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 change with the (t) as `T(t)=T_(0)(1+betat^(1//4))` where `beta` is a constant with appropriate dimension of temperature. the heat capacity of metal is :

Options

A. `(4P(T(t)-T_(0))^(2))/(beta^(4)T_(0)^(3))`
B. `(4P(T(t)-T_(0))^(3))/(beta^(4)T_(0)^(4))`
C. `(4P(T(t)-T_(0))^(4))/(beta^(4)T_(0)^(5))`
D. `(4P(T(t)-T_(0)))/(beta^(4)T_(0)^(2))`

Answer

B

Explanation

`T(t)=T_(0)(1+betat^(1//4))`
`rArr[(T(t)-T_(0))/(BT_(0))]^(4)=t`
Differentiating w.r.t. time
`(1)/(beta^(4)T_(0)^(4))4[T(t)-T_(0)]^(3)(dT(t))/(dt)=1`...(i)
Also `P=(dQ)/(dt)=(CdT)/(dt)rArr(dT)/(dt)=(P)/(C)` (C – heat capacity)
Putting in (i) :
`C=(4P[T(t)-T_(0)]^(3))/(beta^(4)T_(0)^(4))`