he static Maxwell equations needed revision when the electric and magnetic fieldsE and B were allowed to depend on time. Show that the expression E = −∇φ isno longer consistent with the Maxwell equations because the electric field E is nolonger irrotational. Why does B = curl A continue to hold? Here A is the vectorpotential

The Maxwell's equations in differential form are:

1. Gauss's Law: ∇ • E = ρ/ε₀
2. Gauss's Law for Magnetism: ∇ • B = 0
3. Faraday's Law: ∇ x E = - ∂B/∂t
4. Ampere's Law with Maxwell's Addition: ∇ x B = μ₀J + μ₀ε₀ ∂E/∂t

The expression E = -∇φ is derived from the electrostatic condition where the electric field E is conservative, i.e., its curl is zero (∇ x E = 0). This is consistent with Faraday's law in electrostatics where there is no time-varying magnetic field.

However, when the electric and magnetic fields E and B are allowed to depend on time, the electric field E is no longer irrotational. According to Faraday's law, the curl of E is equal to the negative rate of change of the magnetic field B with respect to time (∇ x E = - ∂B/∂t). Therefore, E = -∇φ is no longer consistent with the Maxwell equations in this case.

On the other hand, the expression B = curl A continues to hold because it is derived from Gauss's law for magnetism where the divergence of B is always zero (∇ • B = 0). This is consistent with the Maxwell equations regardless of whether the fields E and B depend on time or not. The vector potential A is defined such that its curl gives the magnetic field B.