Q: A thermocole vessel contains 0.5 kg of distilled water at 30° C. A metal coil of area 5 × 10^{-3} m^{2}, number of turns 100 , mass 0.06 kg and resistance 1.6 Ω is lying horizontally at the bottom of the vessel. A uniform, time varying magnetic field is set up to pass vertically through the coil at the time t = 0. The field is first increased from zero to 0.8 T at a constant rate between 0 and 0.2. s and then decreased to zero at the same rate between 0.2 and 0.4 s. This cycle is repeated 12000 times. Make sketches of the current through the coil and the power dissipated in the coil as functions of time for the first two cycles. Clearly indicate the magnitudes of the quantities on the axes. Assume that no heat is lost to the vessel or the surroundings. Determine the final temperature of the water under thermal equilibrium. Specific heat of the metal = 500 Jkg^{-1}K^{-1} and the specific heat of water = 4200 Jkg^{-1}K^{-1} . Neglect the inductance of the coil.

Solution : Emf induced in the coil $\displaystyle E = – \frac{d\phi}{dt} = – n A \frac{dB}{dt}$

Where n = number of turns, A = area, B = magnetic field.

Current in the coil $\displaystyle I_1 = \frac{E}{R} $

$\displaystyle I_1 = – \frac{n A}{R} \frac{dB}{dt} = \frac{0.5(0.8-0)}{1.6(0.2)}$

I_{1} = -1.25 A

Current in the coil when magnetic field is decreased uniformly from 0.8 T to 0 in 0.2 sec.

$\displaystyle I_2 = – \frac{n A}{R} \frac{dB}{dt} = \frac{0.5(0m-0.8)}{1.6(0.2)}$

I_{1} = 1.25 A

Power dissipated P = I^{2} R = (1.25)^{2} (1.6) = 2.5 W

**Graph showing variation of current :**

**Graph showing variation of power :**

Heat dissipated in the time interval from 0 to 0.4 sec.

= I_{1}^{2} R t + I_{2}^{2} R t

= 2 × (1.25)^{2} (1.6) (0.2) = 1 J

Total heat dissipated when the cycle is repeated 12000 times

H = (12000) (1) J = 12000 J

Let ΔT = increase in temperature

Therefore H = (m_{1} c_{1} + m_{2}c_{2})ΔT

ΔT = 5.63 0C

Therefore final temperature T = 35. 63 0C