Question

The drift velocity of free electrons in a conductor is v, when a current. I is flowing in it if both the radius and current are doubled, then drift velocity will be.
A. \[\dfrac{v}{4}\]
B. \[\dfrac{v}{2}\]
C. \[2v\]
D. \[4v\]

Answer 1

Hint:The drift velocity is the rate of distance covered by the charge particle between two consecutive collisions from neighbouring atoms in an atom.

Formula used:
\[{v_d} = \dfrac{J}{{ne}}\]
where \[{v_d}\] is the drift velocity, \[J\] is the current density, n is the number of free electrons per unit volume and e is the charge on the electron.

Complete step by step solution:
When potential is applied across the wire then there is an electric field generated inside the wire which applies force on the free electrons. The applied force on the free electrons makes them move inside the wire. The average velocity gained by the free electron is called the drift velocity of the electron. Using the formula of the drift velocity,
\[{v_d} = \dfrac{J}{{ne}}\]

As the J is the current density of the wire, so it is equal to the electric current flowing per unit cross-sectional area of the wire,
\[J = \dfrac{i}{A}\]
On substituting the expression for the current density in the formula of drift velocity, we get
\[{v_d} = \dfrac{i}{{neA}}\]
\[\Rightarrow {v_d} = \dfrac{i}{{ne\pi {r^2}}}\]
It is given that both the current and the radius get doubled.
\[{i_2} = 2{i_1}\]
\[\Rightarrow {r_2} = 2{r_1}\]
The initial drift velocity is given as v,
\[v = \dfrac{{{i_1}}}{{ne\pi r_1^2}}\]

Let the final drift velocity is \[{v_2}\].
\[{v_2} = \dfrac{{{i_2}}}{{ne\pi r_2^2}} \\ \]
\[\Rightarrow {v_2} = \dfrac{{2{i_1}}}{{ne\pi {{\left( {2{r_1}} \right)}^2}}} \\ \]
\[\Rightarrow {v_2} = \dfrac{{2{i_1}}}{{4ne\pi r_1^2}} \\ \]
\[\Rightarrow {v_2} = \dfrac{1}{2}\left( {\dfrac{{{i_1}}}{{ne\pi r_1^2}}} \right) \\ \]
\[\therefore {v_2} = \dfrac{v}{2}\]
Hence, the final drift velocity is \[\dfrac{v}{2}\].

Therefore, the correct option is B.

Note: The drift velocity depends on the cross-section of the wire (A), the number of free electrons per unit volume (n) and the magnitude of the electric current. It does not depend upon the length of the wire.