Chapter 11: Magnetic Fields

Chapter Learning Objectives: After completing this chapter the student will be able to:  Calculate the curl of a vector field  Explain how we know that there are no magnetic monopoles.  Use Ampere’s Law to calculate magnetic fields near electrical currents.

You can watch the video associated with this chapter at the following link:

Historical Perspective: Nikola Tesla (1856-1943) was a Serbian- American inventor, engineer, and physicist. He is best known for his contributions to and advocacy for alternating current (AC). His understanding of electricity and magnetism was so far ahead of his time that some believed him to be supernatural. Having spent all of his money to prove his theories, he died penniless. The SI unit of magnetic flux density was named after him in 1960.

Photo credit: https://commons.wikimedia.org/wiki/File:N.Tesla.JPG, [Public domain], via Wikimedia Commons.

11.1 Mathematical Prelude: Curl

You will hopefully recall that in previous mathematical preludes, we introduced the divergence, which can be used to determine whether a source or sink of a vector field exists at a particular point, and the gradient, which is the vector derivative of a function. The third (and final) major vector calculus tool is the curl, which is a measure the rotation in a vector field. The curl of a vector field will provide a vector at each point that shows the direction and strength of the field’s rotation at each point.

The curl of a vector field can be calculated by taking the cross product of the del operator with the vector field:

(Equation 11.1)

The rotation of a vector field (and therefore the curl) can also be expressed by a closed loop line integral around each point, taken in the limit as the area of the closed loop approaches zero.

(Equation 11.2)

Figure 11.1 shows a visual representation of Equation 11.2. Notice that the surface area of the closed loop is shrinking closer and closer to the point being considered, so only local effects of rotation are included in the calculation of the curl at that point.

(a) (b) (c)

Figure 11.1. The curl is a line integral (path shown in red, dotted line) of a vector field (shown in black, solid line) around a specified point. The size of the closed loop path is shrunk down to approach zero, so only local effects are included in the calculation of the curl.

Figure 11.2 shows an example of a vector field that has rotation (and therefore curl) and two vector fields that have no rotation and therefore no curl. The black lines represent the vector field, and the red dotted lines represent a closed loop line integral. Notice that in part (a), all of the vector field lines tend to line up with the line integral path. In part (b), some are in the same direction, some are in the opposite direction, and some are perpendicular, but the sum is zero. In part (c), all the vector field lines are perpendicular to the path, so the curl is zero.

(a) (b) (c)

Figure 11.2. Visual representation of curl. (a) The vector field (in black) is always in the same direction as the path (in red, dotted line), so the curl is positive. (b) The vector field is has no rotation, so the curl is zero. (c) The vector field is perpendicular to the path, so the curl is zero.

So, the curl represents the rotation of the vector field. Why is this important, and how can we actually calculate it? We’ll get to why it is important later in this chapter. But to calculate it, we will use the determinant form of the cross product (first introduced back in Equation 1.9) along with the definition of the del operator:

(Equation 11.3)

The best approach to calculating this determinant is to use the diagonal method shown in Figure 1.5. This will result in the following expression:

(Equation 11.4)

Do not memorize Equation 11.4! Doing so would be a waste of time, since it can be easily derived from Equation 11.3.

There are also similar expressions for the curl of a function in cylindrical and spherical coordinates, as shown below:

(Equation 11.5)

(Equation 11.6)

Don’t forget that Equations 11.5 and 11.6 have factors in front of the determinant that must be included once the determinant has been calculated.

Example 11.1: If a vector field F can be represented by the following expression, calculate the curl of F.

Example 11.2: If a vector field B can be represented by the following expression, calculate the curl of B.

11.2 Magnetic Fields

As we have seen, all positive electric charges serve as a source of electric fields, and all negative electric charges serve as a sink of electric fields. This is true whether or not the charges are moving. But if they are moving, they also create magnetic fields as well. Ultimately, all magnetic fields are created by moving charges, even permanent magnets. (More on permanent magnets later!)

But whereas electric field lines point away from positive charges and toward negative charges, magnetic field lines rotate in a circle around moving charges. The cartoon hand is shown to demonstrate the “right-hand grip rule,” which is sometimes just called the “right-hand rule,” even though we already have a (different) right-hand rule for taking the cross product of two vectors. Here, you point the thumb on your right hand in the direction of the current, and your fingers will curl in the direction of the magnetic field.

B B B B Figure 11.3. Magnetic fields circulate around moving charges. The “right-hand grip rule” determines the direction of rotation. (Image: https://openclipart.org/detail/295787/thumbs-up, public domain)

Magnetic fields are only caused by moving charges. If charges don’t move, then no magnetic fields will be created, only electric fields. Magnetic field strength is represented by the vector B, which is called the magnetic flux density. The units of magnetic flux density are Tesla (T), which is equivalent to 1 kg/(A∙s2). Sometimes, the older unit Weber per square meter (Wb/m2) is used, but this is exactly the same as a Tesla.

A Tesla is a very large unit of magnetic flux density. (For comparison, the electromagnets that are used to move cars in junkyards are approximately 1T.) Therefore, we will sometimes use a unit called Gauss, which is equal to 10-4 T. The Earth’s magnetic field is approximately 0.5 Gauss.

You may have noticed in Figure 11.3 that, although the current causes the magnetic field, it is not technically the source of the magnetic field, since the magnetic field does not point outward and away from the current in the same way that electric fields point away from positive charges.

As a matter of fact, magnetic fields do not have a source, nor do they have a sink. Magnetic field lines always form continuous loops, with no beginning and no end. We sometimes say that

5 there is “no such thing as a magnetic monopole,” which is really just a fancy way of saying that magnetic fields always come in complete loops, with a north pole and a south pole. (More on this in the next section.)

In fact, the observation that magnetic fields have no source and no sink is so fundamental that it is actually one of Maxwell’s Equations. We have previously seen that Gauss’s Law is the first of Maxwell’s Equations, and this is the second. The integral form of this law states that a surface integral of magnetic flux density over any closed surface will always be equal to zero, since any magnetic flux that enters the surface absolutely must also leave the surface:

(Equation 11.7)

The point form of this equation is somewhat simpler. It just states that the divergence (source or sink) of the magnetic flux density is zero at every point in space.

(Equation 11.8)

11.3 Permanent Magnets

So now we understand that all magnetic fields come from the motion of charges, but this conflicts with the most common experience you have had with magnets. If all magnetic fields come from moving charges, and causing those charges to move requires a power supply, then how do permanent magnets work? It’s a question that bothered me from the time I took this class until I was about 30 years old.

Remember the simplified picture we have of an atom: a central nucleus consisting of protons and neutrons, with electrons flying in a circle around the nucleus. e-

e- e-

e- Figure 11.4. A simplified model of an atom. Electrons circulate around the nucleus.

Of course, this model is significantly over-simplified, but it is sufficient for our discussion. Notice that the electrons are charged particles, and they are constantly moving, so they are moving charges. This means that every atom has a very, very tiny magnetic field that points straight through the center of the atom. (In fact, the electrons themselves also each have a magnetic field, but we’re ignoring those for the moment.)

Ordinarily, these infinitesimally small magnetic fields tend to point in random directions, so they cancel each other out, as shown below:

e- e-

- e - e e-

- e

e - Figure 11.5. Randomly aligned atoms yield no net magnetic field.

But it turns out that, for reasons that are too complicated to explain here (a combination of chemistry, quantum mechanics, and thermodynamics, some materials demonstrate a phenomenon called domains. In particular, ferromagnetic materials (iron, nickel, and cobalt) have atoms that tend to line up in the same direction over very long distances (approximately one millimeter). Within each domain, all of the atoms line up in the same direction, but the domains are still randomly oriented, so a plain piece of iron won’t have a net magnetic field.

Figure 11.6. Magnetic domains are randomly aligned, so there is still no net magnetic field.

However, if you apply an external magnetic field to a piece of plain ferromagnetic material, the domains will begin to rotate to orient in the same direction as the external field:

Figure 11.7. An external magnetic field causes the domains to start to rotate.

If the magnetic field is strong enough, or if it is applied for a long enough time, the domains will rotate significantly to orient in the same direction as the external field:

Figure 11.8. A strong external field causes the domains to substantially line up in the same direction.

When the external magnetic field is removed, the domains will relax a little bit, but they will still tend to point in the same direction. This “memory effect” that causes the ferromagnetic material to permanently change once a magnetic field has been applied to it is known as “hysteresis.”

Figure 11.9. When the external field is removed, the domains still tend to substantially line up.

Hysteresis is the secret by which permanent magnets can be created. It is also how magnetic computer hard drives work, where an electromagnet induces a permanent change in the disk head at a particular point for each bit to be stored. That change will persist until it is overwritten.

Permanent magnets can be destroyed by heating them to a high temperature, because doing so introduces a great deal of energy into the domains, causing them to rotate out of alignment. Hitting the magnet while it is hot will demagnetize the material even faster for the same reason.

Incidentally, compasses are nothing more than a permanent magnet with a very low friction bearing that allows it to easily turn. It rotates to line up with the Earth’s magnetic field:

Figure 11.10. Earth’s magnetic field. Field lines flow out of the North Pole and into the South Pole. (Image: https://www.kisspng.com/png-earth-planet-clip-art-earth-634134/ , public domain)

The Earth’s magnetic field is caused by the circular motion of the liquid core, just as the circular motion of electrons creates the magnetic fields of permanent magnets. The magnetic flux lines flow out of the planet’s North Pole and into the planet’s South Pole, which is why permanent magnets are said to have north and south poles. The north pole of a magnet is simply the location where magnetic flux lines leave, and the south pole is where they enter.

N S

Figure 11.11. Bar and horseshoe permanent magnets. Field lines flow out of the north pole and into the south pole. (Images: http://www.publicdomainfiles.com/show_file.php?id=13526266824356 and https://pixabay.com/en/magnet-magnetic-poles-magnetism-297698/, public domain.)

In general, magnets (whether permanent or electromagnets) will always line up so that their flux lines point in the same direction. This means that they will line up so that the north pole of one magnet is typically adjacent to the south pole of an adjacent magnet.

11.4 Ampere’s Law

Now we understand how magnetic flux is generated, and in particular we understand how permanent magnets work. We can even determine the direction of magnetic flux lines caused by a current by using the right-hand grip rule. We also understand that magnetic flux lines are always continuous, with no sources or sinks. But how strong is the magnetic flux created by a particular current? The answer lies in Ampere’s Law. Ampere’s Law is very much like Gauss’s Law, except that now we will select a closed path, rather than a closed surface, and we will consider how much current is enclosed within that path. This will then allow us to calculate the B field causes by the current I. The integral form of Ampere’s Law is:

(Equation 11.9)

Notice that, just as with Gauss’s Law, we will select the symmetry of the closed path to match the symmetry of the problem, which means that B will be constant along the entire path. That will allow the integral to be reduced to a multiplication, yielding:

(Equation 11.10)

We can then solve this equation for B, the magnetic flux density:

(Equation 11.11)

The point form of Ampere’s Law allows us to determine magnetic flux density at each point, dependent only on the current density at that point.

(Equation 11.12)

(Ampere’s Law will eventually become the third of Maxwell’s four equations, but the form we are seeing today will have some more features added to it before we can officially declare it to be in its final form.)

The constant 0 essentially determines how much magnetic flux density is created by a given amount of current. The units of 0 are Henries/meter, and the unit of Henry is defined such that m0 has an exact value:

(Equation 11.13)

We will typically use the following value, which is accurate to five significant figures:

(Equation 11.14)

Example 11.3: What is the magnetic flux density surrounding a line of current I? Assume the current is flowing into the page.

Example 11.4: Determine the magnetic flux density for all values of  for a cylindrical conductor with uniform current density and a total current I.

11.5 Solenoids

A solenoid is a coil of wire with a current running through it. As shown in Figure 11.12, the magnetic flux lines created by each turn of the coil combine together to create a powerful flux density inside the coil. Solenoids are used for many industrial applications that require strong electromagnets.

Figure 11.12. A solenoid creates a strong magnetic flux density inside the coil. (Image copyright Paul Nylander, http://bugman123.com, used with permission.)

Example 11.5: Calculate the flux density inside a solenoid with N turns of wire, radius a, and length d with current I flowing through it. You can assume that the length d is much greater than the radius a.

Solenoids can also be bent into a circle, known as a torus or a toroid. Such a device is typically referred to as a toroidal solenoid.

Figure 11.13. A toroidal solenoid. (Image copyright Paul Nylander, http://bugman123.com, used with permission.)

Example 11.6: Calculate the flux density inside a toroidal solenoid with N turns of wire, radius a, and circumference d with current I flowing through it. You can assume that the magnetic flux density is constant inside the coil.

11.5 Summary

 The curl of a vector field can be used to determine how much rotation it is exhibiting. For rectangular coordinates, this can be calculated as follows:

 Magnetic fields are caused by moving charges. The strength and direction of the magnetic field is represented by the magnetic flux density, B, which has units of Tesla.  Magnetic flux lines have no sources and no sinks. They always form continuous loops that return to the point where they began. This can also be stated as the fact that, “There are no magnetic monopoles,” which is the second of Maxwell’s Equations.

 Permanent magnets are created when ferromagnetic material is subjected to a strong external magnetic field, and hysteresis causes it to remain magnetized after the field has been removed. Only iron, nickel, and cobalt (and their alloys) exhibit the necessary magnetic domains to become permanent magnets.  Ampere’s Law can be used to determine the strength of magnetic flux surrounding a current element, and the right-hand grip rule can be used to determine the direction.

 The magnetic flux density surrounding a line of current is:

 The magnetic flux density inside a solenoid or a toroidal solenoid is: