electromagnetic fileds

2. We know that the Electro static potential
Just as above we define vector magnetic potential
We know that
…1
…2
Substitute 1 in 2
The vector identity
…3
Derivation of Boit-Savart’s Law

3. …4
…5
Substitute 5 in 4
…6
Substitute 6 in 3
…7

4. The vector identity
We know that
We know that
Substitute eqn.4 in eqn.3
…..5
Derivation of Ampere’s Circuital Law
…1
…2
Substitute eqn.2 in eqn.1
…3
…..4
We know that
We know that
…..6
Substitute eqn.6 in eqn.5
…..7
Substitute eqn.8 in eqn.7
…..8

5. By using stokes theorem
We know that
Substitute eqn.2 in eqn.1
…1
…2
We know that
…3
.…4
Substitute eqn.4 in eqn.3
.…5
The vector identity
We know that
…6
…7
Substitute eqn.7 in eqn.6
…8
We know that
Substitute eqn.9 in eqn.8
…..10
…..9
We know that
…..11

6. We know that
Substitute eqn.11 in eqn.10
…..12
Substitute eqn.13 in eqn.12
…..13
…..14

8. Let us now consider the force between two elements and
According to Biot-Savart's law, both current elements produce magnetic fields.
So the force on element due to the field
produced by element
The total force on current loop 1 de to current loop 2