A thermocol vessel contains `0.5kg` of distilled water at `30^(@)C`. A metal coil of area `5 xx 10^(-3) m^(2)`, number of turns `100`, mass `0.06 kg` and resistance `1.6 Omega` 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 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 0.4s`. the cycle is repeated `12000` times. Make sketches of the current through the coil and the power dissipated in the coil as function of time for the first two cycles. Clearly indicate the magnitude of the quantities on the axes. Assumes that no heat is lost to the vessel or the surroundings. Determine the final tempreture of water under thermal equilibrium. Specific heat of metal ` = 500 j kg^(-1) K^(-1)` and the specific heat of water `= 4200 j kg^(-1) K^(-1)`. Neglect the inductance of coil.

Explanation

induced emf
`epsilon = -(dphi)/(dt) = -(d)/(dt)(NBA cos0^(@)) = -NA(dB)/(dt)`
`= -100 xx 5xx 10^(-3) xx (0.8)/(0.2) = -2V`
It is negative between `0and 0.2 s` when field is increased and positive between `0.2 and 0.4 s` when field is decreased.
Current induced `i = (epsilon)/(R) = (2)/(1.6) = 1.25 A`
(a)
Power dissipated `=i^(2)R = (1.25)^(2) xx 1.6 = 2.5 W`
Total energy supplied `= Pt = 12,000(2.5 xx 0.4) = 12 xx 10^(3) J`
`12 xx 10^(3) = m_(1)c_(1)Delta theta + m_(2)c_(2)Delta theta`
rarr `= (m_(1)c_(1) +(m_(2)c_(2))Delta theta`
rarr `Delta theta = (12 xx 10^(3))/(m_(1)c_(1) + m_(2)c_(2))`
`= (12 xx 10^(3))/(0.5 xx 4200 + 0.06 xx 500) = (12 xx 10^(3))/(2130) = 5.6^(@)`
`Delta theta = theta_(2) - theta_(1)`
rarr Final tempreture: `theta_(2) = theta_(1) + Deltatheta = 30 + 5.6 = 35.6^(@)C`