Question

A hollow cylindrical box of length 0.5 m and area of cross-section 25 cm² is placed in a three dimensional coordinate system as shown in the figure. The electric field in the region is given by `vecE = 20 xhati`  where E is NC­⁻¹ and x is in metres. Find

(i) Net flux through the cylinder.

(ii) Charge enclosed by the cylinder.

Answer 1

(i) Given,

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 A,

`E_A = 20 xx 0.5 = 10 N C^-1`

Magnitude of lelectic field at cross - section B

E_B = 20 × 1= 20 NC^-1

The corresponding eletric fluxes are 

`phi_A = vecE.Δvecs = 10 xx 20 xx 10^-4 xx cos 180° = - 0.02 N m^2 C^-1`

`phi_B = vecE.Δvecs  = 20 xx 20 xx 10^-4 xx cos0° = 0.04 N m^2 C^-1`

So, the net flux through the cylinder,

`phi = phi _A + phi_B = - 0.02 + 0.04 = 0.02 N m^2 C^-1 xx 10^12C`

(ii) Using Gauss’s law:

`oint  vecE. vecds = q/in_0 => 0.02 = q/(8.85 xx 10^-12) => q =8.85 xx 0.02 xx 10^-12 = 0.177 xx 10^12 C`