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: rea of small element i, for i = 1, 2, 3,..., n
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 radiusvector 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 radiusvector 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
