Physics 122B Electricity and Magnetism

1. Physics 122B Electricity and Magnetism Lecture 20 (Knight: 32.8-.10) More Magnetic Effects Martin Savage

2. Line Integrals Made Easy If B is everywhere perpendicular to the path of integration ds, then: If B is everywhere parallel to the path of integration ds, then: Physics 122B - Lecture 120

3. r r r r r r × × × Bds Bds Bds ò ò ò O O O Ampere’s Law A special case of a line integral is one that runs in a closed path and returns to where it started, i.e., a line integral around a closed curve, which, for a magnetic field, is denoted by: Consider the case of the field at a distance d from a long straight wire: = 2 p r B = m0 I This result is: • independent of the shape of the curve around the wire; • independent of where the current passes through the curve; • depends only on the amount of current passing through the integration path. = m0 I Physics 122B - Lecture 120 Ampere’s Law

4. r r × Bds ò O Example: The Magnetic FieldInside a Current-Carrying Wire A wire of radius R carries current I uniformly distributed across its cross section. Find the magnetic field inside the wire at a distance r<R from the axis. Physics 122B - Lecture 120

5. Symmetry and Long Solenoids Can B have a radial component inside solenoid ? y Rotate 1800 about y axis Original Solenoid Reverse Current Radial B field? Therefore, radial B field components near center are ruled out by symmetry. But we can still have B fields in z and q directions. Physics 122B - Lecture 120

6. The Magnetic Fieldof a Solenoid (1) A solenoid is a helical coil of wire consisting of multiple loops, all carrying the same current. One can think of the fieldof a solenoid by superimposingthe fields from several loops,as shown in the lower figure. On the axis, the three fields will add to make a stronger net field, but outside the loop the fields from loops 1 and 3 will tend to cancel the field from coil 2. When the fields from all the loops are superimposed, the result is that the field inside the solenoid is strong and roughly parallel to the axis, while the field outside is very weak. In the limit of an ideal solenoid the field inside is uniform and parallel to the axis, while the field outside is zero. Physics 122B - Lecture 120

7. r r × Bds ò O The Magnetic Fieldof a Solenoid (2) We can use Ampere’s Law to calculate the field of an ideal long solenoid by choosing the integration path carefully. We choose a rectangular LxW loop, with one horizontal side outside the solenoid and the vertical sides passing through. If the loop encloses N wires, then Ithrough = NI. Therefore, Ampere’s Law says that: 3 4 W 2 1 If n = N/L is the number of turns per unit length, then: The first side in inside and parallel to B, so B1=B. Sides 2 and 4 are perpendicular to B (no radial B), so B2=B4=0. Side 3 is outside the solenoid, so B3=0. Therefore, B = m0NI/L Physics 122B - Lecture 120

8. Example: Generatinga Uniform Magnetic Field We wish to generate a 0.10 T magnetic field near the center of a 10 m long solenoid. How many turns are needed if the wire can carry a maximum current of 10 A? Physics 122B - Lecture 120

9. Solenoids and Bar Magnets As shown in the figures, the magnetic field of a solenoid looks very much like that of a bar magnet. The north pole of the solenoid can be identified using yet another right hand rule. Let the fingers of your right hand curl in the direction of the solenoid currents. Then your thumb will be pointing in the direction of the magnetic field and to the north pole of the solenoid. Physics 122B - Lecture 120

10. Example: Magnetic Levitation A 0.10 T uniform magnetic field is horizontal, parallel to the floor. A segment of 1.0 mm copper wire is also parallel to the floor and perpendicular to the field. What current through the wire in what direction will allow the wire to “float” in the magnetic field? (rCu=8920 kg/m3) Physics 122B - Lecture 120

11. The Force betweenTwo Parallel Wires Parallel wires carrying current in the same direction attract each other. Parallel wires carrying current in opposite directions repel each other. Physics 122B - Lecture 120

12. Example: A Current Balance Two stiff 50 cm long parallel wires are connected at the ends by metal springs. Each spring has an unstretched length of 5.0 cm and a spring constant of k = 0.020 N/m. How much current is required to stretch the springs to a length of 6.0 cm? Physics 122B - Lecture 120

13. Forces on Current Loops Parallel currents in loops attract. Opposite currents in loops repel. Magnetic poles attract or repel because the moving charges in one current producing the pole exert an attractive or repulsive magnetic force on the moving charges in the current producing the other pole. Physics 122B - Lecture 120

14. Torqueson Current Loops Consider the forces on a current loop carrying current I that is a square of length L on a side that is in a uniform magnetic field B. Its area vector makes an angle q with B. Physics 122B - Lecture 120

15. An Electric Motor We can use the torque of a loop in a magnetic field to make an electric motor. The current through the loop passes through a commutator switch, which reverses the current as the loop approaches the equilibrium position. Physics 122B - Lecture 120

16. Measuring Current with Torque The torque on a coil in a uniform field can be used to measure current. The figure shows a galvanometer or current meter. The magnetic field is arranged so that it is always perpendicular to the coil as the coil pivots on low-friction bearings. A spiral spring produces angle-dependent torque that is opposed by the magnetic field induced torque. Therefore, flowing current through the coil produces a rotation and pointer deflection that is proportional to the current. Physics 122B - Lecture 120

17. Question • What is the direction of the current in the loop? • Out at the top of the loop and in at the bottom; • Out at the bottom of the loop and in at the top; • Either direction is OK. Physics 122B - Lecture 120

18. Atomic Magnets A plausible explanation for the magnetic properties of materials is the orbital motion of the atomic electrons. Thefigure shows a classical model of an atomin which a negative electron orbits apositive nucleus. The electron's motionis that of a current loop. Consequently,an orbiting electron acts as a tinymagnetic dipole, with a north pole anda south pole. However, the atoms of most elements contain many electrons. Unlike the solar system, where all of the planets orbit in the same direction, electron orbits are arranged to oppose each other: one electron moves counterclockwise for each electron that moves clockwise. Thus the magnetic moments of individual orbits tend to cancel each other and the net magnetic moment is either zero or very small. Physics 122B - Lecture 120

19. The Electron Spin me The key to understanding atomic magnetism was the 1922 discovery that electrons have an inherent magnetic moment. Perhaps this shouldn't be surprising. An electron has a mass, which allows it to interact with gravitational fields, and a charge, which allows it to interact with electric fields. There's no reason an electron shouldn't also interact with magnetic fields, and to do so, it comes with a built-in magnetic moment. Q=-e me=-9.274x10-24 J/T An electron's inherent magnetic moment is often called the electron spin, because in a classical picture, a spinning ball of charge would have a magnetic moment. This classical picture is not a realistic portrayal of how the electron really behaves, but its inherent magnetic moment makes it seem as if the electron were spinning. An electron also has an intrinsic angular momentum. However, its magnetic moment is twice as large as a spinning sphere of charge with that angular momentum should have. This is due to quantum effects. Physics 122B - Lecture 120

20. Ferromagnetism (1) It happens that in iron (and other elements nearby in the periodic table, e.g., Co and Ni) the spins interact with each other in such a way that atomic magnetic moments tend to all line up in the same direction. Materials that behave in this fashion are called ferromagnetic. The figures show how the spin magnetic moments are aligned for the atoms making up a ferromagnetic solid. In ferromagnetic materials, the individual magnetic moments add together to create a macroscopic magnetic dipole. The material has a north and a south magnetic pole, generates a magnetic field, and aligns parallel to an external magnetic field. In other words, it is a magnet. Physics 122B - Lecture 120

21. Ferromagnetism (2) Although iron is a magnetic material, a typical piece of iron is not a strong permanent magnet. It turns out, as shown in the figure on the right, that a piece of iron is divided into small regions called magnetic domains. A typical domain size is roughly 0.1 mm. The magnetic moments of all of the iron atoms within each domain are perfectly aligned, so that each individual domain is a strong magnet. The picture shows a photograph of domains in iron. Each domain is magnetized in a different direction. Physics 122B - Lecture 120

22. Induced Magnetic Dipoles When an unmagnetized ferromagnetic material is placed in an externally applied magnetic field, magnetic domains in the material that are aligned with the field are energetically favored. This causes such aligned domains to grow, and for domains that are nearly aligned to rotate their magnetic moments to match the field direction. The net result is that a magnetic dipole moment is induced in the material, with a new south pole close to the north pole of the external magnet. If, when the field is removed, some fraction of the magnetic dipole moment remains, the material has become a permanent magnet. Physics 122B - Lecture 120

23. Hysteresis* Some ferromagnetic materials can be permanently magnetized, and “remember” their history of magnetization. The “hysteresis curve” shows the response of a ferromagnetic material to an external applied field. As the external field is applied, the material at first has increased magnetization, but then reaches a limit at (a) and saturates. When the external field drops to zero at (b), the material retains about 60% of its maximum magnetization. Partially magnetized Saturated Unmagnetized Physics 122B - Lecture 120