Gauss Theorem Application

Description:

We know that the electric flux in a closed surface is given by − E.dS cosθ = q/ε₀ .........(1)

In order to find electric field for different distribution of charge (point charge, line charge , sheet of charge etc.) we use Gauss Theorem. It is more convenient method to use than ordinary method  of coulomb’s law.

To find electric field using Gauss Theorem,  following steps are needed to be followed −

Step 1 − Consider a Gaussian surface.

Gaussian Surface

The Gaussian surface has got certain properties −

• It is closed from all sides.
• Electric field at every point is symmetrical
• θ at every point is symmetrical
• May be combination of more than one surface.

Note − The  key point of all Gaussian surface is its symmetry.

Step 2 − Calculate ∮ E→ . dA→

Step 3 − Calculate charge within the closed surface.

Step 4 − Apply Gauss Theorem, put value to calculate E.

Let there is a charged particle ‘q’, then, the electric field due to ‘q’ at a certain distance ‘r’ at point ‘P’ can be found using Gauss Theorem.

Calculating electric field using Gauss Theorem

Step 1

Consider a Gaussian surface in form of a sphere ‘q’ as its centre and ‘R’ as radius, so that it passes through point ‘P’. It is the most symmetric Gaussian surface as the Electric field is radial (perpendicular to charged surface) with same distance to Gaussian surface and also make 0. With area vector A→ at every point on the surface.

Step 2

∮ E dScos θ = E ∮ dS

For a Sphere

∮ dS = 4πr² ............(2)

So, total flux will be Φ_E = E.4πr²

Step 3

The total charge inside the sphere is ‘q’ ..........(3)

Step 4

We know that ∮ E dSCos θ = q/ ε₀

Putting the values −

E.4πr² = q/ ε₀

⇒ E = q/ 4πε₀r²

Or, E =  1/4πε₀ q/r²

This is the electric field in a closed sphere due to a point charge.