Unit 4: Magnetism
Now that we have studied electric charges, potentials, and fields, let's look at the effect of moving charges: magnetism. Thales of Miletus set the stage for the scientific exploration of magnetism back in Ancient Greece, when he could only observe magnetism via the behavior of natural magnets, called lodestones.
Hans Christian Oersted documented the relationship between moving electric charges and magnetism much later, in 1820 when he accidentally discovered that an electric current could deflect a nearby compass needle. James Clerk Maxwell united electrical and magnetic phenomena into four reasonably simple equations, which we know as Maxwell's Equations, 45 years after Oersted made his observation.
The discovery that electrical currents cause magnetic effects led to the invention of the galvanometer, which we have already encountered as the core component of ammeters and voltmeters.
Completing this unit should take you approximately 7 hours.
Upon successful completion of this unit, you will be able to:
- describe the magnetic field associated with a moving charge, a magnetic dipole, a long, straight current-carrying wire, a wire loop, and a solenoid;
- find the force exerted by a magnetic field on a moving, charged particle;
- explain the fundamental difference between a magnetic and a non-magnetic material;
- state Ampere's Law for the force between two wires;
- solve problems involving the motion of a charged particle in a magnetic field; and
- calculate the torque on a current loop.
4.1: Magnets
A magnet is a material or object that creates a magnetic field. While the magnetic field is invisible, it creates a force that pulls on other ferromagnetic materials, such as iron, steel, nickel, and cobalt. It can also attract or repel other magnets. While a magnet attracts these examples of magnetic materials, non-magnetic materials, such as rubber, coins, feather and leather, are not attracted.
Read this text for a deeper microscopic picture of the substances and devices that show magnetism.
4.2: Magnetic Field
In these readings, we encounter a tool you will recognize from our discussion of electric fields: field lines. But their meaning is quite different in this context. In the case of electric field lines, the direction of the line shows you in what direction a positive test charge will be pulled. However, a magnetic field line shows you the direction in which a little compass needle will align itself!
Why couldn't we have made the definitions for electric and magnetic field lines more similar? The answer is that, with magnets you cannot separate positive and negative charges to make a test object that is purely positive. The closest things to "charges" we have in magnetism are the poles of a bar magnet, which are usually called "north" and "south" instead of "positive" and "negative". But, as this video points out, you cannot separate the north and south poles of a magnet from each other.
The fact that bar magnets can deflect compass needles is important because it gave Oersted the crucial hint that electric currents also produce magnetism. He saw that compass needles near a straight wire get deflected as soon as he turned a current on. But when he turned the current off, the compass returned to what it usually does: pointing along the Earth's north-south direction. Read this text to see the magnetic field line patterns we can construct from this behavior of the compass needle.
Watch this video, which begins with a review of the material introduced above. The video shows that a magnetic field not only deflects compass needles or other small bar magnets – it also deflects the motion of charged particles.
4.3: Magnetic Force on Moving Electric Charges
In the video we just watched, it is essential to realize how different magnetism is from electrostatic forces. For example, when you place a magnet near a negatively-charged balloon, there will be no force whatsoever between the two. But when a single negatively-charged electron flies past the same magnet, it will feel a force (even though its charge is much smaller than the charge on the balloon). It is the speed of the electron that makes all the difference. Only moving charges are affected by magnetic fields.
We call the force felt by the moving, charged object the Lorentz force. The logic behind this force is a little different from what we did in electrostatics, because at this point we do not know how to calculate the strength and direction of a magnetic field using a formula analogous to Coulomb's Law (the formula that relates the electric force to charges).
What we are doing instead is a sort of reverse logic. Let's assume a magnetic field has somehow been created. Now try to characterize the strength and direction of that field quantitatively. As with electricity, you can think of the magnetic field as a would-be magnetic force. So, to characterize the field, we need to measure a force. This could be the force that twists a compass needle away from the Earth's north-south direction, or it could be the force that deflects a moving charge.
Read this text to learn how the Lorentz force on a moving charge is related to the magnetic field, by a formula that also involves the speed of the motion. In principle, having a formula like this would make it possible to quantify the magnetic field by using known values of charge and speed in an experiment that measures the force on the charge (see equation 22.2 in the text).
The Lorentz force does not tell us how magnetic fields are created; it tells us what magnetic fields can do. Watch this video to clarify this and compare it with the electric field.
Coming back to the question how the Lorentz force equation allows us to quantify the magnetic field, watch this video, which specifically talks about the standard unit of the magnetic field, the tesla, and how it is related to the unit of force, the newton.
The formula for the Lorentz force is rather complicated – it is probably the trickiest formula you will encounter in this course due to the strange directionality of the magnetic force. Watch this video, which demonstrates that the force on a moving particle is never in the direction of the magnetic field!
Scientists often use a special symbol for the product of velocity and magnetic field when they must include the angle between them, as in the Lorentz formula. It is called the cross product. Watch this video to see this symbol written down next to the more familiar expression you encountered in the text.
Our main takeaway is that the magnetic force is actually zero if the charged particle is moving parallel to the magnetic field lines – no matter how fast it is moving or how strong the magnetic field is. That is the directional information encoded in the cross product, which shows up in the factor sin 𝜃 in the Lorentz force.
As we saw in reading above, the Lorentz force may be complicated, but it also creates some beautiful effects, such as the Aurora near the Earth's poles. The magnetic force can cause a charged particle to move in a circular or spiral path. Cosmic rays are energetic, charged particles in outer space. As they approach Earth, the Earth's magnetic field can force them into spiral paths. Magnetic forces cause protons in giant accelerators to stay in a circular path.
The curved paths of charged particles in magnetic fields form the basis of a number of phenomena that we use analytically, such as in a mass spectrometer. Read this text, which discusses these applications in more detail.
We have seen effects of a magnetic field on free-moving charges. The magnetic field also affects charges moving in a conductor. The Hall-effect principle is named for physicist Edwin Hall. In 1879 he discovered that when a conductor or semiconductor with current flowing in one direction was introduced perpendicular to a magnetic field, he could measure the voltage at right angles to the current path.
The Hall effect manifests itself as a voltage instead of a mechanical force. This makes it suitable as a compact way to measure magnetic fields by directly converting them to a voltage. We already know how to accurately measure voltages, and we can then apply the same technique to measure a magnetic field.
This video also moves on to our next topic, which is also related to the general idea of how magnetic fields deflect moving charges.
4.4: Magnetic Force on a Current-Carrying Wire
Electric currents typically consist of huge numbers of electric charges that move in a coordinated, overall motion. However, unless you see it heat up and start glowing, it is not easy to tell from the outside whether a wire is carrying a current or not. One difficulty is that even the strongest currents will not create any electrostatic force anywhere along the length of wire, at macroscopic distances.
This is because a conductor remains electrically neutral while electrons move through it. Any excess electrons that enter a segment of the wire on one end will simultaneously be made up for by electrons leaving that segment on the other end. Remember, the conductor contains equally many positive charges in the nuclei of its atoms, as there are electrons in it.
This is why magnetism is the best way to detect and quantify how many amperes of current is going through a circuit. It is created by the motion of the negatively-charged electrons that make up the current, whereas the positively-charged nuclei have no magnetic effect because they are not moving! So while the electric influences of electrons and nuclei cancel out as seen from the outside, their magnetic effects do not.
In a galvanometer (the core of an ammeter), a magnetic field is converted into a force that moves a needle. This is done by exploiting the effect discussed in this text – a current-carrying wire feels a force when it is surrounded by a magnetic field.
The law by which a current-carrying conductor is deflected is just an application of the Lorentz force which we discussed earlier, and it follows essentially the same rules. The difference is only that when talking about wires, we usually prefer to replace the speed that appears in the original Lorentz formula by the current, which is directly proportional to the speed. Watch this video for an explanation of this relationship.
Watch this video, which reviews the steps involved in replacing the electron velocity with the current.
Now that we have figured out how the magnetic force on a straight wire depends on the magnetic field and the current, we can now bend the wire into a loop, or coil. This gives rise to two new applications:
Motors are the most common application of magnetic force on current-carrying wires. Motors have loops of wire in a magnetic field. When current is passed through the loops, the magnetic field exerts torque on the loops, which rotates a shaft. Electrical energy is converted to mechanical work in the process.
Meters, such as the galvanometer, are another common application of magnetic torque on a current-carrying loop. This finally answers the question how ammeters actually work (we treated them as black boxes in the discussion of circuits earlier). As with motors, the basic idea is to convert the magnetic force into a twisting action, also called torque. This is how the indicator arrow on a meter is made to rotate to the appropriate position, indicating how many amperes are flowing through the meter.
Read this text, which discusses these applications.
4.5: Magnetic Fields Produced by Currents: Ampère's Law
Until now we have primarily discussed magnetism in situations where the magnetic field is already there, and we place some type of moving object under its influence. Now, we finally return to our question: how are magnetic fields created in the first place, and what determines their strength?
As Oersted observed, currents create magnetic fields. But we would like to have a formula that says how the strength of the field is related to the amount of current.
An electromagnet uses an electric current to create the same magnetic forces we have just discussed. We use electromagnets for everything from a crane in a wrecking yard which lifts scrapped cars, to controlling the beam of a 90-km-circumference particle accelerator, to the magnets in medical imaging machines. But if you look at an electromagnet closely, it is nothing but a loop (or coil) of wire, just like the coils we just saw in motors and meters!
How can we use the same device (a coil) for two different purposes: creating a force on a current, as in a motor, and turning a current into a magnetic field?
If you think about it, the answer is actually not that surprising if you keep Newton's Third Law in mind: action equals reaction. In a motor, a magnet created a force on the current-carrying coil via the magnetic field. By Newton's Third Law, the current-carrying coil must simultaneously be exerting a force on the magnet. That is the magnetic field created by the coil, and it makes the coil an electromagnet.
To quantify the strength and direction of the magnetic field created by flowing currents, it is best to start with the simplest case of a straight wire. As you will see when you read this text, this is the starting point from which we then progress to the magnetic fields of a simple wire loop and finally a solenoid – the kind of wire coil used in real-life electromagnets.
In all of these examples, the magnetic field is proportional to the current.
But the way the magnetic field depends on your position relative to the wires is very much affected by the geometry of the wires.
A solenoid is a long coil of wire (with many turns or loops, as opposed to a flat loop). Because of its shape, the field inside a solenoid can be uniform, and strong. The field just outside the coils is nearly zero. The magnetic field inside of a current-carrying solenoid is uniform in direction and magnitude. Only near the ends does it begin to weaken and change direction, and the field lines look a lot like those of a bar magnet.
Think back to electrostatic fields; the discussion in this section is the closest we have gotten yet to a magnetic version of Coulomb's Law. Recall that Coulomb's Law predicts a field strength (and force) that falls off with the inverse square of the distance from the source charge. For magnetic fields, the dependence on distance from the conductor is different – and in fact it is too complicated to write down as a formula in the case of a circular wire loop or a solenoid.
The simplest behavior is found for a straight wire, and the resulting formula goes back to Oersted himself, except that he had not put together a consistent system of units. The main feature is that the magnetic field in this case decreases inversely with the distance measured perpendicular to the wire. In the form given in the text, this law can be called a special case of Ampère's Law.
4.6: Magnetic Force Between Two Parallel Conductors
Originally, the name Ampère's Law described a discovery André-Marie Ampère made soon after Oersted's first report of magnetism from current-carrying wires. You can view it as an example of Newton's Third Law in action, because it puts the "source" and "recipient" of a magnetic force on equal footing, as they should be if action and reaction are equal.
Read about this experiment and its results in this section.
We can distinguish two cases, which these next two videos treat separately. First, let's send the currents through both wires in the same direction.
Next, we reverse the direction of the current in one of the wires.
What we did in these two examples is combine the Lorentz force law with the law governing the magnetic field produced by a single straight wire. One wire was chosen as playing the role of the "source" of the magnetic field, and the other wire acted as the "recipient" of the field. The latter had its moving charges deflected according the Lorentz force, producing a net force between the two wires that acts a lot like a spring: when the currents are in the same direction, it is as if the spring is stretched and wants to pull the wires together; and when the currents are in opposite directions, it is a if the spring is compressed and pushes the wires apart.
The example of the two parallel wires teaches us that we do not always want, or need, to draw the magnetic field lines created by all of the currents that are present. Here, we only drew the magnetic field lines for one of the wires. This is because the magnetic field is supposed to indicate the would-be magnetic force exerted by that single wire on the other wire, and the field lines are a tool that helps us construct that force. If, on the other hand, you wanted to add a third parallel wire to the setup and find out what the force on it would be, you would need to draw the combined field lines of the two existing wires, and that field would then tell you the force experienced by the third wire.