Question

At constant temperature, specific resistance of a conductor material increases with:
A. Increase in area of cross section
B. Decrease in length
C. Decrease in area of cross section
D. None of the above

Answer 1

We know that the formula for the relationship between drift velocity and time is : ${{v}_{d}}=(e\dfrac{E}{m})T$, Where vd is the drift velocity, e is the charge of electron, E is the field, m is the mass of electron, T is the relaxation time.

Complete answer:
Relaxation time is the time interval between two successive collisions of electrons in a conductor, when current flows.

We know the formula for relationship between drift velocity and time is,
${{v}_{d}}=(e\dfrac{E}{m})T$.
Where,
vd is drift velocity
e is the charge of electron
E is the field
m is the mass of electron
T is the Relaxation time

We can find the average relaxation time from this formula,
The average relaxation time will be,
$T=({{v}_{d}}\dfrac{m}{e})E$,
Now , let, L be the Length of the conductor
A is the Area of the conductor
n is the electron density
then current flowing through the conductor is,
$I=-neA{{v}_{d}}$,
$I=neA(e\dfrac{E}{m})T$
$I=\dfrac{n{{e}^{2}}EA}{m}T$
We know that field can be expressed as,
E=V/L,
Therefore the current, flowing through the conductor is,
$I=\dfrac{n{{e}^{2}}VA}{mL}T$
$\dfrac{V}{I}=\dfrac{mL}{n{{e}^{2}}TA}$
Now from ohm’s law,
V=IR,
R=V/I,
$R=(\dfrac{m}{n{{e}^{2}}T})\dfrac{L}{A}$
$R=\rho \dfrac{L}{A}$,

From the above equation it is clearly visible that ,that resistivity only depends on average relaxation time, which in turn depends on temperature.

So, the correct answer is “Option D”.

Additional Information:
Specific resistance depends upon the material of the conductor.
Specific resistance is the resistance offered per unit length and unit cross-sectional area when a known amount of voltage is applied to a conductor.
The SI Unit of specific resistance is $\Omega -m$ (ohm-meter).

Note:
The formula for specific resistance is$\rho =\dfrac{R\times l}{A}$, but students generally confuse why does it depends on temperature, it is so because we derive and compare the equation of specific resistance and we compare that with the equation derived from average relaxation time.