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

…4

…5

Substitute 5 in 4

…6

Substitute 6 in 3

…7

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

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

We know that

Substitute eqn.11 in eqn.10

…..12

Substitute eqn.13 in eqn.12

…..13

…..14

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