Question

For which of the following dependence of drift velocity $v_d$ on electric field E, is Ohm's law obeyed?

• A. $v_d\propto E^2$
• B. $v_d=E^{1/2}$
• C. $v_d=$ constant
• D. $v_d\propto E$

Answer 1

Answer: (D)

$v_d\propto E$

Assume a conductor of
length $l$,
charge carrier density $n$,
drift velocity $v_d$,
cross sectional area $A$,

It is placed in an uniform electrical field ($E$)
current through the conductor ($i$) $=\frac{n\times l\times A\times q}{t}$

where $q$ is the charge of a carrier(electron) and $t$ is the time taken by carrier to pass through conductor
$\Rightarrow v_d=\frac{l}{t}$
$\Rightarrow i=n\times v_d\times A\times q$ .......(1)
so to obey ohm's law $i=\frac{V}{R}$ .....(2)
$R=\frac{\rho \times l}{A}$ ......(3)
$V=E\times l$ ....(4)

substituting (2),(3),(4) in (1):
$\Rightarrow n\times v_d\times A\times q=\frac{E\times A}{\rho }$
$\Rightarrow E=\rho \times n\times v_d\times q$
Thus $v_d\propto E$ . hence correct option is D