Does time-varying magnetic field induce time varying-electric field?

Question

As we all know that Faraday's law states that the EMF is also given by the rate of change of the magnetic flux: $$\text{emf} = -N \frac{d\Phi}{dt}$$
So if we are applying a time-varying magnetic field(let $dB/dt =$ constant) on a stationary conducting coil then induced electric field across the coil work as a driving force to induce a current in that coil. According to the above formula, induced emf in a coil will be constant if $dB/dt =$ constant, But if the induced electric field is time-varying then induced emf also be time-varying? isn't it? What I want to say is that I learned somewhere in the past that Statement: "A time-varying electric field can not exist without a corresponding time-varying magnetic field and vice versa", But According to Faraday's law, a linear time-varying magnetic field induces a static electric field, So, is that mean the above statement is wrong? Or in other words, Understand with 3 statements written below-

(1) Linearly time-varying electric field {i.e. $dE/dt =$ constant} is capable of inducing static Magnetic field only (not capable of inducing dynamic magnetic field).

(2) Linearly time-varying magnetic field {i.e. $dB/dt =$ constant} is capable of inducing a static electric field only (not capable of inducing dynamic electric field)

(3) "A time-varying electric field can not exist without a corresponding time-varying magnetic field and vice versa" So, statements (1) and (2) can be understood and verified by

Faraday-Maxwell equation $$\oint E\cdot dl = - \frac{d\Phi}{dt}$$ Where $\Phi =$ magnetic flux, verifies the statement (2), and

Ampere-Maxwell equation $$\oint B.ds = \mu_0I + \mu_0\epsilon_0 \frac{d\Phi}{dt}$$ Where $\Phi=$ electric flux , verifies the statement (1). But if statement (3) is Correct then it violates the other two, Please tell me, about the validation of the 3rd statement.

Answer 1

It depends on how magnetic field $B$ or magnetic flux $\Phi$ varies with time $t$ i.e. linearly varying or non-linearly varying with time $t$

Case-1: If the magnetic field $B$ is varying linearly with time i.e. $B=at+b$ (Assuming area of coil $A$ is constant with time $t$) then $$\frac{d\Phi}{dt}=\frac{d(B\cdot A)}{dt}=A\frac{dB}{dt}=aA=\text{constant}\implies \text{emf}=\text{constant}$$ Thus a magnetic field varying linearly with time $t$ induces a constant electric field $E$ as the induced emf is constant.
Case-2: If the magnetic field $B$ is varying non-linearly with time say $B=at^2+bt+c$ (it may also be a sinusoidal function $B=a\sin(\omega t)$ of time $t$) then $$\frac{d\Phi}{dt}=\frac{d(B\cdot A)}{dt}=A\frac{dB}{dt}=A(2at+b)\ne \text{constant}\implies \text{emf}\ne \text{constant}$$ Thus a magnetic field varying non-linearly with time $t$ will induce a time-varying electric field $E$ as the induced emf is time-varying i.e. $\text{emf}=f(t)$.

Answer 2

None of the three claims are correct.

1. A dynamic electric field can obviously exist without $\frac{d\mathbf{B}}{dt}$ being non-zero. In fact, it can exist without even $\mathbf{B}$ being non-zero. The Faraday-Maxwell equation only implies that the curl of the electric field would be zero without a magnetic field. A dynamic electric field can exist without a magnetic field if the current density is non-zero as can be seen by the Ampere-Maxwell equation. For an explicit counter-example, see this post and Section $18.2$ from the link therein.

2. A dynamic magnetic field can obviously exist without $\frac{d\mathbf{E}}{dt}$ being non-zero. A dynamic magnetic field simply requires the curl of the electric field to be non-zero, as can be seen by the Faraday-Maxwell equation.

3. The third claim is doubly incorrect for it's simply the intersection of the first two claims.