Question

A charge Q is uniformly distributed onto a straight wire of length l. Find the electric at a distance r from the center of the rod. The Gaussian surface and the field line are shown in the figure on the right

Answer 1

A Gaussian surface is a closed surface in three-dimensional space through which the flux of a vector field is calculated; usually thegravitational field, the electric field, or magnetic field.^[1] It is an arbitrary closed surface S = ?V (the boundary of a 3-dimensional region V) used in conjunction with Gauss's law for the corresponding field (Gauss's law, Gauss's law for magnetism, or Gauss's law for gravity) by performing a surface integral, in order to calculate the total amount of the source quantity enclosed, i.e. amount of gravitational mass as the source of the gravitational field or amount of electric charge as the source of the electrostatic field, or vice versa: calculate the fields for the source distribution.

Most calculations using Gaussian surfaces begin by implementing Gauss's law (for electricity):^[2]

Thereby Q(V) is the electrical charge contained in the interior, V, of the closed surface.

This is Gauss's law, combining both the divergence theorem and Coulomb's law.

The flux out of the spherical surface S is:

The surface area of the sphere of radius r is

which implies

By Gauss's law the flux is also

finally equating the expression for ?_E gives the magnitude of the E-field at position r: