Presentation: flux laws & devices

Look at this levitating aluminum plate. Just look at it. And the glowing light bulbs in this candelabra mysteriously not plugged into any electrical outlet or battery. Just look at them. (Video link: "Levitating Barbecue! Electromagnetic Induction.")

In contrast to the previous presentation where we analyzed the behavior of generators using right-hand rules, in this presentation we will analyze the behavior of generators using two magnetic flux laws.

First, Faraday's law.

In order to discuss Farady's law, we need to introduce the concept of magnetic flux. For any imaginary or actual area A (such as that enclosed by a wire loop) in the presence of a (uniform magnitude and direction) magnetic field B, the magnetic flux Φ_B is the product of the magnetic field magnitude B and the area A. The perpendicular symbol "⊥" denotes that the maximum value for magnetic flux Φ_B occurs if the magnetic field lines are perpendicular to the surface (for the maximum amount of magnetic field lines passing "through" the surface); and Φ_B would be zero if the magnetic field is parallel to the surface (as no magnetic field lines would actually go "through" the surface).

The units on the right side of this equation are the product of the units of magnetic field B (in teslas) and area A (in meters²), which is defined to be the units of magnetic flux Φ_B, in T·m² or webers.

(There is a similar construct of electric flux Φ_E (defined as the measure of electric field lines passing through a surface) that we have omitted from our discussion of electromagnetism. As it turns out, what happens to the magnetic flux Φ_B is a much richer topic in terms of practical applications, such as building generators and other devices.)

Faraday's law is a statement that an induced emf ε occurs in a wire loop while the magnetic flux Φ_B through it changes, whether the magnetic field gets stronger or weaker, or by changing the orientation of the surface such that more or fewer magnetic field lines go "through" the surface. If the magnetic flux Φ_B is constant or unchanging, then there is no induced emf in the wire loop.

The amount of induced emf can be compounded by the number of coil turns N in the wire loop. If there are more turns, then there is a proportional increase in the induced emf, for a given change in magnetic flux.

The main idea of Faraday's law is that in order to induce an emf in a wire loop (such that current begins to flow), the magnetic flux Φ_B must somehow be changed.

(The negative sign on the right side of this equation is explained by Lenz's law later in this presentation.)

To give you an idea of how Faraday's law works, let's revisit two types of generators from the previous presentation that we analyzed using only right-hand rules of magnetic fields exerting forces on moving charges.

In the slide-rail generator, we have a rod moving to the right, along rails that complete a circuit, in the presence of a magnetic field that points into the plane of this page. In our previous discussion, moving the rod creates an induced emf in the rod itself, which induces current in the rest of the rod-rail circuit. The faster the rod moves, the more induced emf (and current) is produced; for a stationary rod there is no induced emf (and current).

In terms of Faraday's law, we simply note that the amount of area enclosed by the rod-rail circuit that magnetic field lines passes "through" increases steadily as the rod is moved to the right. This means that there is more and more magnetic flux Φ_B (as the area increases in Φ_B = B_⊥·A), which creates an induced emf (and current) in the rod-rail circuit. The faster the rod moves, the faster increase there is in area and flux, and the more induced emf (and current) is produced; for a stationary rod, there is no change in area and flux, so there is no induced emf (and current).

Let's apply Faraday's law again, to a rotating-coil generator. This was not easily analyzed using right-hand rules to determine how induced current would flow while the coil flips over and over in the presence of an external magnetic field. However, Faraday's law notes that the magnetic flux Φ_B is constantly changing, as the amount of magnetic field lines that pass "through" the square wire coil changes as the coil is perpendicular or sideways to the magnetic field lines, and thus an induced emf (and current) is produced in the coil. The faster the coil rotates, the faster the flux changes over time, and the more induced emf (and current) is produced; for a stationary coil, there is no change flux, so there is no induced emf (and current).

Second, Lenz's law.

Lenz's law is responsible for the negative sign in Faraday's law, and this negative sign has an important meaning. There are going to be two possibilities for the direction of a current induced in a wire loop, due to the changes in magnetic flux Φ_B through it. Lenz's law states that direction of this induced current must "oppose" the changes in magnetic Φ_B.

First, recall from a previous presentation how the third right-hand rule (RHR3) relates the magnetic field created by the current in a wire loop. Keep in mind that is the magnetic field created by the induced current in the wire loop, and not the external magnetic field that is responsible for the constant/changing magnetic flux Φ_B that passes "through" the loop.

However, the current induced in a loop (because the external magnetic flux through it is changing over time) will be in the direction that creates a magnetic field that opposes the changes in the external magnetic flux. If the flux through the coil is increasing because the external magnetic field lines are getting stronger, then the resulting induced current in the coil will "fight" this change by creating magnetic field lines in the opposite direction, to "cancel" out the strengthening external magnetic field. If the flux through the coil is decreasing because the external magnetic field lines are getting weaker, then the resulting induced current in the coil will "fight" this change by creating magnetic field lines in the same direction, to "boost" the weakening external magnetic field.

How does the coil "know" the direction that the induced current must flow in order to resist changes in the external magnetic field? Consider trying to change the motion of a heavy brick, by increasing its speed by throwing it, or decreasing its speed by catching it--how does the brick "know" how hard and which direction to press back on you as you try to speed it up, or slow it down? The brick doesn't really know, but it is merely resisting changes in its motion due to its mass (which Newton's law is that?). In an analogous manner, the coil doesn't really know which direction the induced current should flow, it is merely resisting changes in its magnetic flux...and that is Lenz's law.

For purposes of this discussion, we can personify the coil as "hating change," such that the current induced in it "kills" or "boosts" increasing or decreasing flux through it.

A dramatic application of this is where the current in the outer copper coils is ramped up and down rapidly. This creates an increasing and decreasing magnetic field and flux through the metal object (effectively a coil or loop), and this changing flux creates an induced current inside of the metal object. This induced current rapidly heats up the metal object to its melting point. (Video link: "High power induction heater owns ball bearing.")

The clip at the beginning of this presentation has a similar set-up, where there is large coil in the pedestal where current is ramped up and down rapidly, creating a changing magnetic field. This creates a continuously changing magnetic flux through the aluminum plate (effectively a coil or loop), and we can see how the induced currents in the plate continuously "fights" the changing external magnetic field of the pedestal as it levitates in the air. Also the light bulbs in the chandelier are also effectively coils that have currents induced in them due to the changing magnetic flux from the pedestal, which makes the bulbs light up.

Next time turn on your blender next to your radio. While the rapidly changing currents in your blender won't levitate your radio, it will cause the magnetic flux to continuously change through your radio circuitry, inducing currents that will be picked up as rude static.

Third, transformers.

The essential parts of a transformer are the primary coil and secondary coils, each with different numbers of turns. The metal housing ensures that all of the magnetic field created by the primary coil passes through and creates a magnetic flux through the secondary coil. Rapidly changing the current in the primary coil (as is typically done in household alternating current) creates a continuously changing magnetic field that varies the magnetic flux passing through the secondary coil. This means that the secondary coil will then have an induced emf and current produced in it. (Video link: "How to Make The Metal Melter.")

Because the primary and secondary coils have different numbers of turns (N₁ and N₂, respectively), then the voltage in the primary coil, and induced emf in the secondary coil will be different, making the transformer very useful in allowing voltages to be stepped-down, or stepped-up. This does not violate energy conservation, as the amount of energy supplied per time in the primary coil (that is, power) is ideally equal to the energy given per time to the secondary coil, such that any step-down or step-up in voltage will result in a corresponding step-up or step-down in current.

(Keep in mind that these are all time-averaged values, which seem constant, but the instantaneous values of voltages and currents are all continuously changing per time.)

If the primary coil has more N₁ turns compared to the fewer N₂ turns in the secondary coil, then supplying household alternating current with an emf of 120 V to the primary coil (again, this is a time-averaged value) will be stepped-down to an induced emf of 2.1 V (time-averaged reading from the voltmeter) in the secondary coil. This can be very dangerous, as the trickle of current in the primary coil will then be stepped-up to a very large current in the secondary coil, here to be used as a rudimentary arc welder. (Video link: "How to Make The Metal Melter.")

On the other hand, if the primary coil has fewer N₁ turns compared to the many N₂ turns in the secondary coil, then supplying varying current with an emf of 1.5 V to the primary coil (from a AAA battery) will be stepped-up to an induced emf of 220 V in the secondary coil, in order to light up a compact fluorescent light (CFL) bulb. The larger current in the primary coil will be stepped-down to a trickle of current in the secondary coil, but the voltage (and power) requirement of the CFL is still met after stepping-up. (Video link: "Lighting an 11W 220 V CFL using a 1.5 V AAA battery and a CVS disposable camera flash circuit (SD).")

(Note that the battery only provides a direct current, which by itself could not vary the primary coil magnetic field, and not vary the magnetic flux through the secondary coil, so there would be no induced emf in the secondary coil. However, an oscillating circuit will rapidly vary the otherwise direct current from the battery, and the resulting abrupt changes in the current in the primary coil then can create a fluctuating magnetic field and a fluctuating magnetic flux through the secondary coil, which has more windings, and thus a corresponding stepped-up emf. In practice, salvaging these circuits from disposable cameras should be exercised with caution, due to the amount of charge that can be stored in the capacitors within.)