Question

A resistor has initial resistance 'R $_{0}$ ' at \[{{0}^{\text{o}}}C\]. Now, it is connected to an ideal battery of constant emf $=V$. If the temperature coefficient of resistance is $\alpha$, then after how much time, will its temperature be \[{{T}^{\text{o}}}C\]. Mass of the wire is $\mathrm{m}$ and specific heat capacity of the wire is S. (Assume the resistance varies linearly with temperature neglect heat loss to the surrounding and variation of dimensions of wire)
(A) $\dfrac{\mathrm{msR}_{0} \mathrm{T}}{\mathrm{v}^{2}}$
(B) $\dfrac{{{\text{m}}_{0}}\text{S}{{\text{R}}_{0}}}{{{\text{v}}^{2}}}(\dfrac{T}{2})$
(C) $\dfrac{\mathrm{mSR}_{0}}{\mathrm{v}^{2}}\left(\mathrm{T}+\dfrac{\alpha \mathrm{T}^{2}}{2}\right)$
(D) $\dfrac{\mathrm{mSR}_{0}}{\mathrm{v}^{2}} \mathrm{T}(1+\alpha \mathrm{T})$

Answer 1

Hint: The heat capacity or thermal capacity of a material is a physical property defined as the quantity of heat to be supplied to a given mass of a material in order to alter the temperature of the unit. Joule per kelvin is the SI unit of heat power. A detailed property is heat power. The specific heat capacity of a material is the energy needed to lift by one degree Celsius (° C) one kilogramme (kg) of the material.
Formula used: $\dfrac{d Q}{d t}=\dfrac{v^{2}}{R}$

Complete answer:
Specific heat is defined as the quantity of heat that must be absorbed or lost by one gramme of a material to change its temperature by one degree Celsius. This number is one calorie for water.
Here, Rate of heat produced is given by:
$\dfrac{d Q}{d t}=\dfrac{v^{2}}{R}$
(q)= Heat Produced
(t) = Time taken
(v)= constant emf
(r)= The initial Resistance
$\Rightarrow \dfrac{{{v}^{2}}}{{{R}_{0}}(1+\alpha (T-0))}=\dfrac{{{v}^{2}}}{{{R}_{0}}(1+\alpha T)}$
And
$\dfrac{dQ}{dt}=ms\dfrac{dT}{dt}$
$\Rightarrow ms\dfrac{dT}{dt}=\dfrac{{{v}^{2}}}{{{R}_{0}}(1+\alpha T)}$
Now, On Integrating both side we get
$\int_{T=0}^{T=T}{=}\dfrac{{{v}^{2}}}{{{R}_{0}}ms}\int_{t=0}^{t=t}{d}t$
$\Rightarrow T+\dfrac{\alpha {{I}^{2}}}{2}=\dfrac{{{v}^{2}}}{{{R}_{0}}ms}t$
$\therefore (t)=\dfrac{{{R}_{0}}ms}{{{v}^{2}}}\left( T+\dfrac{\alpha {{T}^{2}}}{2} \right)$
\[\therefore \] After Time (t) $t=\dfrac{R_{0} m s}{v^{2}}\left(T+\dfrac{\alpha T^{2}}{2}\right)$ will its temperature be \[{{T}^{\text{o}}}C\]. Mass of the wire is $\mathrm{m}$ and the specific heat capacity of the wire is S.

Option (C) is correct.

Note:
An EMF source, which maintains a constant terminal voltage, independent of the current between the two terminals, is an ideal battery. There is no internal resistance to an ideal battery, and the terminal voltage is equal to the battery emf. The thermal expansion coefficient explains how an object's size varies with a change in temperature. In particular, it calculates at constant pressure the fractional change in size per degree change in temperature, so that lower coefficients describe a lower tendency for change in size.