Ch24 Page 1. Chapter 24 Magnetic Fields and Forces Thursday, March 11, :26 PM

Transcription

1 Ch24 Page 1 Chapter 24 Magnetic Fields and Forces Thursday, March 11, :26 PM

2 Ch24 Page 2 It seems that microscopic electric currents are the ultimate cause of magnetism. For example, each neutron has a little bit of internal magnetism; in technical language, we say that each neutron has a non-zero magnetic dipole moment. In other words, part of the nature of a neutron is that it acts like a very tiny bar magnet. The cause of the magnetism of a neutron is thought to be circulating charges within the neutron; remember that neutrons are composed of three quarks, all of which have electric charge. Each proton also has a magnetic dipole moment, presumably for the same reason as neutrons: Protons are also composed of circulating quarks. Each electron also has a magnetic dipole moment, which is mysterious, because electrons have no known internal structure, as far as our best experimenters can tell so far. If the electron truly is a "point" particle, then this becomes even more mysterious; how on earth can the electron have magnetism, if magnetism is due to circulating electric current within the electron, when the electron has no room inside it for any circulating anything? Recognizing that electric currents create magnetic fields raises the opposite question: Do magnetic fields have any influence on charged particles? The answer is yes, but in a much more complicated way than the way charged particles exert forces on each other. A magnetic field exerts NO force whatsoever on a charged particle at rest. Furthermore, a magnetic field exerts NO force on a charged particle if the charged particle moves parallel to a magnetic field line. The only way a magnetic field exerts a force on a charged particle is if the charged particle has a nonzero component of its velocity perpendicular to the magnetic field. Furthermore, the magnitude of the force exerted by the magnetic field on the charged particle depends on the strength of the magnetic field, the magnitude of the particle's charge, and the magnitude of the component of the particle's velocity that is perpendicular to the field. We'll discuss this in more detail later.

3 Ch24 Page 3 All this strange interaction of charged particles and magnetic fields becomes a bit less mysterious when you approach the subject from the perspective of special relativity. the theme of unification in physics; Maxwell, Einstein Example: Here is the magnetic field pattern for a magnetic dipole; note the similarity to the electric field pattern for an electric dipole. Here is the magnetic field pattern for two bar magnets placed with unlike poles nearby:

4 Here is the magnetic field pattern for two bar magnets placed with like poles nearby: Ch24 Page 4

5 Earth's magnetic field: Ch24 Page 5

6 Ch24 Page 6 "Refrigerator" magnets Magnetic hard-disk drive

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11 Example: The two insulated wires in the figure cross at a 30 angle but do not make electrical contact. Each wire carries a 5.0 A current. Points 1 and 2 are each 4.0 cm from the intersection and are equally distant from both wires. Determine the magnitudes and directions of the magnetic fields at points 1 and 2. Ch24 Page 11

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13 Ch24 Page 13 In using the right-hand rule, you are free to change the order of F, B, and v, as long as you keep them in "cyclic order." (That is, FBv, BvF, and vfb are all in the same cyclic order.) Different textbooks use different orders that nevertheless have the same cyclic order, so you may have learned a different version of the rule in a previous course. Just use whichever cyclic order is convenient for you, as they all work just fine. Some people like to use a variant of the right-hand rule where they curl their fingers through the acute angle

14 Ch24 Page 14 between vectors v and B; then their thumb points in the direction of the cross product of v and B. This is why I like to place my index finger in the direction of v and my middle finger in the direction of B; then either version of the righthand rule leads to my thumb pointing in the direction of the cross product. Also note carefully that to get the direction of the force F, you have to flip the direction of the cross product only if the charge is negative; if the charge is positive, the direction of the force is the same as the direction of the cross product. For this reason, some teachers use a left-hand rule, where they assume that negatively-charged particles are moving, rather than positively-charged particles. I won't do this, but will stick to the usual "conventional current" convention, and always use right-hand rules.

15 Ch24 Page 15 Path of a charged particle in a uniform magnetic field: could be a circle, a helix, or a straight line, depending on the initial direction of the charged particle. There are three possibilities: if the initial velocity of the charged particle is parallel to the magnetic field, the magnetic field does not exert a force on the charged particle, so the particle's trajectory is a straight line, by Newton's first law of motion if the initial velocity of the charged particle is perpendicular to the magnetic field, the particle moves in a circle: if the initial velocity of the charged particle has nonzero components both parallel and perpendicular to the magnetic field, then the particle's path will be a combination of the two types of motion just discussed, and the path ends up being a helix:

16 A natural example of this type of motion of a charged particle in a magnetic field is aurorae in Earth's atmosphere: Ch24 Page 16

17 Ch24 Page 17 The bending of a charged particle in a magnetic field also provides the idea for a mass spectrometer, which separates samples of particles by mass: How does one ensure that the velocities of the injected particles are the same? For one possibility, see the example a little further below. The same phenomenon is used to accelerate charged particles in the particle accelerators used for highenergy physics experiments. The accelerator in the following diagram is called a cyclotron:

18 Ch24 Page 18 The bending of a charged particle in a magnetic field also provides the idea for an electromagnetic flow meter: Example: Velocity selector How does one arrange for the charged particles in a mass spectrometer to enter the region of the magnetic field with the same velocities? There are a number of different "velocity selectors" that can be used; one such is based on using a region of space with crossed electric and magnetic fields, illustrated below. Derive a formula for the speed of a particle that will go straight through the velocity selector without deflection in terms of the strengths of the fields.

19 Ch24 Page 19 Solution: For a positively charged particle injected into the velocity selector, the electric field exerts a downward force of magnitude qe and the magnetic field exerts an upward force of magnitude qvb. The net force on the charged particle is zero when these two forces balance: Thus, if you choose the electric and magnetic fields just right, you can get zero deflection for just the desired speed. Notice that the result is independent of the charge of the particle, so particles with different charges but the same speed can be injected into the mass spectrometer.

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24 Ch24 Page 24 Magnetic Fields Exert Torques on Current Loops Copyright 2007, Pearson Education, Inc., Publishing as Pearson Addison -Wesley. Slide 24-42

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26 CP 24 Problem describes two particles that orbit the earth's magnetic field lines. Calculate the frequency of the circular orbit for (a) an electron with speed m/s, and (b) a proton with speed m/s. (The strength of the earth's magnetic field is approximately T.) Ch24 Page 26

27 Ch24 Page 27 CP 26 A mass spectrometer similar to the one in Figure is designed to separate protein fragments. The fragments are ionized by the removal of a single electron, then they enter a 0.80 T uniform magnetic field at a speed of m/s. If a fragment has a mass that is 85 times the mass of the proton, determine the distance between the points where the ion enters and exits the magnetic field. CP 45 The two 10-cm-long parallel wires in the figure are separated by 5.0 mm. For what value of the resistor R will the force between the two wires be N?

28 CP 47 An electron travels with a speed of m/s between two parallel charged plates, as shown in the figure. The plates are separated by 1.0 cm and are charged by a 200 V battery. What magnetic field strength and direction will allow the electron to pass between the plates without being deflected? Ch24 Page 28

29 CP 56 A 1.0-m-long, 1.0-mm-diameter copper wire carries a current of 50.0 A towards the East. Suppose we create a magnetic field that produces an upward force on the wire exactly equal in magnitude to the wire's weight, causing the wire to "levitate." What are the magnetic field's magnitude and direction? Ch24 Page 29

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