Question

In a wire of circular cross section with radius r free electrons travel with a drift velocity v when a current i flows through a wire. The current in another wire of half the radius and of the same material when the drift velocity is 2v

Answer 1

Hint:First with finding the relation between the current flows through the wire, radius of wire and the drift velocity of the electrons. Then put the value of radius in that relation in both the cases that is first wire of radius r and another wire of half the radius of first wire.

Formula used:
Current flowing through the given wire;
$I = neA{v_d}$
Where, n is no. of free electrons.
A is the area of the cross-section of the wire.
${v_d}$ is the drift velocity of electrons flowing through the given wire.
And e is the charge of the electron flowing in the wire.

Complete step by step solution:
We know that the current will be given by:
$I = neA{v_d}$
As the material for two wires is the same hence the value of n will also be same for both.
Therefore, current in wire 1 of radius r and drift velocity of v,
${I_1} = neAv$..........(equation 1)
Current in wire 2 of radius $\dfrac{r}{2}$ is,
${I_2} = ne\pi(r/2)^2v_d$
$\Rightarrow {I_2} = ne\left( {\pi \dfrac{{{r^2}}}{4}} \right){v_d}$
By putting value of drift velocity that is 2v here, we get;
${I_2} = ne\dfrac{A}{4}\left( {2v} \right)$
$\therefore {I_2} = \dfrac{{neAv}}{2} = \dfrac{{{I_1}}}{2}$

Hence, current in wire 2 will be half of the current flowing in wire 1.

Note: Here the value of the number of free electrons and the charge of the electrons is same for both the wire it may not be the case all the time. So, be careful if its different value is given in the question because then the final answer will differ from the current one.