Question

The current of a conductor flowing through a conductor in terms of the drift speed of electrons is (the symbols have their usual meanings)

• A. $I = \sqrt{n Ae} v_{d}$
• B. $I = A \sqrt{n e} v_{d}$
• C. $I = \frac{n e}{A v_{d}}$
• D. $I = n e A v_{d}$

Answer 1

Answer: (D)

Relation between current and drift velocity : Consider a conductor of length $l$ and area of cross – section $A$ having $n$ electrons per unit length, as shown in the figure.
The volume of the conductor = $Al$
$\therefore $ Total number of electrons in the conductor
= Volume $\times $ electron density = $Aln$
If $e$ is the charge of an electron, then
total charge contained in the conductor,
$Q = e n A l$

Let a potential difference V is applied across the conductor. The resulting electric field in the conductor is given by
$E = \frac{V}{l}$
Under the influence of this field E, free electrons begin to drift in a direction opposite to that of the field. Time taken by electrons to cross – over the conductor is
$t = \frac{l}{v_{d}}$
where, $v_{d}$ is the drift velocity of electrons.
Therefore, the current through the conductor is given by
$I = \frac{Q}{t} = \frac{e n A l}{I / v_{d}}$
or $I = n e A v_{d}$
$\Rightarrow I \alpha v_{d}$ $\left[\right. \because n , e , A$ are all constant]
Hence, Current density is proportional to drift velocity.