Conceptual difficulty with Electromagnetic induction

[ak71]

Hi all,

I have a conceptual difficulty with the idea of induction of a current in a wire due to a changing magnetic field.

Initially I was very comfortable with the idea. Upon seeing the typical setup of a square wire being rotated in a region of uniform magnetic field, I was happy to accept the outcome although I could not grasp the inner-workings of the process.

I encountered a difficult though upon seeing a different arrangement. A wire loop, being wrapped around a solenoid also has a current induced in it as the B-field inside the solenoid is increased or decreased. This pointed out a hole in my understanding. Whilst I am happy when the wire itself is bathed in B-field, the presence of a B-field only affecting the surface enclosed by the wire but not the wire itself (for the B-field outside a solenoid is near zero) confused me. I understand that that is the definition mathematically (Faraday's Law), but I can’t make the jump from the mathematics to the physics.

If someone could explain to me how a changing field in an effectively independent region of space can induce a current along a wire which itself ‘feels’ no change in field, it would be much appreciated.

Thank you very much for taking the time to read this!

 

[aralbrec]

If someone could explain to me how a changing field in an effectively independent region of space can induce a current along a wire which itself ‘feels’ no change in field, it would be much appreciated.

The wire does in fact feel the change in field, which you will see in a moment. The best way to understand it is with Stoke's theorem but first let's take a little detour.

One of Maxwell's equations, $\nabla \times E = -\partial B / \partial t$, says that a changing magnetic field at some point in space creates a changing electric field around that point. The curl operator is the same as taking the line integral of E around the point where B is changing. Notice how I am using the word 'around' because there is an infinitesimal space displacement involved here. This induced changing E field in space in turn creates a changing magnetic field as described by another of Maxwell's equations, $\nabla \times H = -\partial D / \partial t$ (I've assumed no current is flowing in free space), and again we see curl and therefore H will have a small space displacement from E. That changing H/B field in turn creates another E/D field and so on. This is the mechanism that allows EM waves (ie light) to travel through space. It's also how the changing magnetic field encircled by a wire ultimately induces an E field that makes contact with the wire.

To get at Faraday's Law, it's easiest to invoke Stoke's Theorem which I am sure you have seen illustrated before in your textbooks. Stoke's Theorem is a mathematical theorem that allows us to move from the point form of those curl equations to an integral form:

$\nabla \times E = -\partial B / \partial t \Rightarrow \oint E \cdot dL = -\int\int \partial B / \partial t \cdot dA = - \frac{\partial}{\partial t} \int\int B \cdot dA = - \frac{\partial \phi}{\partial t}$

So at some point with changing B field, the path integral of E around that point (ie the voltage) will equal the rate of change of flux through the area enclosed by the path. I am still talking about an infinitesimal path but from here you can tessellate the surface enclosed by the wire with these loops and, on adding up all the path integrals E of the infinitesimal loops making up that surface, adjacent sides will cancel except where those loops touch the wire. I'm sure you've seen this diagram before in your textbook.

So the mechanism is EM field propagation and the mathematical description of the voltage induced in the wire from Faraday's Law comes from applications of Stoke's Theorem. You can take note that EM fields propagate at the speed of light so the changing flux enclosed by a loop of wire does not in fact induce a voltage in the wire instantly but most of the time this doesn't matter -- assuming it's instant is the same assumption made in circuit analysis where E fields are assumed to establish instantly throughout a circuit.

 

[VantagePoint72]

That's a good explanation aralbrec, but I just want to highlight the punchline since the OP may be missing the forest for the trees in some of your details. Also, this exact scenario is a great way of illustrating the deeper unification of electricity and magnetism.

In the case of moving the circuit through the magnetic field (either rotating the loop or, a bit simpler, pulling a loop through an inhomogeneous field) the Lorentz force law is what causes current to flow. The electrons in the circuit have some velocity $\vec{v}$ which means they are subject to a force given by $q\vec{v} \times \vec{B}$. As long as the total flux through the loop isn't constant, there will be a net electromotive force and current will flow. I.e. the electrons are being pushed around the circuit by the magnetic force.

On the other hand, in the example you describe the electrons aren't moving (at least, not at first) so even if the magnetic field extended to the wire it wouldn't matter—since the magnetic field can't exert a force on stationary charges! Instead, the changing magnetic field induces an electric field which does reach the wire. The electric field lines around a solenoid on AC current looks like the magnetic field around a straight wire of AC current (ignoring the radiative effects aralbrec mentioned, which is justified by something called the "quasistatic approximation" you may have learned about). So as you can see, this electric field would run parallel to a loop of wire around the solenoid, and so the electrons are pushed around the loop by the electric field (alternating directions, of course).

So electromagnetic induction is actually two different effects, depending on if the circuit is moving or the magnetic field is changing. What's remarkable is that they both give the same effect! For instance imagine instead of rotating a loop of wire in a homogeneous magnetic field, you held the wire in place and rotated whatever was causing the magnetic field in the other direction. In the first case, the magnetic field pushes the electrons around the loop, and in the latter case the induced electric field does. But you'll measure the same current in both cases! Of course, we later find out that this isn't an accident: special relativity both requires the same outcome both times since motion is relative, and it explains the seemingly odd coincidence by unifying the electric field and the magnetic field into a single electromagnetic field. Whether you see an electric field or a magnetic field depends on your velocity, but the outcome of any experiments you do will be the same—as it must be!