The diagram above shows two wires; wire 1 and wire 2. The charge carriers in wir

Question

The diagram above shows two wires; wire 1 and wire 2. The charge carriers in wire 1 (of circular cross section and radius R) have a drift speed down the wire that is not constant across the wire. Instead, the drift speed rises linearly from zero at the circumference (r = R) to v0 at the center (r = 0), according to vd(r) = v0(1-r/R). A second wire (wire 2) has the same radius, the same density of charge carriers and a constant drift speed given by vd(r) = f*v0. Evaluate the ratio of the current carried by wire 1 to the current carried by wire 2, when f = 0.570.

Explanation / Answer

for the second wire,

i=V*f*x (where x is the resistance)

also,

x=rho*l/A

also A=piR^2

for the first wire,

let us consider a thin section of thickness dr and radius r.so,

resistance=rho*L/(2*pi*r*dr)

since they are in parallel,

1/Req=summation of 2pi*r*dr/(2ho*L)

also,

V=ir

so,i=V/r

so,

di=V*2pi*r*dr/(rho*L)

so integratin from 0 to R

,i=Vo*2*pi*(r-r^3/R^2)dr/(rho*L)

=Vo*2*pi*(R^2*0.5-0.25R^2)/(rhi*L)

=Vo*0.5*A/(rho*L)

so,

i2/i1=f/0.5

=0.5/0.5

=1

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