Question

(i) Derive an expression for drift velocity of electrons in a conductor. Hence deduce Ohm’s law.
(ii) A wire whose cross-sectional area is increasing linearly from its one end to the other, is connected across a battery of V volts. Which of the following quantities remain constant in the wire?
A. drift speed
B. current density
C. electric current
D. electric field
Justify your answer.

Answer 1

Answer:

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Answer 2

Given:

The drift velocity of electrons in a conductor.

To find:

(i) Derive an expression for drift velocity of electrons in a conductor. Hence deduce Ohm’s law.

(ii) A wire whose cross-sectional area is increasing linearly from its one end to the other is connected across a battery of V volts. Which of the following quantities remain constant in the wire?

A. drift speed  B. current density  C. electric current  D. electric field

Solution:

(i)

Consider an electron of mass m and charge e moving inside a conductor with a drift velocity Vd when an electric field E is applied. (E = V/l)

Force on electron = -Ee

Acceleration of each electron = − Ee/m

Mechanical force on electron = ma

Velocity created due to this acceleration = [Ee/m] (τ)

where τ is the time span between two consecutive collisions.  

This ultimately becomes the drift velocity in steady-state.

vd =  [Ee/m] (τ) = (e/m)τ(V/l)

∴ vd = (e/m)τ(V/l)

We know that current in the conductor i = neA Vd  

i = neA × e/m τ V/l

∴ i = ne²Aτ V/ml

⇒ i = V/R

where, R = ne²Aτ /ml

∴ i ∝ V

This is as per Ohm's Law.

(ii)

A wire whose cross-sectional area is increasing linearly from its one end to the other is connected across a battery of V volts.  

As the current does not depend on the area of cross-sections of the wire,  

We know, I = dq/dt, the rate of flow of charge, where as the drift speed, current density and electric field depends on the increasing area of cross-section with the following formula:

Drift speed = vd = I/Ane

Current density = I/A

Electric field - J/σ

The following quantities remain constant in the wire: drift speed, current density and electric field.