Question

A rectangular conductor of length I and area of cross section A and electron density n; is shown below.

1. When the face Y is given positive potential and X negative potential what will happen to the electrons inside the block 

2. What is meant by drift velocity? How is it related to the field inside the metal?

3. Deduce an expression connecting intensity of electric field and drift velocity.

4. Under the application of an electric field do all the electrons move in a same direction? Explain

Answer 1

1. Electrons will be accelerated towards the side Y.

2. Drift Velocity (v_d):

The average velocity acquired by an electron under the applied electric field is called drift velocity.

Explanation :

When a voltage is applied across a conductor, an electric filed is developed. Due to this electric field electrons are accelerated. But while moving they collide with atoms, lose their energy and are slowed down. This acceleration and collision are repeated through the motion. Hence electrons move with a constant average velocity. This constant average velocity is called drift velocity.

3. Expression for drift velocity :

Let ‘V’ be the potential difference across the ends of a conductor. This potential difference makes an electric field E. Under the influence of electric field E, each free electron experiences a Coulomb force.
F = -eE
or ma = -eE

a = \(\frac{-eE}{m}\)  ......(1)

Due to this acceleration, the free electron acquires an additional velocity. A metal contains a large number of electrons. For first electron, additional velocity acquired in a time τ,

v₁ = u₁ + aτ₁

where u₁ is the thermal velocity and τ is the relaxation time.

Similarly the net velocity of second, third…… electron

v₂ = u₂ + aτ₂

v₃ = u₃ + aτ₃

v_n = u_n + aτ_n

∴ Average velocity of all the ‘n’ electrons will be

V_av = 0 + aτ (∴ average thermal velocity of electron is zero) where

where V is the average velocity of electron under an external field. This average velocity is called drift velocity. ie. drift velocity V_d = aτ ……(2)

Substitute eq (1) in eq (2)

V_d = \(\frac{-eE}{m}\tau\)

4. Electrons will continues its random thermal motion even in the presence of electric field.