**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 × 10 m^{-4}, μ_{o}= 4π × 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 ,****On putting all the given values we get**

**e = − 7.54 × 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**

**(a) 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 × 0.08 × 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 )**

**(b)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 , **

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

**e = 100 V .**