Question

the current in a wire is doubled. what happens to (a) the current density, (b) the conduction-electron density, (c) the mean time between colisions, and (d) the electron drift speed? are each of these doubled, halved, or unchanged? explain.

Answer 1

Concepts and reason

The question is based upon the current density the conduction electron density, the mean time collision between the electron and the electron drift speed in a current carrying wire.

For the quantities, current density, conduction electron density, the mean time between the collisions and the electron drift speed, use their expression.

Fundamentals

Consider a current carrying wire with cross sectional area A. The current density J in the wire is given by following expression:

$J = \frac{I}{A}$

Here, I is the current flowing through the wire.

The conduction electron present in unit volume of the wire is defined as conduction electron density.

The mean time $\tau$between the collisions or mean free time is given as,

$\tau = \frac{{{m_{\rm{e}}}}}{{\rho {e^2}n}}$

Here, ${m_{\rm{e}}}$ is the mass of electron, $\rho$is the resistivity of the material, e is the charge on the electron and n is the free electrons per unit volume in the material.

The electron drift speed ${v_{\rm{d}}}$ is given by following expression:

${v_{\rm{d}}} = \frac{I}{{A{n_{\rm{e}}}e}}$

Here, ${n_{\rm{e}}}$is the electron current density.

(a)

Find the factor by which the current density changes, when the current flowing through the wire doubled.

The current density J in the wire is given by following expression:

$J = \frac{I}{A}$

From this relation, the current density is directly proportional to the current flowing through the wire keeping the area constant.

$J \propto I$

If the current in the wire is doubled, then the new current density $J'$ in the wire is,

$\begin{array}{l}\\\frac{{J'}}{J} = \frac{{I'}}{I}\\\\J' = \frac{{I'}}{I}J\\\end{array}$

Here, $I'$is the new current in the wire.

Substitute 2I for $I'$in equation $J' = \frac{{I'}}{I}J$.

$\begin{array}{c}\\J' = \frac{{2I}}{I}J\\\\ = 2J\\\end{array}$

(b)

Find the factor by which the conduction electron density changes, when the current flowing through the wire doubled.

The conduction current density of the wire is its characteristic property and independent of the current passing through the wire.

Hence, the conduction electron density is unaffected, even if the current flowing through it is doubled.

(c)

Find the factor by which the mean time between the collisions changes, when the current flowing through the wire doubled.

The mean time $\tau$between the collisions or mean free time is given as,

$\tau = \frac{{{m_e}}}{{\rho {e^2}n}}$

From this relation, it is seen that mean time is independent of current through the wire.

Hence, the mean time between the collisions remains unchanged, even if the current is doubled.

(d)

Find the factor by which the electron’s drift speed changes, when the current flowing through the wire doubled.

The electron drift speed in the wire is given by following expression:

${v_d} = \frac{I}{{A{n_e}e}}$

From this relation, the drift speed of the electrons is directly proportional to the current flowing through the wire keeping the rest of the quantities constant.

${v_d} \propto I$

If the current in the wire is doubled, then the new drift speed ${v_d}'$ in the wire is,

$\begin{array}{l}\\\frac{{{v_d}'}}{{{v_d}}} = \frac{{I'}}{I}\\\\{v_d}' = \frac{{I'}}{I}{v_d}\\\end{array}$

Here, $I'$is the new current in the wire.

Substitute 2I for $I'$as follows:

$\begin{array}{c}\\{v_d}' = \frac{{2I}}{I}{v_d}\\\\ = 2{v_d}\\\end{array}$

Ans: Part a

Part a

Answer

The current density of the wire is doubled, if the current flowing through the wire is doubled.

Part b

Part b

Answer

The conduction- electron density in the wire is unchanged, even if the current flowing through the wire is doubled.

Part c

Part c

Answer

The mean time between the collisions remains unchanged even if the current flowing through the wire is doubled.

Part d

Part d

Answer

The drift speed of the wire is doubled, if the current flowing through the wire is doubled.