If A is the vector potential for the magnetic field produced by a steady current I in aclosed-loop filament, show that div A = 0 off the filament

The vector potential A for a magnetic field produced by a steady current I in a closed-loop filament is given by the Biot-Savart law:

A = μ₀/4π ∫(I dl' / |r - r'|)

where:

• μ₀ is the permeability of free space,
• dl' is an infinitesimal element of the current loop,
• r is the position vector in the field,
• r' is the position vector of the current element,
• |r - r'| is the distance between the field point and the current element.

The divergence of A is given by:

div A = ∇ • A

= ∇ • [μ₀/4π ∫(I dl' / |r - r'|)]

= μ₀/4π ∫[∇ • (I dl' / |r - r'|)]

= μ₀/4π ∫[I ∇ • (dl' / |r - r'|)]

The divergence of 1/|r - r'| is zero everywhere except at r = r'. However, since we are considering points off the filament (r ≠ r'), the divergence of 1/|r - r'| is zero. Therefore, the divergence of A is also zero:

div A = 0

This shows that the divergence of the vector potential A is zero off the filament.