Questions
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.
Solution
(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`
∴ vd = 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.
