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

To solve the problem step by step, we will analyze the relationship between resistance, temperature, and heat generated in the resistor.

### Step 1: Understand the relationship between resistance and temperature
The resistance \( R \) of the resistor at a temperature \( T \) can be expressed as:
\[ R(T) = R_0 (1 + \alpha T) \]
where \( R_0 \) is the initial resistance at \( 0^\circ C \), and \( \alpha \) is the temperature coefficient of resistance.
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