Question

Question 6. Consider two stationary point charges in space. Derive the Coulomb’s law from the Gauss’s law.

Answer 1

Gauss's Law states that the electric flux through a gaussian surface is equals 1/εotimes the total charge enclosed( q_in ) in it.

i.e., Φ = q_in /εo

where εo is the permittivity of the free space.

Let +q₁ be a charge placed at a point O. Let us take a point P which lies at distance r from the point O. now, consider a gaussian surface of a sphere of radius r and centre O such that point P lies on the surface of the sphere. Thus, the intensity of electric field on the surface at all the points will be equal in magnitude and will be directed radially outward.

∴ The electric flux passing through the spherical surface.

Φ=

Where S is surface area i.e., S=4πr²= Surface area of sphere.

Φ=E.S.  

  =E.4πr² .........(i)

And from the Gauss law,

Φ=​​q_in​/εo   .....(ii)

Hence, from equation (i) and (ii), we get

E.4πr²=​​q₁​/εo

E=q₁​/4πεo​r²

now let us consider another charge q₂ placed at point P. ,

then, the Force on q₂ , F= q₂​E

F=q₁q₂​/4πεo​r²

this is the inverse square law as given by coluomb for two stationary charges.