Question

proof of columb law with the help of gauss law?

Answer 1

consider, a area element dS around any point P on the Gaussian surface both E and dS are directed radially outward,that is ,the angle between E and dS is zero.

The flux passing through the area element dS ,that is,

d φ =E.dS= EdS cos 00=EdS

Hence, the total flux through the entire Gaussian sphere is obtained as,

Φ=∫EdS

Or                                φ=E∫dS

But ∫dS is the total surface area of the sphere and is equal to 4πr2,that is,

Φ=E(4πr2)                                                                  (1)

But acc. to guass las for electrostatic

Φ=q/ε0 (2)

Where q is the charge enclosed within the closed surface

By comparing equation (1) and (2) ,we get

E(4πr2)=q/ε0

Or  , E=q/4πε0r2 (3)

The equation (3) is the expression for the magnitude of the intensity of electric field E at a point,distant r from the point charge +q.

In vector form,  E=1/4πε0 q/r2=1/4πε0qr/r3

In a second point charge q0be placed at the point at which the magnitude of E is computed ,then the magnitude of the force acting on the second charge q0would be

F=q0E

By substituting value of E from equation (3),we get

F=qoq/4πε0r2 (4)

The equation (4) represents the Coulomb’s Law and it is derived from gauss law.

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