Question

Consider an infinitely long line of charge having uniform charge per unit length 5.4 µC/m. Determine the total electric flux through a closed right circular cylinder of length 1.7 m and radius 80 m that is parallel to the line charge, if the distance between the axis of the cylinder and the line of charge is 10 m. The permittivity of free space is 8.8542 × 10−12 C 2 /N · m2 . Answer in units of N · m2 /C.

Answer 1

Electric Flux, or φ, can be found by multiplying the electric field with the area of a surface. What makes this problem difficult is that the axis of the cylinder and the axis of the line with charge are n ot both centered at the orgin, so the electric field is not the same at all locations on the cylinder.

Luckily, through Gauss Law, we also know that flux = q_enc/ε. Therefore, all we need to know is the amount of charge enclosed in our surface ( the cylinder ).

We know the linear charge density of the line, 5.4 μC/m. Therefore, if we multiply by meters ( the length of the line enclosed ) then we will have the charge in coulombs.

The line enclosed by the clyinder has the same length as the cylinder, which is 1.7 m, so:

λL = q

( 5.4 e-6 C )( 1.7 m ) = q_enc

q_enc = 9.18e-6 C

Now, we use Gauss Law to find find the flux:

φ = q_enc/ε

φ = 9.18e-6 C / 8.854e-12

φ = 1.037e6 Nm²/C