Gauss Law or Gauss Theorem : Solid Angle and Electric Flux

Gauss Law :

The electric flux Φ_E passing through a bonded surface is 1 / Ɛ₀ times the total charge q enclosed by that surface.

Before we discuss about gauss law, we must be clear few physical terms here like solid angle and electric flux.

Vector Area

A plane area such as A, has magnitude as well as direction. The direction of the area is the direction of the normal drawn outwards on the area. Thus, area A can be represented by a vector A along the normal drawn outward on it, whose length represents the magnitude A of the area. thus

where n̂ is the unit vector along the normal drawn outwards. Vector A is called area Vector.

Solid Angle

We know that the arc, of a circle subtends an angle at the center of the circle. This is called the plane angle. Its unit is Radian (rad). 1 radian is an angle that subtends an arc of length equal to the radius of a circle at the center of that circle.

Similarly, the angle subtended by the area of a spherical surface at the center of the sphere is called a solid angle. And it is denoted by Ω.

Solid Angle Unit

Let the center of a sphere be O and the radius be r. We consider an area element dA on its surface. If the points on the boundary of this area element are joined with the center O of the sphere, then the lines so drawn subtend a solid angle dΩ at the center O.

Since the spherical area dA is directly proportional to the square of the radius r², the conclusion dA/r² is a constant. This result is called the subtend solid angle dΩ by the area dA at the center O of the sphere.

The unit of solid angle is called steradian (sr).

if dA = r² then dΩ = 1

Therefore, the 1 steradian is the solid angle subtended at the center of the sphere by a part of the surface of the sphere whose area is equal to the square of the radius of the sphere.

The area on the entire surface of the sphere is A (= 4pir²). Hence the solid angle subtend at the center of the sphere by the entire surface :

Now , let an area element dA be at a distance r from a point O. Let n be the unit ventor along the line joining point O to the area element dA. Let the angle between n and the vector dA be .

If a sphere of radius r is drawn with point O as center, then the projection of the area element dA on the surface of the sphere will be dAcosθ. Hence, the solid angle subtended by the area element dA at the center O of the sphere.

Electric Flux

Electric flux is a property of an electric area. An electric field can be represented by electric lines of force. Where the electric lines of force are close together, the electric field is strong and where the lines of force are far away, the electric field is weak.

The electric flux on an imaginary surface located in an electric field is a measure of the number of lines of force passing through that surface. It is denoted by Φ_E.

The non-uniform electric field is represented by electric lines of force. Suppose an imaginary arbitrary surface is located in this region and dA is a surface element. The surface element dA is so small that it can be considered flat and the electric field on it can be considered ‘similar’.

The surface element dA can be represented by an area vector dA whose magnitude is the area dA of the surface element and whose direction is along the outward normal on the surface element.

Let the electric area be E at the position of the surface element dA. Then the scalar product E.dA is called the electric flux passing through the element.

Electric flux passing through the entire surface: