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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.

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  • $\begingroup$ What exactly is your confusion? Currently, your question is hard to be answered beyond a simple "yes/no". $\endgroup$ – Dvij D.C. May 17 '20 at 21:29
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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.

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  • $\begingroup$ @IamthehopeoftheUniverse They are using the Ampere-Maxwell equation, not ignoring it. If you can explain your objection in a more articulate manner, it'd be useful. $\endgroup$ – Dvij D.C. May 19 '20 at 1:12
  • $\begingroup$ @IamthehopeoftheUniverse Yes, and I explained that even with $\frac{d\mathbf{E}}{dt}$ being constant, the magnetic field doesn't need to be static. It can vary with time, as evident from the equation $\frac{\partial\mathbf{B}}{\partial t}=-\nabla\times\mathbf{E}$. In other words, the time-dependence of the magnetic field is dependent upon the curl of the electric field, not the time-derivative of the electric field. $\endgroup$ – Dvij D.C. May 19 '20 at 1:22
  • $\begingroup$ @IamthehopeoftheUniverse No, the post that I linked to is specifically considering the case where $\mathbf{B}=0$. So what they wrote is perfectly correct. $\endgroup$ – Dvij D.C. May 19 '20 at 1:25
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    $\begingroup$ All Maxwell equations are universally valid for all electromagnetic situations. That's the whole point. Comments are not for elongated discussions, so I won't be responding further in the comments on the same topic. I moved the discussion to chat as you can see and I'd be glad to respond there. $\endgroup$ – Dvij D.C. May 19 '20 at 1:45
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    $\begingroup$ Can the downvoter explain? $\endgroup$ – Dvij D.C. Aug 19 '20 at 23:29
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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. $\text{emf}=f(t)$.

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  • $\begingroup$ {magnetic field varying linearly with time t induces a constant electric field E as the induced emf is constant} this is the whole issue, I think time-varying magnetic field induces a time-varying electric field. $\endgroup$ – I am the hope of the Universe May 18 '20 at 0:00
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    $\begingroup$ @IamthehopeoftheUniverse: Yeah, absolutely you are right. Only linearly varying magnetic field induces constant electric field. $\endgroup$ – Harish Chandra Rajpoot May 18 '20 at 0:04
  • $\begingroup$ could you give me any references regarding, except Faraday's law $\endgroup$ – I am the hope of the Universe May 18 '20 at 0:05
  • $\begingroup$ I read Indian books of physics like NCERT, ABC, HC Verma etc. in my college. But I really don't know which book will surely help you learn. May be some PhD scholar may guide you better than I. $\endgroup$ – Harish Chandra Rajpoot May 18 '20 at 0:10
  • $\begingroup$ I think Maxwell's 3rd equation gives a clear picture ∮E⋅dl = - dΦB/dt, this is itself the modified version of Faraday's law of induction. Actually I was aiming to verify "Faraday's law of induction" by having some external references which will tell the behavior of the induced electric field by the time-varying magnetic field and vice versa. Well, I would love to know if there are any, otherwise, It was a waste of time for both of us. $\endgroup$ – I am the hope of the Universe May 18 '20 at 14:31

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