Electric Field Vector in terms of Scalar and Vector Potentials

In the case of a magnetic field,
    div. B = 0
Since the divergence of any curl is zero,
    B = curl A = ∇x A
Here A is called the magnetic vector potential. It has physical importance where magnetic field varies with time.
The time variation of magnetic field, from Eq. (2), is
    ∂B/∂t = ∂(∇xA)/∂t
Interchanging the space and time operators, we get
    ∂B/∂t = ∇ x ∂A/∂t
The differential form of Faraday’s law is
    ∇ x E = - ∂B/∂t
. : Eq. (4) becomes ∇ x (E + ∂A/∂t) = 0

From vector identity, we find that the curl free field must be gradient of a scalar potential Φ. We write
    E + ∂A/∂t = - grad Φ
Or    E = - ∂A/∂t –grad Φ

Eq. (8) suggest that for time dependent magnetic field, we may think of – grad Φ as the contribution of E due to coulomb field and - ∂A/∂t as the contribution due to electromagnetic induction.