to produce a current which opposes the change in magnetic flux that produced it

16. Suppose we bring a bar magnet near a coil and the North Pole of the bar magnet is facing the coil. In what direction the current is induced in the coil?

Ans. Current will be induced in the anticlockwise direction in the coil. 

PDF Summary - Class 12 Physics Electromagnetic Induction Notes (Chapter 6)

1. Magnetic Flux:

1. The magnetic flux associated with an area placed in a magnetic field is equal to the total number of magnetic lines of force passing naturally through that area.

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2. Net flux through the surface $\phi=\oint \overrightarrow{\mathrm{B}} \cdot \mathrm{d} \overrightarrow{\mathrm{A}}=\mathrm{B} \mathrm{A} \cos \theta$

( $\theta$ is the angle between area vector and magnetic field vector) 

If $\theta=0^{\circ}$ then $\phi=B A$, If $\tau$ $\theta=90^{\circ}$ then $\phi=0$

3. Unit and Dimension i Magnetic flux is a scalar quantity. It's S.I. the unit is Weber (Wb), CGS unit is Maxwell or Gauss $\times \mathrm{cm}^{2}$ ($1 w b=108$ Maxwell).

4. Other units: 

$\mathrm{Tesla} \times \mathrm{m}^{2}=\frac{\mathrm{N} \times \mathrm{m}}{\text { Amp }}=\frac{\text { Joule }}{\text { Amp }}=\frac{\text { Volt } \times \text { Coulomb }}{\text { Amp }}$

$=$ Volt $\times$ sec $=\mathrm{ohm} \times$ Coulomb $=$ Henry $\times$ Ampere

It's dimensional formula, $[\phi]=\left[\mathrm{ML}^{2} \mathrm{~T}^{-2} \mathrm{~A}^{-1}\right]$

2. Faraday's Laws of Emi

(1) First law: 

An induced emf is formed in a circuit whenever the number of magnetic lines of force (magnetic flux) travelling through it changes. The induced emf lasts only as long as the flux is changing or being cut.

(2) Second law: 

The induced emf is calculated from the rate of change of magnetic flux in the circuit, i.e. $\mathrm{e}=-\frac{\mathrm{d} \phi}{\mathrm{dt}} \cdot \operatorname{For} \mathrm{N}$ turns $\mathrm{e}=-\frac{\mathrm{Nd} \phi}{\mathrm{dt}}$; Negative sign indicates that induced $\operatorname{emf}(e)$ opposes the change of flux.

Various Methods of Producing induced E.M.F.

We have learnt that e.m.f. is induced in a circuit, whenever the amount of magnetic flux linked with the circuit is changed. As $\phi=B A \cos \theta$, the magnetic flux $\phi$ can be changed by changing $B, A$ or $\theta$ 

Hence there are three methods of producing induced e.m.f

1. By adjusting the magnetic field B's magnitude,

2. By altering region $A$, i.e., shrinking, stretching, or modifying the coil's shape.

3. By changing the angle between the direction of $B$ and the normal to the surface area $A$, i.e. modifying the surface area and magnetic field's relative orientation.

3. Lenz's Law

The induced emf/induced current direction is determined by this law.

The direction of induced emf or current in a circuit, according to this law, is such that it opposes the source that generates it. The law of conservation of energy underpins this rule.

(1) When the $N$-pole of a bar magnet advances towards the coil, the flux associated with the loop increases, causing an emf. Induced current flows through the loop circuit since it is closed.

(2) Because the approaching north pole is the cause of this induced current, the induced current in the loop is directed in such a way that the front face of the loop behaves like the north pole. Therefore induced current as seen by the observer $O$ is an anticlockwise direction.

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(3) The cause of generated emf in the coil can also be referred to as relative motion if the loop is free to move. As a result, the relative motion between the two objects works against the cause.

The loop and the incoming magnet should be in opposition. As a result, the loop will begin to move in the direction of the magnet is moving.

(4) It is critical to keep in mind that whenever the reason of induced the new motion is always in the direction of the emf.

4. Eddy Current

When a changing magnetic flux is given to a large piece of conducting material, it induces circling currents known as eddy currents. Eddy currents have huge magnitudes and heat up the conductor because the bulk conductor's resistance is usually low.

(1) These are circulating currents, similar to water eddies.

(2) The "Foucault current" is named after Focault's experimental hypothesis.

(3) In a metallic block, the generation of eddy currents results in the loss of electric energy in the form of heat.

(4) As a result of the lamination and slotting processes, the resistance channel for eddy current circulation increases, weakening and lowering them and also reducing losses caused by them.

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Eddy Current Applications: 

Although eddy currents are generally unwelcome, they do have some helpful applications, as listed below.

(i) Dead-Beat Galvanometer: 

When a current is delivered via its coil, a deadbeat galvanometer's pointer comes to rest in the final equilibrium position instantaneously, with no oscillation around the equilibrium position.

This is accomplished by winding the coil around a metallic frame, which induces significant eddy currents that give electromagnetic damping.

(ii) When the train is running, the wheel moves in the air; when the train is stopped by electric brakes, the wheel is made to move in an electromagnet created field. Eddy currents created in the wheels as a result of the changing flux work against the cause and bring the train to a halt.

(iii) Induction Furnace: 

The heat of Joule causes a metal item to melt when it is placed in a rapidly changing magnetic field.

(iv) Speedometer: 

In an automobile's speedometer, a magnet is geared to the vehicle's main shaft and rotates in accordance with the vehicle's speed. Hair springs are used to secure the magnet in an aluminium cylinder. When the magnet rotates, it produces eddy currents in the drum and drags it through an angle, which indicates the speed of the vehicle on a calibrated scale.

(v) Energy Metre: 

The armature coil of an energy meter has a metallic aluminium disc that rotates between the poles of a pair of permanent horseshoe magnets. The current induced in the disc as the armature spins tend to oppose the motion of the armature coil. Deflection is proportional to the energy consumed due to this braking effect.

5. Induced Charge Flow

When a current is induced in the circuit due to the flux change, charge flows through the circuit and the net amount of charge which flows along the circuit is given as: 

$\mathrm{q}=\int \mathrm{idt}=\int \frac{1}{\mathrm{R}}\left|\frac{\mathrm{d} \phi}{\mathrm{dt}}\right| \mathrm{dt}=\frac{1}{\mathrm{R}} \int \mathrm{d} \phi$ $\mathrm{q}=\frac{|\Delta \phi|}{\mathrm{R}}$ and $\mathrm{q}=\mathrm{N} \frac{|\Delta \phi|}{\mathrm{R}}$ for $\mathrm{N}$ tums.

6. Induced Electric Field

It is non-conservative and non-electrostatic in nature. Its field lines are concentric circular closed curves.

A time varying magnetic field dt dB always produces an induced electric field in all space surrounding it. Induced electric field $\left(E_{i n}\right)$ is directly proportional to induced emf so $e=\oint \vec{E}_{\text {in }} \cdot \overrightarrow{d \bar{\ell}} \quad \ldots$ (i)

From Faraday's second laws $\quad \mathrm{e}=-\frac{\mathrm{d} \phi}{\mathrm{dt}} \quad$ wu (ii)

From (i) and (ii) 

$\quad \mathrm{e}=\oint \overrightarrow{\mathrm{E}}_{\mathrm{in}} \cdot \mathrm{d} \overrightarrow{\mathrm{d}}=-\frac{\mathrm{d} \phi}{\mathrm{dt}}$. 

This is known as an integral form of Faraday's laws of EMI.

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A uniform but time-varying magnetic field $B(t)$ exists in a circular region of radius ' $a$ ' that is directed into the plane of the paper as shown, and the magnitude of the induced electric field $\left(E_{\text {in }}\right.$ ) at point $P$, which is located at a distance $r$ from the circular region's centre, is calculated as follows: 

$\oint \vec{E}_{\text {in }} \cdot \overrightarrow{\mathrm{d}} \vec{\ell}=\mathrm{e}=\frac{\mathrm{d} \phi}{\mathrm{dt}}=\mathrm{A} \frac{\mathrm{dB}}{\mathrm{dt}}$

i.e. \[\text{E}(2\pi r)=\pi {{\text{a}}^{2}}\frac{\text{dB}}{\text{dt}}\] 

where \[r\ge a\text{ }\!\!~\!\!\text{ or }\!\!~\!\!\text{ }E=\frac{{{a}^{2}}}{2r}\frac{dB}{dt};{{E}_{\text{in }\!\!~\!\!\text{ }}}\propto \frac{1}{r}\] 

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$\text { when } \mathrm{r}<\mathrm{a} ; \mathrm{E}=\frac{\mathrm{r}}{2} \frac{\mathrm{dB}}{\mathrm{dt}} ; \mathrm{E}_{\mathrm{n}} \propto \mathrm{r}$

7. Dynamic (motional) Emi Due to Translatory Motion:

(1) Consider a conducting rod of length $l$ moving with $a_{\infty}$ uniform velocity $\vec{v}$ perpendicular to a uniform magnetic field $\vec{B}$, directed into the plane of the paper. Let the rod be moving to the right as shown in the figure. The conducting electrons also move to the right as they are trapped within the rod.

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Conducting electrons experiences a magnetic force $F_{m}=e v B$. So they move from $P$ to $Q$ within the rod. The end $P$ of the rod becomes positively charged while end $Q$ becomes negatively charged, hence an electric field is set up within the rod which opposes the further downward movement of electrons i.e. an equilibrium is reached and in equilibrium $F_{c}=F_{m}$ i.e. $e E=e v B$ or $E=v B$ Induced $\operatorname{emf} \mathrm{e}=\mathrm{E} l=\mathrm{Bv} l\left[\mathrm{E}=\frac{\mathrm{V}}{\ell}\right]$

(2) If rod is moving by making an angle theta with the direction of magnetic field or length. Induced $\operatorname{emf} \mathrm{e}=\mathrm{B} v l \sin \theta$

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(3) Motion of conducting rod on an inclined plane: When conductor starts sliding from the top of an inclined plane as shown, it moves perpendicular to its length but at an angle $(90-\theta)$ with the direction of magnetic field.

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Hence induced emf across the ends of conductor $\mathrm{e}=\operatorname{Bv} \sin (90-\theta) l=\operatorname{Bv} l \cos \theta$

The forces acting on the bar are shown in the following figure. The rod will move down with constant velocity only if $\mathrm{F}_{\mathrm{m}} \cos \theta=\mathrm{mg} \cos (90-\theta)=\mathrm{mg} \sin \theta$

$\Rightarrow B i l \cos \theta=m g \sin \theta$

$\Rightarrow \mathrm{B}\left(\frac{\mathrm{Bv}_{\mathrm{T}}(\cos \theta}{\mathrm{R}}\right) \ell \cos \theta \mathrm{mg} \sin \theta=>\mathrm{v}_{\mathrm{T}}=\frac{\mathrm{mgR} \sin \theta}{\mathrm{B}^{2} \ell^{2} \cos ^{2} \theta}$

8. Motional Emi in Loop by Generated Area:

If a conducting rod passes along two parallel conducting rails as illustrated in the diagram, the phenomena of induced emf may also be explained using the idea of the generated area (the area swept by a conductor in a magnetic field while it moves).

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As shown in the figure in time t distance travelled by conductor $=v t$ Area generated $A=l v t.$ 

Flux linked with this area $\phi=B A=B l v t$

(1) Induced current:

$\mathrm{i}=\frac{\mathrm{e}}{\mathrm{R}}=\frac{\mathrm{Bv} \ell}{\mathrm{R}}$

(2) Magnetic force : 

Conductor $P Q$ experiences a magnetic force in opposite direction of it's motion and

$\mathrm{F}_{\mathrm{m}}=\mathrm{Bi} \ell=\mathrm{B}\left(\frac{\mathrm{Bv} \ell}{\mathrm{R}}\right) \ell=\frac{\mathrm{B}^{2} \mathrm{v} \ell^{2}}{\mathrm{R}}$

(3) Power dissipated in moving the conductor : 

For uniform motion of rod $\mathrm{PQ}$, the rate of doing mechanical work by external agent or mech. Power delivered by external source is given as

$P_{\mathrm{mect}}=P_{e x}=\frac{d W}{d t}=F_{\mathrm{ext}} \cdot v=\frac{B^{2} v \ell^{2}}{R} \times v=\frac{B^{2} v^{2} \ell^{2}}{R}$

(4) Electrical power : 

Also electrical power dissipated in resistance or rate of heat dissipation across resistance is given as

$P_{\text {tecmal }}=\frac{H}{t}=i^{2} R=\left(\frac{B v \ell}{R}\right)^{2} \cdot R ; \quad P_{\text {tecmal }}=\frac{B^{2} v^{2} \ell^{2}}{R}$

(It is clear that $P_{\text {mech. }}=P_{\text {teemmal }}$ which is consistent with the principle of conservation f energy.)

(5) Motion of conductor rod in a vertical plane : 

If conducting rod released from rest (at $t=0$ ) as shown in figure then with rise in it's speed $(v)$, induces $\operatorname{emf}(e)$, induced current $(i)$, magnetic force $\left(F_{m}\right)$ increases but it's weight remains constant. Rod will achieve a constant maximum (terminal) velocity,

$\mathrm{v}_{\mathrm{T}}$ if $\mathrm{F}_{\mathrm{m}}=\mathrm{mg}$

So, $\frac{B^{2} v_{\mathrm{T}}^{2} \ell^{2}}{R}=m g \quad \Rightarrow v_{T}=\frac{m g R}{B^{2} \ell^{2}}$

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Special Cases:

The motion of train and aeroplane in earth's magnetic field.

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Induced emf across the axle of the wheels of the train and it is across the tips of the wing of the aeroplane is given by $e=B l v$ where $l=$ length of the axle or distance between the tips of the wings of plane, $B_{v}=$ vertical component of earth's magnetic field and $v=$ speed of train or plane.

9. Motional Emi Due to Rotational Motion-

(1) Conducting Rod : 

A conducting rod of length 1 whose one end is fixed, is rotated about the axis passing through it's fixed end and perpendicular to its length with constant angular velocity $\omega$. Magnetic field $(B)$ is perpendicular to the plane of the paper. emf induced across the ends of the rod where $v=$ frequency (revolution per sec) and $T$ = Time period.

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(2) Cycle Wheel: 

A conducting wheel each spoke of length $l$ is rotating with angular velocity in a given magnetic field as shown below in fig. Due to flux cutting each metal spoke becomes an identical cell of emf e (say), all such identical cells connected in parallel fashion $e_{n e t}=e$ (emf of single cell). Let $\mathrm{N}$ be the number of spokes hence $\mathrm{e}_{\mathrm{nat}}=\frac{1}{2} \mathrm{~B} \omega \ell^{2} ; \omega=2 \pi \mathrm{V}$

Here $\mathrm{e}_{\text {ac }} \propto \mathrm{N}^{0}$ i.e. total emf does not depend on number of spokes $N$.

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(3) Faraday Copper Disc Generator : 

A metal disc can be assumed to made of uncountable radial conductors when metal disc rotates in transverse magnetic field these radial conductors cuts away magnetic field lines and because of this flux cutting all becomes identical cells each of emf ' $e$ '

where $\quad \mathrm{e}=\frac{1}{2} \mathrm{~B} \omega \mathrm{r}^{2}$

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(4) Semicircular conducting loop i If a semi-circular conducting loop (ACD) of radius ' $r^{\prime}$ ' with center at $O$, the plane of loop being in the plane of paper. The loop is now made to rotate with a constant angular velocity $\omega$, about an axis passing through $O$ and perpendicular to the plane of paper. The effective resistance of the loop is $R$.

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(5) In time t the area swept by the loop in the field i.e. region II

$A=\frac{1}{2} r(r \theta)=\frac{1}{2} r^{2} \omega t ; \frac{d A}{d t}=\frac{r^{2} \omega}{2}$

Flux link with the rotating loop at time t $_{2} \phi=B A$

Hence induced emf in the loop in magnitude $|e|=\frac{\mathrm{d} \phi}{\mathrm{dt}}=\mathrm{B} \frac{\mathrm{d} \mathrm{A}}{\mathrm{dt}}=\frac{\mathrm{B} \omega \mathrm{r}^{2}}{2}$ and induced current i $=\frac{|\mathrm{e}|}{\mathrm{R}}=\frac{\mathrm{B} \omega \mathrm{r}^{2}}{2 \mathrm{R}}$

10. Periodic Emi

Suppose a rectangular coil having $N$ turns placed initially in a magnetic field such that magnetic field is perpendicular to its plane as shown-

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Where,

$\omega$-Angular speed

$v$-Frequency of rotation of coil $R$-Resistance of coil

For uniform rotational motion with, the flux linked with coil at any time $t$, $\phi=\mathrm{NBA} \cos \theta=\mathrm{NBA} \cos \omega \mathrm{t}$

$\phi=\phi_{0} \cos \omega$ there $\phi_{0}=\mathrm{NBA}=$ maximum flux

(1) Induced Emf in Coil : 

Induced emf also changes in periodic manner that's why this phenomenon called periodic EMI

$\mathrm{e}=-\frac{\mathrm{d} \phi}{\mathrm{dt}}=\mathrm{NBA} \omega \sin \omega \mathrm{t}=e=e_{0} \sin \omega t$ where $e_{0}=e m f$

amplitude or max. emf $=\mathrm{NBA} \omega=\phi_{0} \omega$

(2) Induced Current : 

At any time $t, i=\frac{e}{R}=\frac{e_{0}}{R} \sin \omega t=i$

$\sin \omega t$ where $i_{0}=$ current amplitude or max. current

$\mathrm{i}_{0}=\frac{\mathrm{e}_{0}}{\mathrm{R}}=\frac{\mathrm{NBA} \omega}{\mathrm{R}}=\frac{\phi_{0} \omega}{\mathrm{R}}$

11. Inductance-

(1) Inductance is an electrical circuit property that opposes any change in the circuit's current.

(2) Inductance is a feature of electrical circuits that is intrinsic. Whether we desire it or not, it will always be found in an electrical circuit.

(3) The inductance of a straight wire carrying electricity with no iron portion in the circuit will be lower.

(4) Inductance in an electrical circuit is akin to inertia in mechanics, because inductance opposes any change in current in the circuit.

11.1 Self Induction-

Whenever the electric current passing through a coil or circuit changes, the magnetic flux linked with it will also change. As a result of this, in accordance with Faraday's laws of electromagnetic induction, an emf is induced in the coil or the circuit which opposes the change that causes it. This phenomenon is called 'self-induction' and the emf induced is called back emf, current so produced in the coil is called induced current.

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(1) Coefficient of self-induction; Number of flux linkages with the coil is proportional to the current i, i.e. $N \phi=L i$ ( $\mathrm{N}$ is the number of turns in coil and $N$ total flux linkage). Hence $L=\frac{\mathrm{N} \phi}{\mathrm{i}}=$ coefficient of self-induction. .

(2) If $i=1 a m p, N=1$ then, $L=\phi$ i.e. the coefficient of self induction of a coil is equal to the flux linked with the coil when the current in it is $1 \mathrm{amp}$.

(3) By Faraday's second law induced $\operatorname{smf}$, $\mathrm{e}=-\mathrm{N} \frac{\mathrm{d} \phi}{\mathrm{dt}}$ Which gives $\mathrm{e}=-\mathrm{L} \frac{\mathrm{di}}{\mathrm{dt}} ; \mathrm{If} \frac{\mathrm{di}}{\mathrm{dt}}=\mathrm{amp} / \mathrm{sec}$ then $|\mathrm{e}|=\mathrm{L} .$ Hence coefficient of self induction is equal to the emf induced in the coil when the rate of change of current in the coil is unity.

(4) Units and dimensional formula of 'J'i, It's S.I. unit $\frac{\text { weber }}{\mathrm{Amp}}=\frac{\text { Tesla } \times \mathrm{m}^{2}}{\mathrm{Amp}}=\frac{\mathrm{N} \times \mathrm{m}}{\mathrm{Amp}^{2}}=\frac{\text { Joule }}{\mathrm{Amp}^{2}}=\frac{\text { Coulomb } \times \text { volt }}{\mathrm{Amp}^{2}}=\frac{\text { volt } \times \sec }{\mathrm{amp}}=\mathrm{ohm} \times \mathrm{sec}$

But practical unit is henry (H). It's dimensional formula $[\mathrm{L}]=\left[\mathrm{ML}^{2} \mathrm{~T}^{-2} \mathrm{~A}^{-2}\right]$

(5) Dependence of self-inductance (L) ' ' L'does not depend upon current flowing or change in current flowing but it depends upon the number of turns $(N)$, Area of cross-section $(A)$ and permeability of medium $(\mu)$.

' $L^{\prime}$ does not play any role till there is a constant current flowing in the circuit. ' $L$ ' comes into the picture only when there is a change in current.

(6) Magnetic potential energy of inductor: In building a steady current in the circuit, the source emf has to do work against of self-inductance of coil and whatever energy consumed for this work stored in magnetic field of coil this energy called as magnetic potential energy $(U)$ of coil.

$\begin{aligned} \mathrm{U}=\int_{0}^{\mathrm{i}} \text { Lidi } &=\frac{1}{2} \mathrm{Li}^{2} \\ \mathrm{U}=& \frac{1}{2}(\mathrm{Li}) \mathrm{i}=\frac{\mathrm{N} \phi \mathrm{i}}{2} \end{aligned}$

Also,

(7) The various formulae for $\mathbf{L}$

11.2 Mutual Induction -

Whenever the current passing through a coil or circuit changes, the magnetic flux linked with a neighbouring coil or circuit will also change. Hence an emf will be induced in the neighboring coil or circuit. This phenomenon is called 'mutual induction'.

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(1) Coefficient of Mutual Induction :

Total flux linked with the secondary due to current in the primary is $\mathrm{N}_{2} \phi_{2}$ and $\mathrm{N}_{2} \phi_{2} \propto i 1 \Rightarrow N_{2} \phi_{2}=M i_{1}$ where $N_{1}-$ Number of turns in primary; $N_{2}$ - Number of turns in secondary; 2 - Flux linked with each turn of secondary; $i_{1}$ - Current flowing through primary; $M$-Coefficient of mutual induction or mutual inductance.

(2) According to Faraday's second law emf induces in secondary $\mathrm{e}_{2}=-\mathrm{N}_{2} \frac{\mathrm{d} \phi_{2}}{\mathrm{dt}} ; \mathrm{e}_{2}=-\mathrm{M} \frac{\mathrm{di}_{1}}{\mathrm{dt}}$

(3) If $\frac{\mathrm{di}_{1}}{\mathrm{dt}}=\frac{1 \mathrm{Amp}}{\mathrm{sec}}$ then $\left|\mathrm{e}_{2}\right|=\mathrm{M} .$ Hence coefficient of mutual induction is equal to the emf induced in the secondary coil when rate of change of current in primary coil is unity

(4) Units and dimensional formula of $M:$ Similar to self-inductance (L)

(5) Dependence of mutual inductance:

(i) Number of turns $\left(N_{1}, N_{2}\right)$ of both coils

(ii) Coefficient of self-inductances $\left(L_{1}, L_{2}\right)$ of both the coils

(iii) Area of cross-section of coils

(iv) Magnetic permeability of medium between the coils $\left(\mu_{r}\right)$ or nature of the material on which two coils are wound

(i) Distance between two coils (As $d$ increases so $M$ decreases) 

(ii) Coupling factor ' $K$ ' between primary and secondary coil.

(vi) Orientation between primary and secondary coil (for 90 orientation no flux relation $M=0$ )

(6) Relation between $M, L_{1}$ and $L_{2}:$ For two magnetically coupled coils $\mathrm{M}=\mathrm{K} \sqrt{\mathrm{L}_{1} \mathrm{~L}_{2}}$ where $\mathrm{k}$ - coefficient of coupling or coupling factor which is defined as; $\mathrm{K}=\frac{\text { Magnetic flux linked in secondary }}{\text { Magnetic flux linked in primary }}$ $0 \leq \mathrm{K} \leq 1$

(7) The various formulae for $M$:

11.3. Combination of Inductance

(1) Series. If two coils of self-inductance $L_{1}$ and $L_{2}$ having mutual inductance are in series and are far from each other, so that the mutual induction between them is negligible, then net self-inductance $\mathrm{L}_{\mathrm{s}}=\mathrm{L}_{1}+\mathrm{L}_{2}$

When they are situated close to each other, then net inductance $\mathrm{L}_{\mathrm{s}}=\mathrm{L}_{1}+\mathrm{L}_{2} \pm 2 \mathrm{M}$

(2) Parallel if two coils of self-inductances $L_{1}$ and $L_{2}$ having mutual inductance are connected in parallel and are far from each other, then net inductance $\mathbf{L}$ is $\frac{1}{L_{p}}=\frac{1}{L_{1}}+\frac{1}{L_{2}} \Rightarrow L_{p}=\frac{L_{1} L_{2}}{L_{1}+L_{2}}$

When they are situated close to each other, then

$L_{p}=\frac{L_{1} L_{2}-M^{2}}{L_{1}+L_{2} \pm 2 M}$

11.4. Growth and Decay of Current in lR Circuit

When a circuit with a pure inductor $L$ and a resistor $R$ in series with a battery and a key is closed, the current through the circuit climbs exponentially until it reaches a maximum value (steady-state). When a circuit is opened from its steady state, the current flowing through it diminishes exponentially.

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(1) The value of current at any instant of time t after closing the circuit (i.e. during the rising of current) is given by $i=i_{0}\left[1-e^{-\frac{k}{L}^{1}}\right] ;$ where $i_{0}=i_{\max }=\frac{E}{R}=$ steady state current.

(2) The value of current at any instant of time t after opening from the steady-state condition (i.e. during the decaying of current) is given by $i=i_{0} e^{-\frac{R}{L}}$

(3) Time constant $(\tau) \dot{i}$ It is given as $\tau=\frac{L}{R}$; It's unit is second. In other words, the time interval, during which the current in an inductive circuit rises to $63 \%$ of its maximum value at make, is defined as the time constant or it is the time interval, during which the current after opening an inductive circuit falls to $37 \%$ of its maximum value.

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(4) Behaviour of inductor $i$ The current in the circuit grows exponentially with time from 0 to the maximum value $i\left(=\frac{E}{R}\right) .$ Just after closing the switch as $i=$ 0, inductor act as open circuit i.e. broken wires and long after the switch has been closed as $i=i_{0}$, the inductor act as a short circuit i.e. a simple connecting wire.

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11.5. Lc Oscillation:

When a charged capacitor $C$ having an initial charge $q_{0}$ is discharged through an inductance $\mathrm{L}$, the charge and current in the circuit start oscillating simple harmonically. If the resistance of the circuit is zero, no energy is dissipated as heat. We also assume an idealized situation in which energy is not radiated away from the circuit. The total energy associated with the circuit is constant.

Frequency of oscillation is given by: $\omega=\frac{1}{\sqrt{\mathrm{LC}}} \frac{\mathrm{rad}}{\mathrm{sec}}$

$\mathrm{Or}=\frac{1}{2 \pi \sqrt{\mathrm{LC}}} \mathrm{Hz}$

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The oscillation of the $L C$ circuit is an electromagnetic analogue to the mechanical oscillation of a block-spring system.

11.6. Dc Motor-

It's a machine that turns electrical energy into mechanical energy.

(1) Principle: It is based on the fact that current-carrying capacity of a current-carrying capacity of a current-carrying capacity. When a coil is placed in a magnetic field, it produces torque. This coil is rotated by torque.

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(2) Construction: It is made up of the components shown in the diagram.

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$\mathrm{ABCD}=$ Armature coil, $\mathrm{S}_{1}, \mathrm{~S}_{2}=$ split ring commutators

$\mathrm{B}_{1}, \mathrm{~B}_{2}=$ Carbon brushes, $\mathrm{N}, \mathrm{S}=$ Strong magnetic poles

(3) Working : Force on any arm of the coil is given by $\vec{F}=i(\vec{\ell} \times \vec{B})$

The force applied on $A B$ will be perpendicular to the paper's plane and pointing inwards. On the, $C D$, the force will be equal and opposite. When viewed from the top, the coil turns in a clockwise direction. Due to computation, the current in AB reverses, but the force on $A B$ and $C D$ remains in the same direction, causing the coil to rotate in the same direction.

(4) Back Emf in Motor: Due to the rotation of the armature coil in a magnetic field a back emf is induced in the circuit. Which is given by $e=E-i R. $ Back emf directly depends upon the angular velocity of the armature and magnetic field B. But for constant magnetic field B, value of back emf e is given by e $\propto \omega$ or $\mathrm{e}=\mathrm{k} \omega$, $(\mathrm{e}=\mathrm{NBA} \omega \sin \omega t)$

(5) Current in the Motor: $i=\frac{E-e}{R}=\frac{E-k \omega}{R}$ When motor is just switched on i.e. $=0$ so $e=0$ hence $i=\frac{E}{R}=$ maximum and at full speed, is maximum so back emf e is maximum and i is minimum. Thus, maximum current is drawn when the motor is just switched on which decreases when the motor attains the speed.

(6) Motor Starter: When the motor is started, a tremendous current passes through it, potentially burning it out. As a result, a starter is used to securely start a dc motor. Its purpose is to put an appropriate resistance into the circuit when the motor is turned on. When the motor is running at maximum speed, the resistance steadily lowers until it reaches zero.

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The value of starting resistance is maximum at time $t=0$ and its value is controlled by spring and electromagnetic system and is made to zero when the motor attains its safe speed.

(7) Mechanical power and Efficiency of $d c$ motor :

Efficiency $\eta=\frac{P_{\text {mechanical }}}{P_{\text {supplied }}}=\frac{P_{\text {owt }}}{P_{\text {w }}}=\frac{e}{E}=\frac{\text { Back em.f. }}{\text { Supply voltage }}$

(8) Uses of dc motors: They are used in electric locomotives, electric ears, rolling mills, electric cranes, electric lifts, dc drills, fans and blowers, centrifugal pumps and air compressors, etc.

11.7. Dc Generator:

If the current produced by the generator is direct current, then the generator is called dc generator. $\mathrm{dc}$ generator consists of (i) Armature (coil) (ii) Magnet (iii) Commutator (iv) Brushes

In dc generator commutator is used in place of slip rings. The commutator rotates along with the coil so that in every cycle when the direction of ' $e$ ' reverses, the commutator also reverses or makes contact with the other brush so that in the external load the current remains in some direction giving dc.

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Tips and Tricks:

1. If a bar magnet moves towards a fixed conducting coil, then due to the flux changes an emf, current and charge are induced in the coil. If the speed of the magnet increases then the induced emf and induced current increase but the induced charge remains the same.

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Induced parameter: $e_{1}$, $i_{1}$, $q_{1}$

$e_{2}\left(>e_{1}\right)$, $i_{2}\left(>i_{1}\right)$, $q_{2}\left(=q_{1}\right)$

2. Can ever electric lines of force be closed curves? Yes, when produced by a changing magnetic field.

3. No flux cutting $\longrightarrow$ No EMI

4. Vector form of motional $\mathrm{emf}: \mathrm{e}=(\overrightarrow{\mathrm{v}} \times \overrightarrow{\mathrm{B}}) \vec{\ell}$

5. In motional $\operatorname{smf} \vec{v}, \vec{B}$ and $\vec{l}$ are three vectors. If any two vector are parallel - No flux cutting.

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6. A piece of metal and a piece of non-metal are dropped from the same height near the surface of the earth. The non-metallic piece will reach the ground first because there will be no induced current in it.

7. If an aeroplane is landing down or taking off and its wings are in the east-west direction, then the potential difference or emf will be induced across the wings. If an aeroplane is landing down or taking off and its wings are in the north-south direction, then no potential difference or emf will be induced.

8. When a conducting rod moving horizontally on the equator of earth no emf induces because there is no vertical component of the earth's magnetic field. But at poles, BV is maximum so maximum flux cutting hence emf induces.

9. When a conducting rod falls freely in earth's magnetic field such that it's length lies along the East-West direction then induced emf continuously increases w.r.t. time and induced current flows from West - East. $10.1$ henry $y=109 \mathrm{emu}$ of inductance or $109 \mathrm{ab}$ - henry.

11. Inductance at the ends of a solenoid is half of it's the inductance at the centre $\left(\mathrm{L}_{\text {end }}=\frac{1}{2} \mathrm{~L}_{\text {conte }}\right)$

12. A thin, long wire composed of a high-resistivity material acts primarily as a resistance. It does, however, contain some inductance and capacitance. As a result, obtaining a pure resistor is challenging. Pure capacitors and inductors are similarly difficult to come by.

13. A resistive circuit with no capacitive or inductive elements contains some inductance because of the intrinsic presence of self-inductance in all electrical circuits.

Self-inductance can be reduced by doubling back the coil on itself, as in the coils of a resistance box.

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14. It is not possible to have mutual inductance without self-inductance but it may or may not be possible self-inductance without mutual inductance.

15. If main current through a coil increases $(i \uparrow) s o \frac{d i}{d t}$ will be positive (+ve), hence induced emf e will be negative (i.e. opposite $\mathrm{gmf}$ ) $=E_{\text {net }}=E-e$

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16. Due to the high inductance of the circuit, a high temporarily induced $\mathrm{emf}$ is sometimes produced when the key is suddenly opened, resulting in sparking at the key position. A capacitor is put across the key to prevent sparking.

17. Due to the high inductance of the circuit, a high temporarily induced $\mathrm{emf}$ is sometimes produced when the key is suddenly opened, resulting in sparking at the key position. A capacitor is put across the key to prevent sparking.

18. Resistance can exist with or without inductance, but inductance cannot exist without resistance.

19. An inductor's circuit behavior differs significantly from that of a resistor. while a resistor opposes the current i, an inductor opposes the change $\frac{d i}{d t}$ in the circuit.

20. In RL-circuit with dc source the time taken by the current to reach half of the maximum value is called half life time and it is given by

$\mathrm{T}=0.693 \frac{\mathrm{L}}{\mathrm{R}}$

21. The dc motor is a versatile energy conversion device that can be used in a variety of applications. It can handle loads that require a lot of starting torque, as well as accelerating and decelerating torque.

22. Ohm's law can be utilized when a source of emf is linked across the two ends of the primary winding or the two ends of the secondary winding alone. However, because the primary and secondary windings are not electrically coupled, ohm's law should not be applied to the transformer as a whole.

23. Even when the transformer's secondary circuit is open, it pulls a current termed no load primary current to supply no load Cu and iron losses.

24. Transformer has highest possible efficiency out of all the electrical machines.

Electromagnetic Induction Class 12 Notes

List of topics to be covered in this chapter are as follows.

• Electromagnetic Induction

• Faradays’ law

• Lenz’s law

• Eddy Currents

• Induced EMF & Current

• Self & Mutual Inductance

Let us go through these topics one-by-one.

• Electromagnetic Induction

When changing current is passed through a closed coil, varying magnetic flux develops in it. Due to this flux, an emf is induced in the coil. This induced emf generates an induced current in it. So, this phenomenon of generating induced emf or current because of changing flux is called the Electromagnetic Induction.

• Faraday’s Law

First law:- Faraday’s law is the fundamental law of induction. It states that electric currents can be induced in closed coils when exposed to varying magnetic fields (or flux).  This induced emf lasts as long as there is a changing magnetic flux.

Second Law:- The magnitude of induced emf in a circuit is directly proportional to the rate of change of flux. It is given by:

\[E \propto \frac{d \Phi}{dt} \Rightarrow E = - \frac{d \phi}{dt}\]

Where,

The proportionality constant is unity, and the negative sign indicates Lenz’s law.

The CGS unit of ф = Maxwell, and 1 Weber = 10⁸ Maxwell

Now, let us understand Lenz’s law.

• Lenz’s Law

Lenz’s law states that the direction of induced emf or current is always in such a way that opposes the cause for which it is generated.

Lenz’s law works in accordance with the law of conservation of energy.

• Eddy Currents

Eddy currents are also called Foucault’s currents. This current flows through a closed loop conductor.

These are loops of current induced within the conductor because of varying magnetic flux, as per Faraday’s law of Electromagnetic Induction.

The magnitude of Eddy Currents ‘I’ is given by:

\[I = - \frac{e}{R} = \frac{d \Phi}{R dt}\], where R is the resistance.

What are the Properties of Eddy Currents?

Physics chapter 6 class 12 notes explain the Lenz law of electromagnetic induction where the current twirls in a manner that creates a magnetic field. The nature of eddy currents is the opposing type, which causes a significant loss of energy. This current converts kinetic energy or another crucial energy form into heat. 

The Loss of Energy Can Be Useful for Specific Functions Like

Train brakes- When the braking system is used, the metal wheels create a magnetic field via eddy current. This magnetic interaction is linked with an applied field which lowers the speed of the train. When the wheels move faster, the smooth stopping motion is created with a reduction of braking force.

Few galvanometers have a permanent centre with a nonmagnetic metallic matter. It is seen that an oscillating coil affects Eddy’s current, which further brings motion to rest.

Furthermore, an induction furnace helps in the production of alloys by melting layers of metal. This eddy current generating within metals also produces high temperature or heat.

Notes of Electromagnetic Induction

• Self-Induction

When a closed coil is exposed to a changing current, an additional current or induced current is generated within it. So, this changing current is the primary current, which may increase or decrease. However, this induced current opposes any change in the primary current. This behaviour of a coil is called the self-induction.

1. The coefficient of self-inductance (L)  of a long solenoid is given by:

\[L = \mu_{0} n^{2}AI\]

2. If the solenoid is any other magnetic material, then

\[L = \frac{\mu_{0}/mu_{r} N^{2} A}{I}\]

3. Self-inductance of a toroid is:

\[L = \frac{\mu_{0} \mu_{r} N^{2} A}{2 \pi r}\]

4. Energy stored in an inductor = \[\frac{1}{2} LI^{2}\]

• Mutual Induction

In this concept, there are two coils; when changing current passes through the primary coil, the magnetic flux passes through both primary and secondary coils. An increase in the magnitude of changing current produces an induced emf in the secondary coil (S-coil). This, in turn, generates an induced current in the S-coil.

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1. The coefficient of mutual inductance (K) is given by:

\[K = \frac{M}{\sqrt{L_{1}L_{2}}}\]

2. Mutual inductance of two long coaxial  solenoids (Sa and Sb) is given by:

\[M = \frac{\mu_{0} N_{a} N_{b} A}{l}\]

Importance of CBSE Class 12 Physics Chapter 6 Electromagnetic Induction Notes

This chapter introduces the basic and advanced concepts of electromagnetic induction to the students. They will learn the scientific principles of current electricity and its magnetic effects in detail in this chapter and will use their knowledge to solve exercise problems. To make this preparation better, the experts have compiled the ideal notes to follow. These notes have been composed by following the latest syllabus of CBSE Class 12 Physics to include all the topics properly.

The notes for this chapter are a concise version of all the content. It means students will find the precise description and elaboration of the topics included. They will learn the meaning and definition of various scientific terms included in this chapter along with the formulas to derive them. The stepwise representation of the formula derivation will also help students to focus on the principles and to strengthen their conceptual foundation.

Hence, these notes become a part and parcel of the study material for Class 12 Physics Chapter 6. Referring to these notes will make preparation easier and better.

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Tips to prepare NCERT Class 12 Physics Chapter 6 Electromagnetic Induction

1. Focus on the theoretical portion of this chapter first. Understand the concepts, laws, principles, formulas and derivations well.

2. You can use the concise notes developed by the experts to prepare this chapter faster.

3. Once you are done with the theoretical portion, Proceed to solve the exercise questions to test your preparation level.

4. Compare your answers to the NCERT solutions given.

5. Clarify all the doubts related to the exercise questions by referring to the solutions given here.

6. Check where you need to study more and do the needful.

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8. Check the notes before an exam for a quick glance at all the important topics of this chapter and take your preparation to the next level.

Download CBSE Class 12 Physics Chapter 6 Electromagnetic Induction Notes PDF

Get the free PDF version of these notes here and complete your study material for this chapter. The easier version of these notes will also aid in resolving doubts faster. Focus on how to compile answers by studying these notes to score more in the CBSE board exam and competitive exams. 

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