Physics 122B Electricity and Magnetism

1. Physics 122B Electricity and Magnetism Lecture 18 (Knight: 32.2,3,7) Magnetic Fields and Forces February 16, 2007 (28 Slides) Martin Savage

2. Mid Term 2 Results Physics 122B - Lecture 18

3. Lecture 18 Announcements • Lecture HW Assignment #6 has been posted on Tycho and is due at 10 PM next wed. . • Requests for regrades of Exam 2 (see Syllabus for procedure) will be accepted until noon Friday. Physics 122B - Lecture 18

4. Experiments with Magnetism:Experiment 1 Tape a bar magnet to a cork and allow it to float in a dish of water. The magnet turns and aligns itself with the north-south direction. The end of the magnet that points north is called the magnet’s north-seeking pole, or simply its north pole. The other end is the south pole. Physics 122B - Lecture 18

5. Experiments with Magnetism:Experiment 2 Bring the north poles of two bar magnets near to each other. Then bring the north pole of one bar magnet near the south pole of another bar magnet. When the two north poles are brought near, a repulsive force between them is observed. When the a north and a south pole are brought near, an attractive force between them is observed. Physics 122B - Lecture 18

6. Experiments with Magnetism:Experiment 3 Bring the north pole of a bar magnet near a compass needle. When the north pole is brought near, the north-seeking pole of the compass needle points away from the magnet’s north pole. Apparently the compass needle is itself a little bar magnet. Physics 122B - Lecture 18

7. Experiments with Magnetism:Experiment 4 Use a hacksaw to cut a bar magnet in half. Can you isolate the north pole and the south pole on separate pieces? No. When the bar is cut in half two new (but weaker) bar magnets are formed, each with a north pole and a south pole. The same result would be found, even if the magnet was sub-divided down to the microscopic level. Physics 122B - Lecture 18

8. Experiments with Magnetism:Experiment 5 Bring a bar magnet near an assortment of objects. Some of the objects, e.g. paper clips, will be attracted to the magnet. Other objects, e.g., glass beads, aluminum foil, copper tacks, will be unaffected. The objects that are attracted to the magnet are equally attracted by the north and south poles of the bar magnet Physics 122B - Lecture 18

9. Experiments with Magnetism:Experiment 6 Bring a magnet near the electrode of an electroscope. There is no observed effect, whether the electroscope is charged or discharged and whether the north or the south pole of the magnet is used. Physics 122B - Lecture 18

10. Conclusions from Experiments • Magnetism is not the same as electricity. Magnetic poles are similar to charges but have important differences. • Magnetism is a long range force. The compass needle responds to the bar magnet from some distance away. • Magnets have two poles, “north” (N) and “south” (S). Like poles repel and opposite poles attract. • Poles of a magnet can be identified with a compass. A north magnet pole (N) attracts the south-seeking end of the compass needle (which is a south pole). • Some materials (e.g., iron) stick to magnets and others do not. The materials that are attracted are called magnetic materials. Magnetic materials are attracted by either pole of a magnet. This is similar in some ways to the attraction of neutral objects by an electrically charged rod by induced polarization. Physics 122B - Lecture 18

11. Compasses and Geomagnetism The north (N) pole of a compass needle is attracted to the geographical north pole of the Earth and repelled by its geographic south pole. Apparently, the Earth is a large magnet, and one with a magnetic south pole near the Earth’s geographic north pole. The reasons for the Earth’s magnetismare complex, involving standing currents inthe Earth’s molten interior. The Earth’s magnetic poles are separated somewhat from the geographic poles, and move around some as a function of time. There is evidence in the geological record that the Earth’s field has reversed directions at varying intervals spaced by millions of years. This is not well understood. Physics 122B - Lecture 18

12. The Discoveryof the Magnetic Field In April of 1820, the Danish physicist Hans Christian Oersted was giving a evening lecture in which he was demonstrating the heating of a wire when an electrical current passed through it. He noticed that a compass that was nearby on the table deflected each time he made the current flow. Until that time, physicists had considered electricity and magnetism to be unrelated phenomena. Oersted discovered that the “missing link” between electricity and magnetism was the electric current. Hans Christian Oersted (1777- 1851) Physics 122B - Lecture 18

13. Current Effect on Compass Place some compasses around a wire. When no current is flowing in the wire, all compasses point north. When current flows in the wire, the compasses point in a ring around the wire. The Right-Hand Rule: Grip the wire so that the thumb of your right hand points in the direction of the current. Then your fingers will point in the direction that the compasses point. This is the direction of the magnetic field created by the current flow. Physics 122B - Lecture 18

14. Vector Conventions Example: field around a current. For discussions of magnetism, we will need a three-dimensional perspective. , but we will use two-dimensional diagrams when we can. To get the 3rd dimension into a two-dimensional diagram, we will indicate vectors into and out of a diagram by using crosses and dots, respectively. Rule: a dot (·) means you are looking at the point of an arrow coming toward you; a cross (´) means you are looking at the tail feathers of an arrow going away from you. Physics 122B - Lecture 18

15. The Magnetic Field • Definition of the magnetic field: • The magnetic field at each point is a vector, with both a magnitude, which we call the magnetic field strength B, and a 3-D direction. • A magnetic field is created at all points in the space surrounding a current carrying wire. • The magnetic field exerts a force on magnetic poles. The force on a north pole is parallel to B, and the force on a south pole is anti-parallel to B. Physics 122B - Lecture 18

16. Magnetic Field Lines Field Map The magnetic field can be graphically represented as magnetic field lines, with the tangent to a given field line at any point indicating the local field direction and the spacing of field lines indicating the local field strength. The field line direction indicates the direction of force on an isolated north magnetic pole. B-field lines never cross. Field Lines B-field line spacingindicates fieldstrength weak strong B-field lines always form closed loops. Physics 122B - Lecture 18

17. Two Kinds of Magnetism? We jumped from a discussion of the magnetic effects of permanent bar magnets to the magnetic fields produced by current carrying wires. Are there two distinct kinds of magnetism? No. As we will see in the lectures that follow, the two manifestations of magnetism are actually two aspects of the same fundamental magnetic force. Physics 122B - Lecture 18

18. Question • The magnetic field at point P points • Up. • Down. • Right. • Into the page. • Out of the page. Physics 122B - Lecture 18

19. Moving Charges Bav x r This is the Biot-Savart Law. It is an inverse-square law, like Coulomb’s Law, but it is more complicated because it depends on the angle between the velocity v and radius vector r from the moving charge to the point of observation. The force constant m0/4p= 10-7 T m/A, where Tesla or T are the magnetic field units. 1 T = 1 N A-1m-1. Note that B = 0 when q = 00 or 1800 and is maximum at 900. Physics 122B - Lecture 18

20. Field of a Moving Charge All charges create E Fields, but only moving charges create B fields. Physics 122B - Lecture 18

21. Example: The Magnetic Fieldof a Proton A proton moves on the x axis with velocity vx = 1.0 x 107 m/s. As it passes the origin, what are the magnetic fields at Cartesian coordinate locations (1,0,0), (0,1,0), and (1,1,0) in mm? Position (1,0,0) is along the x axis, so q=0 and Bx=0. Position (0,1,0) is along the y axis, so q=900 and r=1 mm. Physics 122B - Lecture 18

22. Superposition Magnetic fields obey the principle of superposition. The magnetic fields from a number of moving charges add vectorially to produce a net magnetic field. In particular, the magnetic fields produced by individual electrons moving in a current-carrying wire add to produce a magnetic field that is characteristic of the net current. Physics 122B - Lecture 18

23. Multiplying Vectors Given two vectors: There are two ways of mutiplying these vectors: Dot Product (Scalar Product) Cross Product (Vector Product) A·B is B times the projectionof A on B, or vice versa. (determinant) Physics 122B - Lecture 18

24. Note that the r in the numerator is r (unit vector), not r (vector)! ^ Cross Product Formof the Biot-Savart Law The Biot-Savart Law can be represented more compactly using a vector cross product. This automatically gives a B field that is perpendicular to the plane of the charge velocity and radius vector to the point at which the field is being evaluated. Physics 122B - Lecture 18

25. Example: The Magnetic Fieldof a Electron An electron is moving to the right, as shown in the diagram. What is the direction of the electron’s magnetic field at the position indicated by the black dot? · Physics 122B - Lecture 18

26. Question 2 v · • The positive charge is moving straight out of the page. What is the magnetic field at the position of the dot? • Up. • Down. • Left • Right. • Out of the page. Physics 122B - Lecture 18

27. r r ò B.dl = m0 I Ampere’s Law From Biot-Savart Law : Current penetrating the surface enclosed by closed loop integration Magnetic analogue of Gauss’s Law Physics 122B - Lecture 18

28. r r ò B.dl = |B| 2 p r = m0 I B-field of Infinite Straight Current Carryng Wire Cylindrical symmetry, magnitude of B is same at any azimuthal angle r dl B m0 I |B| = 2 p r I Physics 122B - Lecture 18

29. Ampere’s Experiment When Ampere heard of Oersted’s results, he reasoned that if a current produced a magnetic effect, it might respond to a magnetic effect. Therefore, he measured the force between two parallel current-carrying wires. He found that parallel currents create an attraction between the wires, while anti-parallel currents create repulsion. André Marie Ampère (1775 – 1836) Physics 122B - Lecture 18

30. Magnetic Force A current consists of moving charges. Ampere’s experiment implies that a magnetic field exerts a force on a moving charge. This is true, although the exact form of the force relation was not discovered until later in the 19th century. The force depends on the relative directions of the magnetic field and the velocity of the moving charge, and is perpendicular to both.. Physics 122B - Lecture 18

31. Magnetic Forceon Moving Charges • Properties of the magnetic force: • Only moving charges experience the magnetic force. There is no magnetic force on a charge at rest (v=0) in a magnetic field. • There is no magnetic force on a charge moving parallel (a=00) or anti-parallel (a=1800) to a magnetic field. • When there is a magnetic force, it is perpendicular to both v and B. • The force on a negative charge is in the direction opposite to vxB. • For a charge moving perpendicular to B (a=900) , the magnitude of the force is F=|q|vB. Physics 122B - Lecture 18

32. Example: Magnetic Force on an Electron A long straight wire carries a 10 A current from left to right. An electron 10 cm above the wire is traveling to the right with a speed of 1.0x107 m/s. What is the magnitude and direction of the force on the electron. Direction of F is up. Electron will move away from the wire. Physics 122B - Lecture 18

33. Cyclotron Motion Consider a positive charged particle with mass m and charge q moving at velocity v perpendicular to a uniform magnetic field B. The particle will move in a circular path of radius rcyc because of the force F on the particle, which is: Physics 122B - Lecture 18

34. Example: The Radius of Cyclotron Motion An electron is accelerated from rest through a potential of 500 V, then injected into a uniform magnetic field B. Once in the magnetic field, it completes a half revolution in 2.0 ns. What is the radius of the orbit? Physics 122B - Lecture 18

35. Lecture 18 Announcements • Lecture HW Assignment #6 has been posted on Tycho and is due at 10 PM next wed. . • Requests for regrades of Exam 2 (see Syllabus for procedure) will be accepted until noon Friday. Physics 122B - Lecture 18