# How does a time varying magnetic field confined in a cylindrical region produces induced electric field even outside the cylindrical region?

We know from Faraday's Law of Electromagnetic Induction that due to Time Varying Magnetic Field (TVMF), a non conservative electric field will be induced.

Now if we consider a cylindrical region in which the magnetic field is varying with time then outside the region of cylinder there is no magnetic field and no variation of it either, so how does the electric field gets induced there?

I understand that in the interiority of the cylindrical region, the electric field will be induced as there is TVMF, but how it gets induced outside it?

Isn't it true that Induced electric field should only generate at the location of time variation of magnetic field?

Isn't it strange TVMF can produce Induced Electric Field even at a distant location?

Is it a law of nature kind of thing that TVMF can produce induce electric field at a distant location?

But that is like claiming you can't have a magnetic field generated where there isn't any current density - which is obviously untrue.

e.g. for a steady current, we write $$\nabla \times \vec{B} = \mu_0 \vec{J}$$

Imagine the current is in a long wire. The current density $$\vec{J}$$ is zero outside the wire, yet there is a "induced" magnetic field outside the wire (with a zero curl).

In an exactly analogous way, Faraday's law says $$\nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t},$$ with now the changing B-field playing the role of current density and the E-field playing the role of the B-field. i.e. The Maths is the same, just the symbols have changed. Therefore it is perfectly possible to have an electric field where there is no changing magnetic field.

A curl-free electric field can exist even in the absence of a changing magnetic field. Only the curl of the E-field requires a changing B-field, not the electric field itself.

• I would guess that the OP doesn't know vector calculus. Perhaps an explanation from the perspective of vector potential could help build the OP's intuition. – S. McGrew Nov 23 '19 at 20:50
• @Rob: Thanks for your response. So, Is it a law of nature kind of thing that TVMF can produce induce electric field at a distant location? – Devansh Mittal Nov 24 '19 at 4:59
• I understand that the magnetic field can exist at a place where there is no current density, but why do we make this analogy to explain the effects of TVMF? How current density and TVMF are analogous? They are two different things, producing two different results and the results could be well different. – Devansh Mittal Nov 24 '19 at 5:01

Outside the solenoid, I assume this is the system that you have mind, is a vector potential. It falls off as 1/r and is directed tangentially. When the current through the solenoid is varied, dA/dt acts as an electric field.

Faraday's Law states that a time-varying $$\textbf{magnetic flux}$$ through a surface induces a electric field in said surface's border.

Now, take a circumference concentric with the cylinder in where there's a TVMF. The flux - "how much the magnetic field goes through " - the surface delimited by the circumference, varies with time. In a way, if the magnetic field is changing, you can think as if the number of magnetic field lines per volume unit is changing, and thus, the amount of lines piercing the circle (inside the cylinder) is varying. Outside the cylinder there's no magnetic field and so no line pierces the surface. However, no matter how large you take the radius of the circumference, if it's concentric to the cylinder, you'll always have the magnetic field lines piercing (in the intersection of your circle and the cylinder)

Then, around your circumference, you have an induced electric field, even outside the cylinder. As the radius increases, since the flux is constant, the induced $$\vec{E}\rightarrow 0$$.