Problem Statement: Determine the flux crossing 1mm by 1mm area on cylindrical surface sheet of r=20m, ø=55, z=5m if D=4xax 2(1-y)at 4zazC/m^2?

Solution:
To determine the flux crossing 1mm by 1mm area on a cylindrical surface sheet, we need to use Gauss's law. Gauss's law relates the electric flux through a closed surface to the charge enclosed within the surface.

Gauss's Law:
The total electric flux through any closed surface is proportional to the electric charge enclosed within the surface.

Electric Flux:
The electric flux through a surface is defined as the product of the electric field and the area of the surface perpendicular to the field.

Cylindrical Surface:
A cylindrical surface is a surface that is formed by a straight line moving parallel to a fixed axis and passing through a curve in a plane perpendicular to the axis.

Steps to Determine the Flux:
To determine the flux crossing 1mm by 1mm area on a cylindrical surface sheet, we need to follow the below steps:

1. Determine the charge enclosed within the surface.
2. Determine the electric flux through the surface.
3. Calculate the flux crossing 1mm by 1mm area on the surface.

Charge Enclosed within the Surface:
To determine the charge enclosed within the cylindrical surface sheet, we need to integrate the charge density over the volume enclosed by the surface.

The charge density is given by D = 4xax 2(1-y)at 4zazC/m^2

The charge enclosed within the cylindrical surface sheet is given by:
Q = ∫∫∫ D dv

As the surface is cylindrical, we can use cylindrical coordinates to integrate over the volume.

Q = ∫∫∫ D r dr dø dz

Q = ∫∫∫ 4xax 2(1-y)at 4zazC/m^2 r dr dø dz

Q = 32aC

Therefore, the charge enclosed within the cylindrical surface sheet is 32aC.

Electric Flux through the Surface:
To determine the electric flux through the cylindrical surface sheet, we need to use Gauss's law.

The electric flux through the surface is given by:
Φ = Q/ε0

Where, ε0 is the permittivity of free space.

ε0 = 8.854 x 10^-12 C^2/Nm^2

Φ = (32aC)/(8.854 x 10^-12 C^2/Nm^2)

Φ = 3.61 x 10^13 Nm^2/C

Therefore, the electric flux through the cylindrical surface sheet is 3.61 x 10^13 Nm^2/C.

Flux Crossing 1mm by 1mm Area on the Surface:
To determine the flux crossing 1mm by 1mm area on the cylindrical surface sheet, we need to use the formula for electric flux.

The electric flux through an area is given by:
Φ = E*A

Where, E is the electric field and A is the area of the surface perpendicular to the field.

As the surface is cylindrical,