An electric field is uniform, and in the positive x-direction for positive x, and uniform with the same magnitude , but in the negative x-direction for negative x. It is given that
`vec(E) = 200 hat(i) N//C for x gt 0 and vec(E) = -200 hat(i) N//C` for x gt 0. A right circular cylinder of length 20 cm and raidus 5cm has its center at the origin and its axis along the x-axis so that one face is at `x = +10 cm` and the other is at `x = -10 cm`.
(a) What is the net outward flux through the side of the cylinder ? (b) What is the net outward flux through the cyclinder ? (c) what is net charge inside the cylinder ?

Explanation

(a) Here, `l = 20 cm, r = 5cm = 0.05m`.
As is clear from Fig, on the left face L, `vec(E) and Delta vec(S)` are parallel.

`phi_(L) = vec(E) . Delta vec(S) = -200 hat(i) Delta vec(S)`
`= -200[pi (0.05)^(2)] cos 0^(@)`
`= 1.57 N m^(2) C^(-1)`.
Again, on the right face, `vec(E) and Delta vec(S)` are parallel.
`:. phi_(R) = vec(E) . Delta vec(S) = + 200 hat(i).Delta vec(S)`
`= 200 [pi (0.05)^(2)] cos 0^(@)`
`1.57 N m^(2) C^(-1)`
(b) Net outward flux through the cylinder,
`phi = 1.57 + 1.57 + 0 = 3.14 N m^(2) C^(-1)`
(c) By Gauss's Theorem,
`q = in_(0) phi = 8.85xx10^(-12)xx3.14`
`2.78xx10^(-11) C`