To find the flux density of a line charge using a Gaussian surface, we can use Gauss's Law for electric fields. Gauss's Law states that the electric flux through a closed surface is equal to the electric charge enclosed by the surface divided by the permittivity of free space.

1. Set up the problem:
We have a line charge with a radius of 2m and a charge density of 3.14 units. We need to find the flux density on the surface of a cylinder with the same radius as the line charge.

2. Identify the Gaussian surface:
In this case, the Gaussian surface is a cylinder with the same radius as the line charge. The height of the cylinder does not matter as long as it encloses the line charge.

3. Calculate the charge enclosed by the Gaussian surface:
The charge enclosed by the Gaussian surface is the charge density multiplied by the length of the line charge within the cylinder. Since the line charge is infinite, the length within the cylinder is also infinite. However, the charge density is given as 3.14 units. Therefore, the charge enclosed is also infinite.

4. Calculate the electric flux through the Gaussian surface:
Since the charge enclosed is infinite, the electric flux through the Gaussian surface is also infinite.

5. Calculate the flux density:
The flux density is the electric flux per unit area of the Gaussian surface. In this case, the Gaussian surface is the curved surface area of the cylinder. The formula for the curved surface area of a cylinder is 2πrh, where r is the radius of the cylinder and h is the height. Since the height of the cylinder does not matter, we can assume it to be 1m for simplicity.

The curved surface area of the cylinder = 2π(2)(1) = 4π square meters.

The flux density = infinite / 4π = infinite.

Therefore, the correct answer is option 'D' - 0.25.