Chapter 18, Problem 18Q

To determine

To explain:

If the number of free electrons per atom in one wire is double of that in the other, whereas both are geometrically identical with the same atomic density, compare their drift velocities and their electric currents.

Answer to Problem 18Q

Solution:

The drift velocities are the same but the current in one is double of the other.

Formula used:

  •The average drift velocity depends on the structural properties of the material and the external electric field $\left(\overrightarrow{E}\right)$, as

                 $\overrightarrow{v_d}=-\frac{e\tau }{m}\overrightarrow{E}$

    Where, e is the charge over one electron and $v_d$ is drift velocity of the free electrons in the wire at that particular temperature, τ is the average relaxation time between two collisions of free electrons and m is the mass of one electron.

  •The current in a wire of cross-sectional area A, having n number of free electrons per unit volume, is defined as,

                 $I=Ane{v}_d$

    Where, e is the charge over one electron and $v_d$ is drift velocity of the free electrons in the wire at that particular temperature.

Explanation of Solution

As it is given that the number of free electrons per atom in the material of one wire is twice of that in the other and also because the density of atoms in both the wires are similar, the number of free electrons per unit volume in one wire will also be twice of that in the other, i.e.,

                 $n_1=2n_2$

But because the density of atoms in both the wires is similar, the average relaxation time between two collisions of free electrons will be the same, i.e.

                 ${\tau }_1={\tau }_2$

Therefore, from the formula $\overrightarrow{v_d}=-\frac{e\tau }{m}\overrightarrow{E}$, we can conclude that the average drift velocities in the two wires would be same, i.e.,

                 $v_{d_1}=v_{d_2}$

But as $n_1=2n_2$, therefore from the formula $I=Ane{v}_d$, we can conclude that the current in one wire would be twice of that in the other, i.e.,

                 $I_1=2I_2$