Coulomb Gauge

Suppose we are assuming a field satisfying the condition
            ∇.A = 0                                    … (1)

This conforms to a Gauge transformation which is known as Coulomb gauge.
The Coulomb gauge restricts the divergence of A as
        Div A = ∇.A = 0

The equation for scalar potential V is
    ∇²V + ∂(∇.A)/∂t = - 1/ε₀ρ                                … (2)

Using Eq. (1), Eq. (2) reduces to Poisson’s equation
    ∇²V = - 1/ε₀ ρ                                        … (3)

The advantage of the Coulomb gauge is that the scalar potential is particularly simple to calculate.
Using Coulomb gauge condition Eq. (1), equation of the vector potential reduces to
    (∇²A – μ₀ ε₀ ∂²A/∂t²) = - μ₀J + μ₀ ε₀ ∇(∂V/∂t)                        … (4)   


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