Understanding Vector and Scalar Potentials of a Moving Current Loop

How to find the vector and scalar potentials of a small current loop moving with constant velocity in the laboratory frame?

What are the implications of taking the limit v0​≪c in the formulae?

Vector and Scalar Potentials Calculation:

When a small current loop moves with constant velocity v0​ in the laboratory frame, the vector potential A(r) and the scalar potential φ(r) can be found using the Biot-Savart law and the Coulomb's law.

After taking the limit v0​≪c, it can be deduced that the moving loop possesses both a magnetic dipole moment and an electric dipole moment.

Explanation:

To find the vector potential A(r) and the scalar potential φ(r) in the lab frame for a small current loop moving with constant velocity v0​, we can use the Biot-Savart law and the Coulomb's law.

The vector potential A(r) is given by:

A(r) = (μ₀/4π) ∫ (J(r')/|r - r'|) dτ' where μ₀ is the permeability of free space, J(r') is the current density, r is the observation point, and r' is the position vector of the current element.

The scalar potential φ(r) is given by:

φ(r) = (1/4πε₀) ∫ (ρ(r')/|r - r'|) dτ' where ε₀ is the permittivity of free space and ρ(r') is the charge density.

Taking the limit v0​≪c, we can analyze the terms in the vector and scalar potentials. In this limit, the terms involving the velocity v0​ become negligible compared to the terms involving the distance r - r'.

The vector potential A(r) and the scalar potential φ(r) simplify to:

A(r) = (μ₀/4π) ∫ (J(r')/r) dτ'

φ(r) = (1/4πε₀) ∫ (ρ(r')/r) dτ'

Now, let's deduce that the moving loop possesses both a magnetic dipole moment and an electric dipole moment.

A magnetic dipole moment is given by: μ = (1/2) ∫ (r' x J(r')) dτ' where x denotes the cross product.

In the limit v0​≪c, the term involving the velocity v0​ becomes negligible compared to the term involving the current density J(r'). Therefore, the magnetic dipole moment simplifies to: μ = (1/2) ∫ (r' x J(r')) dτ'

An electric dipole moment is given by: p = ∫ (r' ρ(r')) dτ'

In the limit v0​≪c, the term involving the velocity v0​ becomes negligible compared to the term involving the charge density ρ(r'). Therefore, the electric dipole moment simplifies to: p = ∫ (r' ρ(r')) dτ'