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

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.

• What exactly is your confusion? Currently, your question is hard to be answered beyond a simple "yes/no".
– ACat
May 17, 2020 at 21:29

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. $$\text{emf}=f(t)$$.

• {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. May 18, 2020 at 0:00
• @IamthehopeoftheUniverse: Yeah, absolutely you are right. Only linearly varying magnetic field induces constant electric field. May 18, 2020 at 0:04
• could you give me any references regarding, except Faraday's law May 18, 2020 at 0:05
• 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. May 18, 2020 at 0:10
• 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. May 18, 2020 at 14:31

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.

• @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.
– ACat
May 19, 2020 at 1:12
• @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.
– ACat
May 19, 2020 at 1:22
• @IamthehopeoftheUniverse No, the post that I linked to is specifically considering the case where $\mathbf{B}=0$. So what they wrote is perfectly correct.
– ACat
May 19, 2020 at 1:25
• 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.
– ACat
May 19, 2020 at 1:45
• Can the downvoter explain?
– ACat
Aug 19, 2020 at 23:29