Applications of induction

Faraday’s law is not just a theoretical nicety. Induction of electric fields is central to many technologies, such as:

  • Power generation: Most electricity is generated by moving a loop of wire through a magnetic field (this is known as a dynamo).
  • Metal detectors: these generate a changing magnetic field, then any metal objects experience induced currents which in turn produce a response magnetic field. It is this response magnetic field that is detected. (Note that despite the central role of magnetism, the metal doesn’t need to be magnetic, it just needs to let currents flow from the induced electric field.)
  • Transformers: Induction is used to change the voltage of an alternating current, by generating a magnetic field from one coil which induces an emf in another coil (with a different number of turns).
  • Sparkplugs: An extreme version of a transformer is a sparkplug, in which a rapidly changing current in one coil induces a very high voltage in another coil, causing a spark to ignite fuel in an engine.
  • Induction charging: Induction is used to charge electric toothbrushes, phones, and even electric cars.
  • Induction cookers: Induction is used to heat pans without heating the stove. A coil underneath the pan creates a magnetic field, and eddy currents within the pan itself cause it to heat (YF 29.6).
  • Magnetic braking: Induction is used to slow down some trains, rollercoasters and industrial machinery using eddy currents (YF 29.6).

More specifically, these are applications of mutual induction, where the magnetic field is generated in one system A, and the induced electric field is used by another system B. For power generation, A is a permanent magnet and B is a coil. For transformers, A and B are separate coils. And so on.

Self-induction

There is another type of induction, called self-induction, where A and B are actually the same coil.

We have seen that if a current goes around a loop or coil, it makes a magnetic field within it. But:

  • if the current is changing with time, the magnetic field within the loop must also change with time;
  • by Faraday’s law this generates an electric field;
  • by Lenz’s law (or the sign in Faraday’s law), this electric field will oppose the change in current.

This is known as self-induction. The self-inductance (or often we just say ‘inductance’) $L$ is defined as

\[\mathcal{E} = - L \frac{\dd I}{\dd t}\,.\]

This is just a definition of $L$, so we have to show that it is an appropriate one by calculating the inductance $L$ for a coil, as follows.

  • We will assume the coil is long compared to its radius, and we denote its length $l$ (although we earlier called it $L$, obviously this would now be very confusing.)
  • Starting with the magnetic field inside, we have $\vec{B} = \mu_0 I n \hat{\vec{i}}$ (we derived this in lecture 13; remember $n$ is the number of loops per unit length).
  • The flux is therefore $\Phi_B = \mu_0 I n A$ where $A$ is the area of a single loop of the coil.

  • In each loop of the coil, the emf induced according to Faraday’s law is

    \[\mathcal{E}_{\mathrm{loop}} = - \frac{\dd \Phi_B}{\dd t} = -\mu_0 n A \frac{\dd I}{\dd t}\]

  • There are $nl$ loops in total, so the total emf is

    \[\mathcal{E} = - \mu_0 n^2 A l \frac{\dd I}{\dd t}\]

  • Comparing with the definition of inductance $L$ we have

    \[L = \mu_0 n^2 A l\]

If one wants a big inductance $L$, one can:

  • increase the volume of the conductor (increase $A$ and/or $l$);
  • increase the number of loops per unit length $n$;
  • place a material inside the coil with a high permeability $K_m$, in which case one should replace $\mu_0$ with $\mu = K_m \mu$ in the above expression.

Self-inductance has many applications, some of which are covered in the ‘circuits’ part of PHYS1122. These include:

  • preventing damaging current spikes in circuits (inductors resist sudden changes in current);
  • filtering out high-frequency signals (e.g. in loud-speakers with separate ‘tweeters’ and ‘woofers’);
  • generating electronic oscillations (by having a capacitor and an inductor together in a circuit).

Inductors and capacitors are highly complementary devices for electronics:

  • Capacitors allow short-lived currents or high frequencies to flow unimpeded, but long-lived currents i.e. low frequencies are blocked;
  • Inductors allow long-lived currents or low frequencies to flow unimpeded, but sudden changes i.e. high frequencies are blocked.

Energy storage in inductors

Imagine suddenly shutting off the current to a circuit that includes an inductor.

  • The inductor will generate a large emf in the opposite direction to the current, and this will drive a current in the opposite direction.
  • This is used, for example, to generate sparks in sparkplugs.
  • It also routinely generates very high voltages when you switch inductive electric equipment (many times the mains voltage) which safe electrical installations must be able to handle.
  • The energy for generating these sparks, good or bad, comes from the inductor itself.

The derivation of the energy stored is in YF 30.3, and we will outline it in the lecture too.

  • To do so, we imagine connecting an inductor to a constant source of voltage $V$, so that $V = L {\rm d}I/{\rm d}t$
  • Then the current increases linearly with time (${\rm d}I/{\rm d}t = V/L = $ constant)

  • Using this and the energy cost of moving charge around the circuit (${\rm d} U = V {\rm d} Q$), we will reach the result that the energy stored once a current $I$ is flowing is:

    \[U = \frac{1}{2} L I^2.\]
  • This energy must be stored in the magnetic field, just as we saw in Lecture 7 that capacitors store their energy in the electric field.

  • Using the exact same approach as taken for capacitors, we will see in the lecture that the energy density $u$ of the magnetic field is therefore:

    \[u = \frac{B^2}{2 \mu_0}\,.\]
  • As with the electric field result, this is a general result about energy associated with magnetic fields themselves (though the proof of this will appear next year).

Ampère-Maxwell law

We have been studying consequences of Faraday’s law. Alongside Gauss’ law for the electric and magnetic fields, this is one of the four Maxwell’s equations. But one Maxwell equation is still missing – something to tell us the strength of the magnetic field, for which Ampère’s law is the obvious candidate.

We need to modify Ampère’s law to finish our description of electromagnetism:

  • One of the signs that our description of magnetism is incomplete mentioned last lecture is that Ampère’s law sometimes leads to seemingly absurd results, as follows:
    • Consider a capacitor charging. The magnetic field around the gap between plates would be predicted by Ampère’s law to be zero, but the magnetic field just to either side of the plates is large due to the current flowing.
    • This is weird but isn’t strictly speaking an inconsistency
    • It can be made into an actual contradiction by considering e.g. a tilted loop that passes through the region where there is a magnetic field, but does not actually intersect the current.
    • The conclusion is that Ampère’s law must be incomplete.
  • This is historically the argument that Maxwell used to deduce an alteration to Ampère’s law, without any direct experimental evidence at all! Theoretical reasoning at its very best.1
  • Maxwell imagined that even in insulators (or a perfect vacuum), perhaps currents could exist momentarily when the electric field was changing:
    • The momentary currents are called ‘displacement currents’, imagining that insulators are full of charges that can move a short distance.
    • This is rather like the modern vision of a dielectric but Maxwell imagined it could happen in a vacuum too.
    • Nowadays we don’t think of the displacement current as a true current, but the name has stuck.

  • In the lecture, we will use the example of a parallel-plate capacitor to show that the magnetic field is made continuous by imagining this displacement current flowing through a surface is equal to

    \[I_{\rm disp} = \epsilon_0 \frac{\dd \Phi_E}{\dd t}\]

    where $\Phi_E$ is the electric flux through the surface.

  • The fixed Ampère law (aka the Ampère-Maxwell law) is then:

    \[\oint \vec{B} \cdot d\vec{l} = \mu_0 (I_{\rm enc} + I_{\rm disp}) = \mu_0 \left( I_{\rm enc} + \epsilon_0 \frac{\dd \Phi_E}{\dd t} \right).\]

    where $I_{\rm enc}$ is the true physical current enclosed by the loop.

  • Maxwell obtained the equation by arguments about consistency, along the lines above.
    • Obviously, this doesn’t guarantee the result correctly describes reality.
    • The fact that it’s correct was only confirmed by experiments well after Maxwell’s death (see next lecture).
    • In the modern view, the displacement current term isn’t much to do with real currents; it arises completely naturally from the formulation of electromagnetism as a gauge theory (which you will see in future years).
  • Note that the Ampère-Maxwell law is dynamic in the same sense as Faraday’s law.
    • Both laws involve time derivatives.
    • Forces therefore depend not only on where charged objects are at a given point in time, but also how they came to be there.
    • This is a huge break from Newtonian physics, in which forces are generated directly by objects at a given instant in time.
    • It is not only practically important, but it also paves the way for modern notions of cause and effect in which there should be no such thing as instant action at a distance.

Footnotes

  1. Though, before we get too overexcited, it’s worth acknowledging that theoretical reasoning can become dangerous. In Maxwell’s own words:

    I have no reason to believe that the human intellect is able to weave a system of physics out of its own resources without experimental labour. Whenever the attempt has been made it has resulted in an unnatural and self-contradictory mass of rubbish.

    He made theoretical leaps of the imagination but only where they were really strongly justified after a lot of studying experimental results. ↩