A long wire that is at rest in the Earth reference frame initially carries no cu

Question

A long wire that is at rest in the Earth reference frame initially carries no current. Observer E in this reference frame measures linear charge densities of En = 45.00 C/m for the (negative) electrons in the wire and Ep = 60.00 C/m for the positive ions. A current is created as electrons are made to move through the wire at speed v, M moves along with the electrons at their speed v.

Part A

What value must v have in order for M to measure an electric field of magnitude E = 3.25 × 107 N/C at a distance 0.0100 m above the wire?

Explanation / Answer

Electric field due to charged wire,

E = 2k lambda/r

3.25e7 = 2*9e9* lambda/0.01

lambda = 0.01*3.25e7/[2*9e9]

= 18.06*10^-6 C

= 18.0556 uC

but the net charge in earth's frame = 60-45 = 15 uC

Lorents factor = 18.0556/15

1/ sqrt(1- v^2/c^2) = 18.0556/15

(15/18.0556)^2 = 1 - (v/c)^2

v/c = sqrt(1-0.6901)

v = 0.556c

= 1.67*10^8 m/s answer

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