Intermediate Electromagnetism and Electromagnetic Induction
DC Electric Circuits
As an electric current is passed through a coil of wire, it creates a magnetic field. If the magnitude of this current changes over time, so will the strength of the magnetic field.
We also know that a magnetic field flux that changes over time will induce a voltage along the length of a wire coil. Explain how the complementary principles of electromagnetism and electromagnetic induction manifest themselves simultaneously in the same wire coil to produce self-induction.
Also, explain how Lenz’s Law relates to the polarity of the coil’s self-induced voltage.
∫f(x) dx Calculus alert!
In a simple resistor circuit, the current may be calculated by dividing applied voltage by resistance:
Although an analysis of this circuit probably seems trivial to you, I would like to encourage you to look at what is happening here from a fresh perspective. An important principle observed many times in the study of physics is that of equilibrium, where quantities naturally “seek” a state of balance. The balance sought by this simple circuit is equality of voltage: the voltage across the resistor must settle at the same value as the voltage output by the source:
If the resistor is viewed as a source of voltage seeking equilibrium with the voltage source, then current must converge at whatever value necessary to generate the necessary balancing voltage across the resistor, according to Ohm’s Law (V = IR). In other words, the resistor’s current achieves whatever magnitude it has to in order to generate a voltage drop equal to the voltage of the source.
This may seem like a strange way of analyzing such a simple circuit, with the resistor “seeking” to generate a voltage drop equal to the source, and current “magically” assuming whatever value it must to achieve that voltage equilibrium, but it is helpful in understanding other types of circuit elements.
For example, here we have a source of DC voltage connected to a large coil of wire through a switch. Assume that the wire coil has negligible resistance (0 Ω):
Like the resistor circuit, the coil will “seek” to achieve voltage equilibrium with the voltage source once the switch is closed. However, we know that the voltage induced in a coil is not directly proportional to current as it is with a resistor - instead, a coil’s voltage drop is proportional to the rate of change of magnetic flux over time as described by Faraday’s Law of electromagnetic induction:
vcoil = Instantaneous induced voltage, in volts
N = Number of turns in wire coil
[(d φ)/dt] = Instantaneous rate of change of magnetic flux, in webers per second
Assuming a linear relationship between coil current and magnetic flux (i.e. φ doubles when i doubles), describe this simple circuit’s current over time after the switch closes.
A very useful method of measuring current through a wire is to measure the strength of the magnetic field around it. This type of ammeter is known as a clamp-on ammeter:
Knowing the principle behind this meter’s operation, describe what current values will be indicated by the three clamp-on ammeters in this circuit:
- Meter A =
- Meter B =
- Meter C =
Lenz’s Law describes the opposition to changes in magnetic flux resulting from electromagnetic induction between a magnetic field and an electrical conductor. One apparatus capable of demonstrating Lenz’s Law is a copper or aluminum disk (electrically conductive, but non-magnetic) positioned close to the end of a powerful permanent magnet. There is no attraction or repulsion between the disk and magnet when there is no motion, but a force will develop between the two objects if either is suddenly moved. This force will be in such a direction that it tries to resist the motion (i.e. the force tries to maintain the gap constant between the two objects):
We know this force is magnetic in nature. That is, the induced current causes the disk itself to become a magnet in order to react against the permanent magnet’s field and produce the opposing force. For each of the following scenarios, label the disk’s induced magnetic poles (North and South) as it reacts to the motion imposed by an outside force:
Combining Lenz’s Law with the right-hand rule (or left-hand rule, if you follow electron flow instead of conventional flow) provides a simple and effective means for determining the direction of induced current in an induction coil. In the following examples, trace the direction of current through the load resistor:
If an electric current is passed through this wire loop, in which position will it try to orient itself?
If this experiment is carried out, it may be found that the torque generated is quite small without resorting to high currents and/or strong magnetic fields. Devise a way to modify this apparatus so as to generate stronger torques using modest current levels and ordinary magnets.
The relationship between magnetic flux and induced voltage in a wire coil is expressed in this equation, known as Faraday’s Law:
e = Instantaneous induced voltage, in volts
N = Number of turns in wire coil
φ = Instantaneous magnetic flux, in webers
t = Time, in seconds
Explain what the mathematical expression [(d φ)/dt] means, in light of what you know about electromagnetic induction. Hint: the [d/d] notation is borrowed from calculus, and it is called the derivative.
Also, explain why lower-case letters are used (e instead of E, φ instead of Φ) in this equation.
If a copper ring is brought closer to the end of a permanent magnet, a repulsive force will develop between the magnet and the ring. This force will cease, however, when the ring stops moving. What is this effect called?
Also, describe what will happen if the copper ring is moved away from the end of the permanent magnet.
Electromechanical watt-hour meters use an aluminum disk that is spun by an electric motor. To generate a constant “drag” on the disk necessary to limit its rotational speed, a strong magnet is placed in such a way that its lines of magnetic flux pass perpendicularly through the disk’s thickness:
Explain the phenomenon behind this magnetic “drag” mechanism, and also explain how the permanent magnet assembly should be re-positioned so that it provides less drag on the disk for the same rotational speed.
One context in which to understand Lenz’s Law is the well-known physical law called the Conservation of Energy, which states that energy can neither be created (from nothing) nor destroyed (to nothing). This well-founded law of physics is the general principle forbidding so-called “over-unity” or “free energy” machines, where energy would supposedly be produced from no source whatsoever.
Demonstrate that if Lenz’s Law were reversed, the Conservation of Energy principle would be violated. In other words, imagine what would happen if the effects of Lenz’s Law were exactly opposite in direction, and show how this would result in more energy produced by a system than what is input to that system.
If the motion of a conductor through a magnetic field induces a voltage in that conductor, it stands to reason that a conductive fluid moving through a pipe can also generate a voltage, if properly exposed to a magnetic field. Draw a picture showing the necessary orientation of the pipe, the magnetic field, and the electrodes intercepting the induced voltage.
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