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:

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) = 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:

Aicharya · Accepted Answer

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

\Rightarrow

Rate of heat absorption by the rod,

\frac{d Q}{d t} = P = H \cdot \frac{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)

\Rightarrow \frac{d T}{d t} = T_{0} \cdot β \cdot \frac{1}{4} t^{- \frac{3}{4}}

Putting it in (1) gives

P = H \cdot T_{0} \cdot β \cdot \frac{1}{4} \cdot t^{- \frac{3}{4}}

\Rightarrow H = \frac{4 P \cdot t^{3 / 4}}{T_{0} \cdot β}

\Rightarrow H = 4 P \frac{{(T - T_{0})}^{3}}{T_{0}^{4} \cdot B^{4}}

Now using equation

(2)

T - T_{0} = T_{0} β \cdot t^{1 / 4}

\Rightarrow t^{3 / 4} = T_{0} {(\frac{T - T_{0}}{T_{0} β})}^{3}

A current carrying wire heats a metal rod. The wire provides a constant power P to the rod. The metal rod is…