If E(t,x,y,z) and B(t,x,y,z)represent the electric and magnetic fields at point (x,y,z) at time t, a basic principle of electromagnetic theory says th

nicekikah 2021-02-21 Answered
If E(t,x,y,z) and B(t,x,y,z)represent the electric and magnetic fields at point (x,y,z) at time t, a basic principle of electromagnetic theory says that ×E=Bt. In this expression ×E is computed with t held fixed and Bt is calculated with (x,y,z) fixed.
Use Stokes
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avortarF
Answered 2021-02-22 Author has 113 answers

Given information:
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 CEdr=tSBndσ.
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 CEdr
The basic principle of the electromagnetic theory is ×E=Bt.
Stokes' Theorem:
Let S be a piecewise cmooth oriented surface having a piece smooth boundary curve C. Let F=Mi+Nj+Pk be a vector field whose components have continuous first partial derivatives on an open region containing S. Then the circulation of F around C in the direction counter clockwise with respect to the surface's unit normal vector n equals the integral of the curl vector field ×F over S:
C Fdr=S(×F)ndσ
Calculation:
Assume that S is a surface in which the Stoke's theorem holds. Then CEdr=S(×E)ndσ ...(1)
By the basic principle of the electromagnetic theory, ×E=Bt.
Then equation(1) becomes,
CEdr=S(×E)ndσ
=SBtndσ
=tSBndσ
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 CEdr=tSBndσ is derived by using Stoke's theorem.

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