Electromagnetic Induction

Q: Answer the following :A metallic rod PQ of length l is rotated with an angular velocity \omega about an axis passing through its mid-point (O) and perpendicular to the plane of the paper, in uniform magnetic field \vec{B} , as shown in the figure. What is thee potential difference developed between the two ends of the rod, P and Q ?

Given,Length = l Magnetic field = BEmf induce in O Q will be equal to OP .And O will be at high potential and P, Q will at low potential V_{OQ} = V_{OQ} = \dfrac{1}{2} \omega^2 \left(\dfrac{l}{2} \right) \therefore V_P = V_Q \therefore V_{PQ} = O

Q: A bar magnet is moved in the direction indicated by the arrow between two coils PQ and CD. Predict the direction of the induced current in each coil.

According to Lenz’s law the direction of the induced emf is such that it opposes the change in magnetic flux responsible for its production.Since North Pole of bar magnet is receding away from the coil so the right end of the coil will develop South Pole i.e. induced current as seen from the left end will be anticlockwise. Again, since South Pole is pushing towards the second coil so the left end of the coil will develop South Pole in order to repel it and decrease the flux i.e. induced current as seen from the left end will be clockwise

Q: Answer the following:Draw the graph showing variation of the value of the induced emf as a function of rate of change of current flowing through an ideal inductor.

Given, 1 deal inductor circuit having inductance (L)Ref. image ILet at Any instant i current is flowing through the inductor,emf induce, \varepsilon = L \dfrac{d i}{dt} \therefore \dfrac{di}{dt} = \dfrac{1}{L} \varepsilon __(I) y = mx __(II)On comparing equation (I) & (II)Ref. image II

Q: Define mutual inductance between a pair of coils. Derive an expression for the mutual inductance of two long coaxial solenoids of same length wound one over the other.

Mutual inductance, between a pair of coils, equals the magnetic flux, linked with one of them due to a change of unit current flowing in the other. Let r_1 and r_2 be radii of inner (let it be primary P) and outer (let it be secondary S) co-axial solenoids respectively, and n_1 and n_2 be number of turns per unit length of the two solenoids. Let N_1 and N_2 be total number of turns in two solenoids and each of length l. Let the secondary solenoid carry current I_2 . This current sets up magnetic flux \phi_1 through the primary (inner) solenoid. The total flux linkages with the primary solenoid are given by N_1\phi_1 = M_{12} I_2 where , M_{12} is the mutual inductance of the primary (inner) solenoid with respect to the secondary (outer) solenoid.Magnetic field at the centre of the secondary solenoid due to a current I2 is given by B_2 = \mu_0 n_2 I_2 .The total flux linkages with the primary solenoid are given by N_1\phi_1=(n_1l)B_2I_1 N_1\phi_1=(\mu_0\pi n_1n_2lr_1^2)I_2 M_{12}=\frac{N_1\phi_1}{I_2}={\mu_0\pi n_1n_2lr_1^2} Similarly M_{21}=\frac{N_2\phi_2}{I_1}={\mu_0\pi n_1n_2lr_1^2}

Q: Define self-inductance of a coil. Obtain the expression for the energy stored in an inductor L connected across a source of emf.

Self-inductance of a coil is the property of the coil in which it opposes the change of current flowing through it.Self induction of long solenoid of inductance L, A long solenoid is one which length is very large as compared to its cross section area. the magnetic field inside such a solenoid is constant at any point and given by B=\frac { { \mu }_{ 0 }NI }{ l } { \mu }_{ 0 }=\quad absolute\quad magnetic\quad permeability\\ N=\quad total\quad number\quad of\quad turns Magnetic flux through each turn of solenoid \phi =B\times area\quad of\quad each\quad turn\\ \phi =\dfrac { { \mu }_{ 0 }NI }{ l } \times A\\ total\quad flux=flux\times total\quad number\quad of\quad turns\\ N\phi =N(\frac { { \mu }_{ 0 }NI }{ l } \times A)\quad \quad ..........(1) If L is the coefficient of inductance of solenoid N\phi =LI\quad \quad \quad \quad \quad \quad ...............(2)\\ from equation 1 and 2 LI=N(\dfrac { { \mu }_{ 0 }NI }{ l } \times A)\\ L=\dfrac { { \mu }_{ 0 }{ N }^{ 2 }A }{ l }\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(3) The magnitude of emf is given by \quad |e|\quad or\quad e\quad =L\dfrac { dI }{ dt } \quad \quad \quad \quad ...............(4)\\ multiplying\quad I\quad to\quad both\quad sides\\ eIdt=LIdI\\ but\quad I=\dfrac { dq }{ dt } \\ Idt=dq Also work done (dW)= voltage X Charge(dq)or dW = eXdq = eIdtsubstituting the values in equation 4 dW = LIdtBy integrating both sides \int _{ 0 }^{ w }{ dW } =\int _{ 0 }^{ { I }_{ 0 } }{ LIdt } \\ W=\dfrac { 1 }{ 2 } L{ { I }_{ 0 } }^{ 2 } this work done is in increasing the current flow through inductor is stored as potential energy (U) in the magnetic field of inductor U=\frac { 1 }{ 2 } L{ { I }_{ 0 } }^{ 2 }

Q: Find the condition under which the charged particles moving with different speeds in the presence of electric and magnetic field vectors can be used to select charged particles of a particular speed.

A charge q moving with velocity v in presence of both electric and magnetic fields experiences a force F=q(E+v\times B) Lets consider a simple case in which electric and magnetic fields are perpendicular to each other and also perpendicular to the velocity of the particle.Electric and magnetic forces are in opposite directions and the values of E and B are adjusted in such a manner that magnitudes of the two forces are equal.In such a situation, qE=qvB or v=\frac{E}{B} .This condition can be used to select charged particles of a particular velocity out of a beam containing charges moving with different speeds(irrespective of their charge and mass). The crossed E and B fields, therefore,serve as a velocity filter. Only particles with speed E/B pass undeflected through the region of crossed fields.

Q: (a) How are eddy currently generated in a conductor which is subjected to n magnetic field?(b) Write two examples of their useful applications.(c) How can the disadvantages of eddy currents be minimized?

a) When a conductor is placed in a time varying magnetic flux the change in flux due to varying magnetic field induces small loops in the conductor and current flows through these loops according to for 2 day's law.these currents are known as eddy currentb) Applications of eddy currenti) Electromagnetic breaking :- Eddy current brakes use the drag force created by eddy current to slow or stop moving objects, for example , electromagnetic brakes are used in railroad cars.ii) Levitation :- separation of aluminium cans from other metals in an eddy current separator ferrous metals cling to the magnet and aluminium are forced away from the magnet.c) The disadvantages of the eddy current can be reduced by making thin strips instead of a block of the core of induction because as the blocks are made of thin strips the no of possible loops reduces.

Q: Draw a labelled diagram of an ac generator. Obtain the expression for the emf induced in the rotating coil of N turns each of cross-sectional area A, in the presence of a magnetic field \overrightarrow{B} .

Let say at the instant t=0 , the plane of the coil is perpendicular to the direction of magnetic field i.e. area vector of coil points in same direction as that of magnetic field. So maximum flux passes through the coil.Flux passing through the coil at t=0 , \phi_{max} = NBA The coil rotates about an axis with constant angular velocity w as shown in the figure. Angle rotated by coil in time t is wt .Flux passing through the coil at that instant, \phi = NBA\cos wt Rate of change of flux \dfrac{d\phi}{dt} = NBA \dfrac{d}{dt} (\cos wt) \therefore \dfrac{d\phi}{dt} = NBA \times (-w\sin wt) We get \dfrac{d\phi}{dt} = -NBA w\sin wt Induced emf in the coil \mathcal{E} = -\dfrac{d\phi}{dt} \implies \mathcal{E} =NBAw \sin wt

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Previous Year Questions

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- Practice Previous Year questions to get a better idea of the CBSE exam.

Calculate the induced emf in a closed loop.

(a) State the principle of an ac generator and explain its working with the help of a labelled diagram. Obtain the expression for the emf induced in a coil having

N

turns each of cross-sectional area

A

, rotating with a constant angular speed

^{'} ω^{'}

in a magnetic field

B

, directed perpendicular to the axis of rotation.
(b) An aeroplane is flying horizontally from west to east with a velocity of

900 k m / h o u r

. Calculate the potential difference developed between the ends of its wings having a span of

20 m

. The horizontal component of the Earth's magnetic field is

5 \times 1 0^{- 4} T

and the angle of dip is

3 0^{\circ}

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Basics problems on gauss law, faraday law and Lenz's law.

Answer the following :
A metallic rod

P Q

of length

l

is rotated with an angular velocity

ω

about an axis passing through its mid-point (O) and perpendicular to the plane of the paper, in uniform magnetic field

B

, as shown in the figure. What is thee potential difference developed between the two ends of the rod, P and Q ?

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A bar magnet is moved in the direction indicated by the arrow between two coils PQ and CD. Predict the direction of the induced current in each coil.

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Solve problems on inductor circuits with varying inductance.

Answer the following:
Draw the graph showing variation of the value of the induced emf as a function of rate of change of current flowing through an ideal inductor.

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Problems On Self And Mutual Inductance

Define mutual inductance between a pair of coils. Derive an expression for the mutual inductance of two long coaxial solenoids of same length wound one over the other.

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Define self-inductance of a coil. Obtain the expression for the energy stored in an inductor L connected across a source of emf.

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Use the relationship between line integral of electric field and surface integral of magnetic field.

Find the condition under which the charged particles moving with different speeds in the presence of electric and magnetic field vectors can be used to select charged particles of a particular speed.

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Describe the applications and disadvantages of eddy currents.

(a) How are eddy currently generated in a conductor which is subjected to n magnetic field?
(b) Write two examples of their useful applications.
(c) How can the disadvantages of eddy currents be minimized?

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Calculate the emf induced in an AC generator.

Draw a labelled diagram of an ac generator. Obtain the expression for the emf induced in the rotating coil of N turns each of cross-sectional area A, in the presence of a magnetic field

B

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