Eduprobe
Question:
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) = T₀(1 + βt¹/⁴) where β is a constant with appropriate dimension while T₀ is a constant with dimension of temperature. The heat capacity of metal is?
4P(T(t) - T₀)³β⁴T₀⁴
4P(T(t) - T₀)⁴β⁴T₀⁵
4P(T(t) - T₀)β⁴T₀²
4P(T(t) - T₀)²β⁴T₀³
Solution:
Correct option is B.
4P(T(t) - T₀)³β⁴T₀⁴
dQ = HdT
dQ/dt = H * dT/dt
P = H * T₀ * β * (1/4) * t⁻³/⁴
P/T₀ * β = t⁻³/⁴ * H
Now, T - T₀ = T₀βt¹/⁴
So t³/⁴ = (T - T₀/T₀β)³
∴H = 4P(T - T₀)³/T₀⁴β⁴
Correct option 2.