State and explain Gauss’s law.
Gauss’s law states that the flux of the electric field through any closed surface S is 1/∈ₒ times the total charge enclosed by S
Let the total flux through a sphere of radius r enclose a point charge q at its centre. Divide the sphere into a small area element as shown in the figure.
The flux through an area element ΔS is
Here, we have used Coulomb’s law for the electric field due to a single charge q.
The unit vector `hatr`is along the radius vector from the centre to the area element. Because the normal to a sphere at every point is along the radius vector at that point, the area element ΔS and `hatr` have the same direction. Therefore
Because the magnitude of the unit vector is 1, the total flux through the sphere is obtained by adding the flux through all the different area elements.
Because each area element of the sphere is at the same distance r from the charge,
Now, S the total area of the sphere equals 4πr². Thus,
Hence, the above equation is a simple illustration of a general result of electrostatics called Gauss’s law