Ohm\'s Law: Electrons of charge e and mass m are moving at constant speed v insi

Question

Ohm's Law: Electrons of charge e and mass m are moving at constant speed v inside a straight conducting wire of length L and cross sectional area A. The number density of electrons inside the wire is n. There is a uniform electric field E and a friction force of magnitude F_f = mv/tau where tau is a constant. What is the acceleration of the electrons? What is the total force on each electron? Equating the magnitude of the electric force to the magnitude of the friction force, find the relation between v and E. Show that the current density j is proportional to E: j = sigma E. Find the proportionality constant sigma, the conductivity, in terms of n, e, m and tau. Express the total voltage difference Vin the wire in terms of the electric field E. Express the total current I in terms of the current density j. Obtain Ohm's Law, V = R I from the relation j = sigma E. Express the "resistance" R in terms of the other parameters.

Explanation / Answer

part a:

as electrons are moving at constant speed, acceleration=time derivative of speed

=0 m/s^2

total force=mass*acceleration=0 N

part b:

as total force is 0,

electric force magnitude is equal to magnitude of friction force

hence q*E=m*v/tau

==>v=q*E*tau/m....(1)

part c:

current =charge /time

time taken to move the distance L =distance / speed=L/v

current density=current/cross setional area

==>j=charge/(time*cross sectional area)

=number density*volume*charge of each electron/(L*A/v)

=n*A*L*e*v/(L*A)

=n*e*v

=n*e*e*E*tau/m

=(n*e^2*tau/m)*E

=sigma*E

hence j is proportional to E

the proportionality constant=j=n*e^2*tau/m

part d:

voltage difference V=electric field * distance=E*L

part e:

I=j*A

part f:

j=sigma*E

==>I/A=sigma*V/L

==>V=I*L/(A*sigma)=I*L/(A*n*e^2*tau/m)=I*L*m/(n*e^2*A*tau)

==>V=I*R

where R=L*m/(n*e^2*A*tau)

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