Gauss' Law   full version

SI units & Physics constants

The Gauss' law is a method, widely used to calculate the electric fields form symmetrically charged objects

Electric flux

Here (all units see here):

A is area of small virtual plane inside the electric field

 is electric field

 is unit vector normal to the plane

 is angle between vectors  and

The electric flux thorough the area A is defined by scalar product

The net electric flux through any surface is defined by the sum

where:

n is total number of element over the surface

is a

 is the electric field at the center of the element i

 is the angle between  and the normal to the plane of the element i

Here the smaller value  the more n, and so the more accurate value of the electric flux is defined

The exact value of net electric flux over a surface with area A is calculated by surface integral

Gauss surface for a given charges is any imaginary closed surface with area A, totally surrounding the charges

Here  is electric field produced by the charges

The net enclosed charge inside the Gauss surface is defined as

where n is total number of charges within the Gauss surface

The Gauss' Law

The Gauss law states that electric net flux through any Gauss surface is directly proportional to the net charge enclosed by the surface

where  is permittivity of vacuum

Electric field from uniformly charged thin spherical shell:

outside the shell      with magnitude

inside the shell     

where:

Q is total charge of the shell

R is radius of the shell

 is position vector of point P where the electric field is defined

Electric field from uniformly charged solid sphere:

outside the sphere      with magnitude

inside the sphere       with magnitude

Electric field from uniformly charged thin line:

  with magnitude

where:

 is linear charge density of the line with length L charged by charge Q

 is radius-vector drawn perpendicular to axis of the line from the axis to the point where the electric field is defined

Electric field from uniformly charged thin cylindrical shell:

outside the shell   with magnitude

inside the shell     

Electric field from uniformly charged solid cylinder:

outside the cylinder    with magnitude

inside the cylinder    with magnitude

where:

 is linear charge density of the cylinder with length L charged by charge Q

R is radius of cylinder

 is radius-vector normal to axis of the line drawn from the axis to the point where the electric field is defined

Electric field near uniformly charged plane

where is surface charge density of the plane with area A charged by charge Q

Conductors in electrostatic field

Electrostatic field is an electric field produced by fixed charges

The conductor is a substance with large number of free electrons which are not bounded to atoms and can freely move through entire substance under the influence of negligible small electric forces

The net electrostatic field inside the conductor is zero

Here  is electrostatic field outside the conductor

A charge Q placed on a conductor is localised only over the surface of the conductor inside its thin surface layer A

The electrostatic field on the surface of a conductor is always directed perpendicular to the surface, so that its tangential component is zero

The surface charge density of the conductor is defined by Gauss' Law

where E is electrostatic field on the surface

Electrostatic polarization of conductor

When an uncharged conductor is inserted into an external electrostatic field, , the electric field lines near the surface of the conductor are distorted, so that they are directed normal to the surface everywhere on the surface. As a result, the electriostatic field induces the opposite charges on the faces of the conductor as shown below