Is Maxwell's equation applicable to any arbitrary Electric vector field and Magnetic vector field?

Question

Suppose if we have a varying electric field $\vec{E}$ and magnetic field $\vec{B}$ that are independent and acting on particular area such that ${\rm curl}\,\vec{E}\neq 0$. Then can we write, by Maxwell's equation, $$\nabla \times \vec{E}=-\frac{\partial\vec{B}}{\partial t}$$ I.e., if we apply a constant magnetic field to an area where an electric field varying with position acts, does the magnetic field starts to evolve in time?

So in simple terms I would like to know whether the Maxwell's equations are applicable to any arbitrary electric field and magnetic field or is it only applicable in case of electromagnetic field.

Answer 1

The equation you cite, which is Faraday's Law, always holds. In fact the four complete Maxwell's equations, which are (in SI units) $$\vec{\nabla}\cdot\vec{E}=\frac{\rho}{\epsilon_{0}}\\ \vec{\nabla}\cdot\vec{E}=0\\ \vec{\nabla}\times\vec{E}=-\frac{\partial\vec{B}}{\partial t}\\ \vec{\nabla}\times\vec{B}=\mu_{0}\vec{J}+\epsilon_{0}\mu_{0}\frac{\partial\vec{E}}{\partial t}$$ all always hold. Together with the Lorentz Force Law, they describe all the physics of charged particles.

However, your terminology seems a bit confused. The electric and magnetic fields together comprise the "electromagnetic field." Any $\vec{E}$ and $\vec{B}$ that together satisfy Maxwell's equations, are an electromagnetic field.

Answer 2

There is no difference between "electromagnetic field" and "electric field and magnetic field". An electric field and a magnetic field are always just two components of an electromagnetic field, witch can be characterized by two vectors $\vec E$, $\vec B$. And electromagnetic field (= electric field + magnetic field) always obey the Maxwell equations.

However, there is indeed some context in which electric and magnetic fields are opposed to a single electromagnetic field. In the general case, when electromagnetic field changes rapidly enough, its electric and magnetic parts are strongly connected and it is impossible to consider them independently. And therefore they say that the electric and magnetic fields are two parts of a single electromagnetic field. But when the frequency of change of the field is small, then the connection between its two components — between the electric and magnetic fields — becomes weak, so that they can be considered as two independent fields.

The separation of the slowly changing electromagnetic field to (practically) independent electric and magnetic fields follows from the Maxwell equations.

Namely, if we take one pair of the Maxwell equations \begin{equation}\nabla\cdot\vec{E}= \rho / \epsilon_0 \\ \nabla\times\vec{E}= - \partial \vec B / \partial t \end{equation} and condition $\partial \vec B / \partial t \approx 0$, we get the following equations for an electric field alone: \begin{equation}\nabla\cdot\vec{E}= \rho / \epsilon_0 \\ \nabla\times\vec{E}=0 \end{equation}

Similarly, another Maxwell equations pair when $\partial \vec E / \partial t \approx 0$ becomes equations for a magnetic field alone.