Question

Two conducting wires X and Y, such that diameter of X is half that of Y and number density of electron in Y is three times that in X are joined in series across a battery. Find the ratio of drift velocity of electrons in wire X to Y.

Answer 1

Hint: Here relation between the physical dimensions and other physical quantities of wire are given. It is also given that the two wires are connected in series, so the current flowing through both the wires will be the same. Use the relation of current flowing through a conducting wire and drift velocity of electrons.

Complete step by step answer:
In this question we are given that there are two conducting wires with names X and Y.
The relation between the diameters of wire X and Y is given as,
Diameter of X is equal to Diameter of Y divided by two. It can be represented in the form of expression as,
${D_x} = \dfrac{{{D_y}}}{2}$
Now the relation between the number density of electron in wire X and Y is given as,
Number density of electrons in Y is equal to three times the number density of electrons in X. It can be represented in the form of expression as,
${n_y} = 3{n_x}$
Now the relation between the area of wire X and Y is given as,
Area of X is equal to Area of Y divided by four. It can be represented in the form of expression as,
${A_x} = \dfrac{{{A_y}}}{4}$
Now it is given in the question that the two conducting wires are connected in series.
As we know that, the current is the series connection remains the same.
So we have, current flowing through the wire X is equal to the current flowing through the wire Y. It can be represented in the form of an expression as,
${I_x} = {I_y}$
As we know that, current flowing through a conducting wire can be expressed as,
$I = nAe{V_d}$
Here ${V_d}$ is the drift velocity of the electrons.
Now equating the current flowing the wire X and Y we have,
${n_x}{A_x}e{V_{dx}} = {n_y}{A_y}e{V_{dy}}$
Putting the relationship of the respective physical quantities we have,
${n_x}{A_x}e{V_{dx}} = 3{n_x} \times 4{A_x} \times e{V_{dy}}$
After cancelling the common elements we have,
$\dfrac{{{V_{dx}}}}{{{V_{dy}}}} = 12$
So, the ratio of drift velocity of electrons in wire X to Y is 12.

Note: It is important to note that current is defined as the rate of flow of charges, electrons or ions through a conductor. The drift velocity is defined as the average speed through these charges, ions or electrons move. So drift velocity is an important aspect of current.The current is directly proportional to the drift velocity of the electrons.