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 $T\left( t \right)={{T}_{0}}\left( 1+\beta {{t}^{{}^{1}/{}_{4}}} \right)$ where $\beta$ is a constant with appropriate dimension while ${{T}_{0}}$ is a constant with dimension of temperature. The heat capacity of metal is?
$\begin{aligned} & \text{A}\text{. }\dfrac{4P{{\left( T\left( t \right)-{{T}_{0}} \right)}^{4}}}{{{\beta }^{4}}{{T}_{0}}^{5}} \\\ & \text{B}\text{. }\dfrac{4P{{\left( T\left( t \right)-{{T}_{0}} \right)}^{3}}}{{{\beta }^{4}}{{T}_{0}}^{4}} \\\ & \text{C}\text{. }\dfrac{4P\left( T\left( t \right)-{{T}_{0}} \right)}{{{\beta }^{4}}{{T}_{0}}^{2}} \\\ & \text{D}\text{. }\dfrac{4P{{\left( T\left( t \right)-{{T}_{0}} \right)}^{2}}}{{{\beta }^{4}}{{T}_{0}}^{3}} \\\ \end{aligned}$

Answer 1

It is said that a current carrying wire heats a metal rod which is inside an insulated container. Since the metal rod is inside a container we can say that no energy is lost during the heating process. We know that the rate of change of heat energy is power. Thus by differentiating heat energy we will get the solution.

Formula used:
$dQ=HdT\left( t \right)$

Complete step by step answer:
In the question we are given a current carrying wire that provides a constant power to a metal rod and heats it. It is said that the metal rod is placed in an insulated container.
We are given the change in temperature with time (t ) as, $T\left( t \right)={{T}_{0}}\left( 1+\beta {{t}^{{}^{1}/{}_{4}}} \right)$ where ‘$\beta$’ and ${{T}_{0}}$ are constants.
By simplifying this equation we will get,
$\Rightarrow T\left( t \right)={{T}_{0}}+{{T}_{0}}\beta {{t}^{{}^{1}/{}_{4}}}$
$\Rightarrow T\left( t \right)-{{T}_{0}}={{T}_{0}}\beta {{t}^{{}^{1}/{}_{4}}}$
Let ‘$dQ$’ be the small amount of heat supplied by the wire to the rod. Then we have,
$dQ=HdT\left( t \right)$, where ‘H’ is heat capacity and ‘T’ is temperature.
Now we can calculate the rate of change of heat with respect to time as,
$\dfrac{dQ}{dt}=H\dfrac{dT\left( t \right)}{dt}$
By substituting $T\left( t \right)={{T}_{0}}\left( 1+\beta {{t}^{{}^{1}/{}_{4}}} \right)$, we will get
$\Rightarrow \dfrac{dQ}{dt}=H\dfrac{d\left( {{T}_{0}}\left( 1+\beta {{t}^{{}^{1}/{}_{4}}} \right) \right)}{dt}$
We know that the rate of change of heat is power. Therefore we get,
$\Rightarrow P=H\dfrac{d\left( {{T}_{0}}\left( 1+\beta {{t}^{{}^{1}/{}_{4}}} \right) \right)}{dt}$
By differentiating the right hand side of the above equation, we will get
$\Rightarrow P=H\times {{T}_{0}}\times \beta \times \dfrac{1}{4}\times {{t}^{{}^{-3}/{}_{4}}}$
From this we will get the equation for heat capacity as,
$\Rightarrow H=\dfrac{4P}{{{T}_{0}}\times \beta \times {{t}^{{}^{-3}/{}_{4}}}}$
From earlier calculation, we have
$T\left( t \right)-{{T}_{0}}={{T}_{0}}\beta {{t}^{{}^{1}/{}_{4}}}$
From this equation we can write,
$\Rightarrow {{t}^{{}^{1}/{}_{4}}}=\dfrac{{{T}_{0}}\beta }{T\left( t \right)-{{T}_{0}}}$
Therefore we will get,
$\Rightarrow {{t}^{{}^{-3}/{}_{4}}}={{\left( \dfrac{{{T}_{0}}\beta }{T\left( t \right)-{{T}_{0}}} \right)}^{-3}}$
By substituting this in the equation for heat capacity, we will get
$\Rightarrow H=\dfrac{4P}{{{T}_{0}}\times \beta \times {{\left( \dfrac{{{T}_{0}}\beta }{T\left( t \right)-{{T}_{0}}} \right)}^{3}}}$
$\Rightarrow H=\dfrac{4P{{\left( T\left( t \right)-{{T}_{0}} \right)}^{3}}}{{{T}_{0}}\times \beta \times {{\left( {{T}_{0}}\beta \right)}^{3}}}$
$\therefore H=\dfrac{4P{{\left( T\left( t \right)-{{T}_{0}} \right)}^{3}}}{{{T}_{0}}^{4}{{\beta }^{4}}}$
Therefore we get the heat capacity of the metal as, $\dfrac{4P{{\left( T\left( t \right)-{{T}_{0}} \right)}^{3}}}{{{T}_{0}}^{4}{{\beta }^{4}}}$.

So, the correct answer is “Option B”.

Note:
Heat is a form of energy which can be transferred between two substances at different temperatures. The direction of the flow of energy is always from the substance at higher temperature to the substance at lower temperature.
Heat capacity is simply the ratio of the heat absorbed by the material to the change in temperature. It can be defined as the total amount of energy required to raise the temperature of a substance by $1{}^\circ C$ or $1K$.