Question

find the electric field a distance r from a line of a positive charge of infinite length and constant charge per unit length lambda ?

Answer 1

The total charge enclosed is q_enc = lL, the charge per unit length multiplied by the length of the line inside the cylinder.

To find the net flux, consider the two ends of the cylinder as well as the side. There is no flux through either end, because the electric field is parallel to those surfaces. On the other hand, the electric field through the side is simply E multiplied by the area of the side, because E has the same magnitude and is perpendicular to the side at all points.

Net flux = E A = E (2pr) L

By Gauss' Law the net flux = q_enc/e_o

Therefore E (2pr) L = lL/e_o

The factors of L cancel, which is encouraging - the field should not depend on the length we chose for the cylinder. Solving for the magnitude of the field gives:

E = l/[ 2pr e_o ]

Because k = 1/4pe_o this can also be written:

E = 2kl/r

The electric field is proportional to the linear charge density, which makes sense, as well as being inversely proportional to the distance from the line.