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.

Options

A. (i) `0.625 Nm^2//C` , (ii) `3.506 xx 10^-12 C`
B. (i) `0.125 Nm^2//C` , (ii) `1.106 xx 10^-12 C`
C. (i) `0.330 Nm^2//C` , (ii) `2.550 xx 10^-12 C`
D. (i) `2.125 Nm^2//C` , (ii) `6.106 xx 10^-12 C`

Answer

B

Explanation

Here, `vec(ds) = 25 cm^(2)`
`= 25xx10^(-4) m^(2)`, along X-axis
`vec(E) = 50xx hat(i), phi = ?, q = ?`
On the left end of the cylinder,
`vec(E_(1)) = 50xx1 NC^(-1)` along X-axis
`:. Phi_(1) = E_(1) ds cos 180^(@) = 50xx25xx10^(-4) (-1)`
`= -0.125 NC^(-1) m^(2)`
On the right end of the cylinder,
` vec(E_(2)) = 50xx2 = 100 NC^(-1)` along X-axis
`:. phi_(2) = E_(2) = ds cos 0^(@) = 180xx25xx10^(-4) (1)`
`=0.250 NC^(-1) m^(2)`.
Net flux through the cyclinder
`phi = phi_(1) + phi_(2) = 0.125 + 0.250`
`= 0.125NC^(-1) m^(2)`
`q = in_(0) phi = 8.85xx10^(-12) xx 0.125`
`= 1.106xx10^(-12) C`