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A current carrying wire heats a metal ro...

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 :

A

`(4P(T(t)-T_(0))^(3))/(beta^(4)T_(0)^(4))`

B

`(4P(T(t)-T_(0))^(2))/(beta^(4)T_(0)^(3))`

C

`(4P(T(t)-T_(0))^(4))/(beta^(4)T_(0)^(5))`

D

`(4P(T(t)-T_(0)))/(beta^(4)T_(0)^(2))`

Text Solution

Verified by Experts

The correct Answer is:
A
dQ=HdT
`(dQ)/(dt)=H.(dT)/(dt)`
`P=H.T_(0).beta.1/4.t^(-3//4)`
`(4P)/(T_(0)beta)=t^(-3//4).H`
Now `T-T_(0)=T_(0)betat^(1//4)`
So, `t^(3//4) =((T-T_(0))/(T_(0)beta))^(3)`
`:. H=(4P(T-T_(0))^(3))/(T_(0)^(4)beta^(4))`
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