Question

Derive Maxwell’s Third Equation.

Answer 1

According to Faraday’s Law, electromagnetic force induced in a closed loop is negative rate of charge of magnetic flux.

`e=(dø)/dt`

Total magnetic flux on any arbitrary surface S

∅=∮𝐵.𝑑𝑠
`e=(dø)/dt`[∮𝐵.𝑑𝑠]=−∮`[(dB)/dt].ds`

The electromotive force is the work done in carrying a unit charge around the closed loop.

𝑒=∮𝐸.𝑑𝑙
∮𝐸.𝑑𝑙=−∮`[(dB)/dt]`.𝑑𝑠

By using Stokes theorem contour integration can be converted to surface integration as

∮𝐸.𝑑𝑙=∮[∇×𝐸].𝑑𝑠

∮[∇×𝐸].𝑑𝑠=−∮`[(dB)/(dt)]`.𝑑𝑠

∮[∇×𝐸+`(dB)/dt`𝑑𝑡].𝑑𝑠=0

𝛁×𝑬=−`(dB)/(dt)`

This is Maxwell’s third equation.