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

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. 4/20/2019 Physics 122B - Lecture 19

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.. 4/20/2019 Physics 122B - Lecture 19

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: 4/20/2019 Physics 122B - Lecture 19

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? 4/20/2019 Physics 122B - Lecture 19

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 4/20/2019 Physics 122B - Lecture 19

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. 4/20/2019 Physics 122B - Lecture 19

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+ e- p+ 4/20/2019 Physics 122B - Lecture 19

The Hall Effect The sign of the mobile charges matters !!! 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 !!! 4/20/2019 Physics 122B - Lecture 19

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? 4/20/2019 Physics 122B - Lecture 19

The Cross Product Form of 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)!] ^ 4/20/2019 Physics 122B - Lecture 19

The Magnetic Field of a Current Consider the charge DQ moving at speed v through a short segment of wire Ds. DQ v = I Dt v = I Dt Ds Dt = I 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 4/20/2019 Physics 122B - Lecture 19

Example: The Magnetic Field of 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. 4/20/2019 Physics 122B - Lecture 19

Example: The Magnetic Field near 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 10-6 W m.) 4/20/2019 Physics 122B - Lecture 19

Example: The Magnetic Field of 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 4/20/2019 Physics 122B - Lecture 19

Example: Matching the 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. 4/20/2019 Physics 122B - Lecture 19

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 4/20/2019 Physics 122B - Lecture 19

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: 4/20/2019 Physics 122B - Lecture 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? 4/20/2019 Physics 122B - Lecture 19

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; 4/20/2019 Physics 122B - Lecture 19

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: 4/20/2019 Physics 122B - Lecture 19

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: 4/20/2019 Physics 122B - Lecture 19

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. 4/20/2019 Physics 122B - Lecture 19