- Gauss’s Theorem Definition: Gauss’s theorem states that the total electric flux through any closed surface is equal to the net positive charge enclosed by that surface.
- Flux and Charge: The flux from an electric charge depends on the quantity of the charge.
- Mathematical Expression: Gauss’s theorem is expressed mathematically using a surface integral involving flux density and the outward vector.
- Component Flux: If a charge is not at the center, the flux lines resolve into horizontal and vertical components.
- Total Flux Calculation: The total electric flux through a closed surface equals the total charge, proving Gauss’s theorem.
A static electric field always surrounds a positive or negative electrical charge, where energy flows as flux. This flux comes from the electric charge, and its amount depends on the charge’s quantity. Gauss’s theorem was introduced to describe this relationship. This powerful theorem in electrical science helps us find the amount of flux through the surface surrounding the charge.
This theorem states that the total electric flux through any closed surface surrounding a charge, is equal to the net positive charge enclosed by that surface.
Suppose the charges Q1, Q2_ _ _ _Qi, _ _ _ Qn are enclosed by a surface, then the theorem may be expressed mathematically by surface integral as
Where, D is the flux density in coulombs/m2 and dS is the outwardly directed vector.
Explanation of Gauss’s Theorem
To explain Gauss’s theorem, let’s look at an example for better understanding.
Consider a charge Q at the center of a sphere, with the flux from the charge normal to the surface. Gauss’s theorem states that the total flux from the charge equals Q coulombs, and this can be proved mathematically. But what happens if the charge is not at the center but at any other point?
When the charge is not at the center, the flux lines are not normal to the surrounding surface. The flux is resolved into two perpendicular components: the horizontal sinθ component and the vertical cosθ component. Summing these components for all charges gives the net result equal to the total charge, proving Gauss’s theorem.
Proof of Gauss’s Theorem
Let us consider a point charge Q located in a homogeneous isotropic medium of permittivity ε.
The electric field intensity at any point at a distance r from the charge is
The flux density is given as,
Now from the figure the flux through area dS
Where, θ is the angle between D and the normal to dS.
Now, dScosθ is the projection of dS is normal to the radius vector. By definition of a solid angle
Where, dΩ is the solid angle subtended at Q by the elementary surface are dS. So the total displacement of flux through the entire surface area is
Now, we know that the solid angle subtended by any closed surface is 4π steradians, so the total electric flux through the entire surface is
This is the integral form of Gauss’s theorem. And hence this theorem is proved.
