A resistor has initial resistance `R_(0)` at `0^(@)C` now it is connected to an ideal battery of constant `emf=v` if the temperature co-efficient of resistance is `alpha`, then after how much time will its temperature be `T^(@)C` mass of the wire is m, specific heat capacity of the wire is S. (Assume the resistance veries linearly with temperature Also neglect heat loss the surrounding)

Options

A. `(msR_(0)T)/(v^(2))`
B. `(m_(0)SR_(0))/(v^(2))(T//2)`
C. `(mSR_(0))/(v^(2))(T+(alphaT^(2))/(2))`
D. `(mSR_(0))/(v^(2))T(1+alphaT)`

Explanation

Rate of heat produced
`(dQ)/(dt)=(v^(2))/(R)=(v^(2))/(R_(0)(1+alpha(T-0)))=(v^(2))/(R_(0)(1+alphaT))and(dQ)/(dt)=ms(dT)/(dt)`
`impliesms(dT)/(dt)=(v^(2))/(R_(0)(1+alphaT))`
`int_(T=0)^(T=T)=(v^(2))/(R_(0)ms)int_(t=0)^(t=t)dt`
`T+(alphaT^(2))/(2)=(v^(2))/(R_(0)ms)t,t=(R_(0)ms)/(v^(2))(T+(alphaT^(2))/(2))`