Exploring Magnetism in Physics: Lectures on Biot-Savart & Ampere's Laws

1. Physics 122B Electricity and Magnetism Lecture 19 (Knight: 32.4-32.6) Biot-Savart and Ampere’s Laws Martin Savage

2. Lecture 19 Announcements • Lecture HW has been posted on Tycho and is due at 10 PM Wednesday, • The deadline for requests for regrades of Exam 2 (see Syllabus for procedure) is noon Monday. Physics 122B - Lecture 19

3. 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 19

4. 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 19

5. 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 19

6. The Cyclotron Accelerator When a charged particle moves in a uniform field, fcyc is independent of both radius and energy. One can “pump” energy into the particle as it cycles, using an electric field that varies with the frequency fcyc. This is the basic principle of the cyclotron, a particle accelerator that can increase the energies of charged particles. The ultimate energy of a cyclotron is determined by the radius R of the magnet: Emax = (RqB)2/2m Physics 122B - Lecture 19

7. Measuring Momentum with Magnetic Deflection* In other words, by measuring the orbit radius of curvature rorbit of a particle with charge q that is deflected by a uniform perpendicular magnetic field B into a circular orbit, one is measuring the momentum p of the particle. The figure shows a “split-pole magnetic spectrograph”, a magnetic analyzer used in nuclear physics research to analyze particles from nuclear reactions into groups with the same momentum. Physics 122B - Lecture 19

8. The Earth’s Van Allen Belt Charged particles tend to spiral in a magnetic field, moving along field lines in a spiral path. When the field lines “pinch” together, the particles “bounce” and reverse the direction of their motion along the field lines. This forms a “trap” that can catch and hold charged particles. The Earth’s magnetic field forms such a trap for electrons and protons emitted by the Sun. These trapped charged particles form the Van Allen radiation belts in the space around the Earth. Energetic electrons and protons leaking from the trap near the Earth’s magnetic poles, ionize the air in the upper atmosphere and produce the Aurora Borealis in the northern sky. p+ p+ e- Physics 122B - Lecture 19

9. The Hall Effect When a charged particle moves in a vacuum, it experiences a force that is perpendicular to its velocity in a magnetic field. In 1879, Edwin Hall, a graduate student at Johns Hopkins Univ., discovered that the same behavior is true of charged particles moving in a conductor. Edwin Herbert Hall(1855 – 1938) The sign of the mobile charges matters !!! Physics 122B - Lecture 19

10. Example: Hall Probe Measurement of Magnetic Field A Hall probe consists of a strip of metallic bismuth that is 0.15 mm thick and 5.0 mm wide. Bismuth is a poor conductor with a charge carrier density of n = 1.35x1025 m-3. The Hall voltage on the probe is measured to be 2.5 mV when the current is 1.5 A. What is the magnetic field, and what is the electric field inside the bismuth? Physics 122B - Lecture 19

11. The 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. [Note that the r in the numerator is r (unit vector), not r (vector)!] ^ Physics 122B - Lecture 19

12. Ds Dt DQ v = I Dt v = I Dt = I Ds The Magnetic Field of a Current Consider the charge DQ moving at speed v through a short segment of wire Ds. Therefore, we can use the Biot-Savart law to find the magnetic field B produced by the wire segment: The Biot-Savart Law for current elements Physics 122B - Lecture 19

13. Example: The Magnetic Fieldof a Long Straight Wire A long straight wire carries current I in the x direction. Find the magnetic field a distance d from the wire. dB all in same direction B is constant along the wire and falls off as 1/d. Physics 122B - Lecture 19

14. Example: The Magnetic Fieldnear a Heater Wire A 1.0 m long 1.0 mm diameter nichrome heater wire is connected to a 12 V battery. What is the magnetic field, B, at 1.0 cm from the wire? (rnichrome = 1.5 x 10-6W m.) Physics 122B - Lecture 19

15. Example: The Magnetic Fieldof a Current Loop A loop of radius R carries a current I. Find the magnetic field at a distance z from the center of the loop along the z axis. dBz Physics 122B - Lecture 19

16. Example: Matchingthe Earth’s Magnetic Field What current is needed in a 5 turn coil 10 cm in diameter to cancel the Earth’s magnetic field at the center of the coil? (BEarth = 5 x 10-5 T) Right-Hand Rule for Loops: Fingers follow current and thumb points along B. Physics 122B - Lecture 19

17. Loops as Magnetic Dipoles A current loop is a magnet. It has two distinct sides, that can be identified as its north and south poles. Magnetic lines of flux come out of the north pole and go into the south pole. A loop suspended from a thread aligns itself with the Earth’s field, with the loop’s north pole pointing north. The side of a loop from which lines of flux come out (N pole) is repelled by the north pole of a bar magnet and attracted by the south pole of a bar magnet. S N Physics 122B - Lecture 19

18. The Magnetic Dipole Moment Remember that the on-axis electric field produced by an electric dipole of electric dipole moment p has the form: The on-axis magnetic field of a current loop has a similar form, where its magnetic dipole moment m is equal to AI: Physics 122B - Lecture 19

19. Example: The Field of a Magnetic Dipole • The on-axis magnetic field 10 cm from a magnetic dipole is 1.0 x 10-5 T. What is the magnetic moment of the dipole? • If this magnetic dipole is created by a single current loop 4.0 mm in diameter, what is the current I the loop? Physics 122B - Lecture 19

20. Question 1 What is the current direction of the loop when looking down on the loop, and which side is the north magnetic pole? (a) Current cw, N pole on top; (b) Current cw, N pole on bottom; (c) Current ccw, N pole on top; (d) Current ccw, N pole on bottom; Physics 122B - Lecture 19

21. Line Integrals (1) A line integral is a special kind of vector integral in which the projection of some vector quantity is projected on a straight or curved linear path connecting two points, and the product of vector’s projection times infinitesimal path distance is summed. The simplest line integral is just the sum over the path length L: We have previously seen this kind of integration in our discussion of work: Physics 122B - Lecture 19

22. Line Integrals (2) Now consider a line integral in the presence of a magnetic field. Divide the path up into line segments of length Ds. At the kth segment the magnetic field is Bk. If B is always in the same direction as ds and constant over the path, then: Physics 122B - Lecture 19

23. Lecture 19 Announcements • Lecture HW has been posted on Tycho and is due at 10 PM Wednesday, • The deadline for requests for regrades of Exam 2 (see Syllabus for procedure) is noon Monday. Physics 122B - Lecture 19