Understanding Electric Field in Hollow Circular Cylinder

Explanation:

To show that the field inside the hollow circular cylinder is zero, we can consider an infinitesimally small Gaussian cylinder surrounding a point on the surface of the conductor. The electric flux crossing the Gaussian surface is zero, which implies that the electric field inside the cylinder is also zero.

To show that the field outside the cylinder is the same as if the charge were all on the axis, we can apply Gauss's law to a larger Gaussian surface enclosing the entire cylinder. The total charge enclosed by the surface is equal to the charge on the axis, and the electric flux through the surface is proportional to the charge on the axis, indicating that the field outside is the same as if the charge were all on the axis.

In the case of a pipe of square cross section with a uniform surface density of charge, neither of the statements hold true. The field inside the square pipe would not be zero because the electric field lines would penetrate the space inside the pipe, unlike in the case of a hollow circular cylinder. Additionally, the field outside the square pipe would be different from that if the charge were all on the axis because the shape and symmetry of the charge distribution is different.