Two of Maxwell's equations contain a path integral on the left side and an area integral on the right. Suppose the area is the surface of a piece of paper at which you are looking and is chosen to point toward you. Then, the path integral is:

Question

Two of Maxwell's equations contain a path integral on the left side and an area integral on the right. Suppose the area is the surface of a piece of paper at which you are looking and  is chosen to point toward you. Then, the path integral is:

Quizplus · Accepted Answer

One of the two equations is Gauss's law, which relates the flux of an electric field through a closed surface to the charge enclosed within it. Since the electric field is pointing towards you and you are looking at the surface of the paper, the charge must be inside the paper. Therefore, the path integral must be counterclockwise around the circumference of the paper according to the right-hand rule. The other equation is Faraday's law, which relates the circulation of an electric field around a closed path to the time rate of change of the magnetic flux through an area enclosed by the path. Since the area of the paper is fixed, the path integral must be the same in all directions. However, since the electric field is pointing towards you, the magnetic field must be pointing either to the left or to the right by the right-hand rule. Therefore, the path integral can be from left to right or from right to left depending on the direction of the magnetic field, but it cannot be from top to bottom.

Two of Maxwell's Equations Contain a Path Integral on the Left