A hollow cylindrical box of length 1m and area of cross section `25 cm^(2)` is placed in a three dimensional co-ordinate system as shown in Fig, The electric field in the
region is given by `vec(E) = 50 x hat(i)` , where E is in `NC^(-1)` and x is in meter. Find
(i) Net flux through the cylinder
(ii) Charge enclosed by the cylinder.

Explanation

Given ` vec E = 50 x hat I and Delta s = 25 cm^(2) = 25 xx 10^(-4) m^(2)`
As the electric field is only along the x - axis, so flux will pass only through the cross - section of cylinder.

Magnitude of electric field at cross - section A,
`E_(A) = 50 xx 1= 50NC^(-1)`
Magnitude of electric field at cross - section B,
`E_(B) = 50 xx 2=100 NC^(-1)`
The corresponding electric fluxes are :
`phi_(A) = vec E * vec (Deltas) = 50 xx 25 xx 10^(-4) xx cos 180^(@) = -0.125 Nm^(2)//C`
`phi_(B) = vec E * vec (Deltas) = 100 xx 25 xx 10^(-4) xx cos 0^(@) = 0.25 Nm^(2)//C`
So, the net flux through the cylinder,
`phi = phi_(A) + phi_(B) = -0.125+0.25 = 0.125 Nm^(2)//C`
(ii) Using Gauss's Law:
`oint vec E .vec(ds)=(q)/(epsilon_(0))rArr 0.125`
`=(q)/(8.85 xx 10^(-12))`
`q=8.85 xx 0.125xx10^(-12)=1.1 xx 10^(12)C.`