If E and B are in a constant steady state, explain why the electric field must be irrotational

The electric field E is said to be irrotational when its curl is zero. This means that there is no 'rotation' or 'circulation' of the field lines around any point in the field.

In a steady state, the electric and magnetic fields (E and B) do not change with time. According to Faraday's law of electromagnetic induction, the curl of E is equal to the negative rate of change of B with respect to time. Mathematically, this is expressed as:

curl E = -dB/dt

In a steady state, dB/dt = 0 because B is not changing with time. Therefore, the curl of E is also zero. This means that the electric field E is irrotational in a steady state.

In other words, in a constant steady state, there are no changes in the magnetic field that could induce a 'rotation' in the electric field. Therefore, the electric field must be irrotational.