The functions E(t,x,y,z) nad B(t,x,y,z) represent the electric and magnetic fields at point (x,y,z) at time t.
The Faraday's law is
Here, C represents a wire loop through which current flows counter clockwise with respect to the surface's unit normal n, giving rise to the voltage
The basic principle of the electromagnetic theory is
Let S be a piecewise cmooth oriented surface having a piece smooth boundary curve C. Let
Assume that S is a surface in which the Stoke's theorem holds. Then
By the basic principle of the electromagnetic theory,
Then equation(1) becomes,
Then the voltage around a loop equals the negative of the rate of change of magnetic flux through the loop.
Therefore, the Faraday's law