Q.3. A long solenoid with 15 turns per cm has a small loop of area 2.0 cm2 placed inside the solenoid normal to its axis. If the current carried by the solenoid changes steadily from 2.0 A to 4.0 A in 0.1 s, what is the induced emf in the loop while the current is changing ?

**Sol.** Number of turns per unit length , n = 15 cm^{-1} = 1500 m^{-1}

Area , A = 2 cm^{2} = 2 x 10 m^{-4} , μ_{o} = 4π x 10 ^{-7}

Change in current , dI = 4-2 = 2A ,

Change in time , dt = 0.1 sec

Magnetic field B = μ_{o} n I

Magnetic Flux , φ = B A = μ_{o} n I A

A/c to Formula , induced emf , e = −dφ/dt

e = −d(μ_{o} n I A )/dt

e = − μ_{o}n A (dI/dt )

On putting all the given values we get

e = − 7.54 x 10^{-6}

**Q.4.** A rectangular wire loop of sides 8 cm and 2 cm with a small cut is moving out of a region of uniform magnetic field of magnitude 0.3 T directed normal to the loop. What is the emf developed across the cut if the velocity of the loop is 1 cm s–1 in a direction normal to the (a) longer side, (b) shorter side of the loop? For how long does the induced voltage last in each case ?

**Sol. ** using formula , e = B l v and time t = length of wire / v

(i)along longer side

length , l = 8 cm = 0.08 m , B = 0.3 T , v= 1 cm/s = 0.01 m/s

emf developed , e= B l v = 0.3 x 0.08 x 0.01 = 0.24 mV

time of emf = length of shorter arm/v = 0.02/0.01 = 2 sec.

(since , emf developed as long as loop does not get out the field , i.e. distance travelled by shorter arm )

(ii)along longer side

emf developed , e= B l v =0.3 x 0.02 x 0.01 = 0.06mV

time of emf = length of longer arm/v = 0.08/0.01 =8 sec.

Q 5. A 1.0 m long metallic rod is rotated with an angular frequency of

400 rad s^{–1} about an axis normal to the rod passing through its one end. The other end of the rod is in contact with a circular metallic

ring. A constant and uniform magnetic field of 0.5 T parallel to the

axis exists everywhere. Calculate the emf developed between the

centre and the ring.

Sol . Induced emf , e = Blv

Average velocity of the rod , v = (0 + ωl)/2 = ωl/2

Induced emf , e = Bωl^{2}/2

On putting the values , B = 0.5 T , l = 1 m , ω = 400 rad s^{–1}

e = 100 V .