Question

(a) State Gauss’ law for magnetism.
(b) How this differs from Gauss’ law for electrostatics?
(c) Why is the difference in the two cases?

Answer 1

Hint: We know a magnetic field which has a divergence value of zero. It underlines the fact that magnetic monopoles cannot exist. Only if the volume contains a charge will there be a net electric field emanating from a finite volume.

Complete step by step answer:
(a) The magnetic flux through a closed surface is protected by Gauss’ Law for magnetism. The area vector points out from the surface in this case. Since magnetic field lines are continuous loops, there are as many magnetic field lines going into all closed surfaces as coming out. Therefore, zero is the net magnetic flux across a closed surface.
Net flux \[\phi = \int {B.dA = 0} \].

(b) The Gauss Electrostatics law is a very useful tool for the measurement of electric fields in circumstances of strong symmetry. The rule of Gauss on magnetism is considerably less helpful.

(c) In both laws, there is a distinction because the magnetic field-lines act quite differently from the electric field-lines, which start with positive charges, end with negative charges, and never form closed loops. The claim that electric field-lines never form closed loops, incidentally, follows from the result that the work done to take an electric charge around a closed loop is always zero. This clearly cannot be true if a charge around the path of a closed electric field-line can be taken.

Additional Information:
Gauss’ law: The law of Gauss, also known as the flux theorem of Gauss, is a law in physics relating to the distribution of electric charges to the resulting electric field. In its integral form, it states that, irrespective of how that charge is distributed, the flux of the electric field from an arbitrary closed surface is proportional to the electric charge enclosed by the surface. Although the law alone is insufficient to determine the electric field across a surface enclosing any distribution of charge, in cases where symmetry requires field uniformity, this may be possible. Where no such symmetry exists, the law of Gauss can be used in its differential form, which states that the electric field divergence is proportional to the local charge density.

Note:
The Law of Gauss is a general law which applies to any closed surface. It is an important tool because by mapping the field on a surface outside the charge distribution, it allows the assessment of the amount of enclosed charge. It simplifies the calculation of the electric field for geometries of sufficient symmetry.