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. 1 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. 2 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. 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. 4 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. I 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. 6 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 - 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.