A current carrying wire heats a metal rod. The wire provides a constant power P to the rod. The metal rod is enclosed in an insulated container. It is observed that the temperature T in the metal rod changes with time t as: T t = T 0 1 + β t 1 / 4 where β is a constant with appropriate dimension while T 0 is a constant with dimension of temperature. The heat capacity of the metal is:

Step-by-step solution

Heat required for small increment in temperature of a substance is given by d Q = m . s . d T where, m = mass of substances or s = specific heat of substance Now, heat capacity, H = m . s . ∴ d Q = H . d T ⇒ Rate of heat absorption by the rod, d Q d t = P = H ⋅ d T d t .......(1) But given that, temperature of rod as a function of time is T = T 0 1 + β t 1 / 4 ......(2) ⇒ d T d t = T 0 ⋅ β ⋅ 1 4 t - 3 4 Putting it in (1) gives P = H ⋅ T 0 ⋅ β ⋅ 1 4 ⋅ t - 3 4 ⇒ H = 4 P ⋅ t 3 / 4 T 0 ⋅ β ⇒ H = 4 P T - T 0 3 T 0 4 ⋅ B