Question

Derive an expression for  drift velocity of free electrons.

 Derive an expression for drift velocity of electrons in a conductor. Hence deduce Ohm's law.

Answer 1

(i) Free electrons are in continuous random motion. They undergo  change  in direction at each collision and the thermal velocities are randomly distributed in all directions.

∴ Average thermal velocity,`u=(u_1+u_2+....+u_n)/n " is Zero".....(1)`

The electric field E exerts an electrostatic force ‘−Ee’.

Acceleration of each electron is,`veca=(-evecE)/m " ......(2)"`

Where,

m → Mass of an electron

e → Charge on an electron

Drift velocity,

`vec(v_d)=(vec(v_1)+vec(v_2)+....+vec(v_n))/n`

`vec(v_d)=((vec(u_1)+vecat_1)+(vec(u_2)+vecat_2)+....+(vec(u_n)+vecat_n))/n`

Where,

`vecu_1,vecu_2->` Thermal velocities of the electrons

`vecatau_1,vecatau_2->` Velocity acquired by electrons

τ₁, τ₂ → Time elapsed after the collision

`vec(v_d)=((vec(u_1)+vec(u_2)+...+vecu_n))/n+(veca(vec(t_1)+vec(t_2)+...vec(t_n)))/n`

Since `(vec(u_1)+vec(u_2)+....vec(u_n))/n=0`

∴ v_d = a τ

Where,`t=(t_1+t_2+t_3....t_n)/n " is the average time elapsed"`

Substituting for a from equation (2),

`vec(v_d)=(-evecE)/mt " ...(4)"`

As, `E=V/l`
From (4) we can write

`v_d=(eV)/(ml)τ`
Also,
`I=An""ev_d`
Therefore,
`I=An""e((eV)/(ml)τ)=(An""e^2τ)/(ml) V`

`or V/I=(ml)/(An""e^2τ)=R` .... (5)

As we can see all the parameter on the R.H.S of the equation 5 are constant given temperature. And it is known as Resistance of the electric conductor.