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प्रश्न
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.
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उत्तर
(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.
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