Physics 122 B Electricity and Magnetism Lecture 25

Physics 122 B Electricity and Magnetism Lecture 25 (Knight: 34. 1 to 34. 5) Maxwell’s Equations Martin Savage Physics 122 B - Lecture 25

Lecture 25 Announcements l Midterm 3 is graded and can be picked up at the end of lecture l. The Final Exam is Tuesday June 5 at 2. 30 – 4. 20 pm 12/17/2021 Physics 122 B - Lecture 25 2

About the Final Examination l Final is at 2: 30 p. M on Tuesday (June 5) l l l Total point value of Final = 150 points Tutorial multiple-choice question (20 pts) Lecture multiple-choice questions (60 pts) Lecture long answer questions (25 pts) Lab multiple-choice question (20 pts) Tutorial long-answer question (25 pts) You may bring three 8½” x 11” pages on which you may write anything on both sides. Also be sure to bring a Scantron sheet (pre-filled-out as much as possible) and a calculator with a good battery. l There will be assigned seating. Look up your seat assignment on Tycho before coming to the Final. l 12/17/2021 Physics 122 B - Lecture 25 3

The Series RLC Circuit The figure shows a resistor, inductor, and capacitor connected in series. The same current i passes through all of the elements in the loop. From Kirchhoff’s loop law, E = v. R + v. L + v. C. Because of the capacitive and inductive elements in the circuit, the current i will not in general be in phase with E, so we will have i = Icos(wt-f) where f is the phase angle between current i and drive voltage E. If v. L>v. C then the current i lags E and f>0. If v. C>v. L then i leads E and f<0. 12/17/2021 Physics 122 B - Lecture 25 4

Analyzing an LRC Circuit Draw the current vector I at some arbitrary angle. All elements of the circuit will have this current. 12/17/2021 Draw the resistor voltage VR in phase with the current. Draw the inductor and capacitor voltages VL and VC 900 before and behind the current, respectively. Draw the emf E 0 as the vector sum of VR and VL-VC. The angle of this phasor is wt, where the timedependent emf is E 0 cos wt. Physics 122 B - Lecture 25 The phasors VR and VL-VC form the sides of a right triangle, with E 0 as the hypotenuse. Therefore, E 02 = VR 2+(VL-VC)2. 5

Impedance and Phase Angle We can define the impedance Z of the circuit as: From the phasor diagram , we see that the phase angle f of the current is given by: 12/17/2021 Physics 122 B - Lecture 25 6

Resonance C 12/17/2021 The current I will be a maximum when w. L=1/w. C. This defines the resonant frequency of the system w 0: Physics 122 B - Lecture 25 7

Example: Designing a Radio Receiver An AM radio antenna picks up a 1000 k. Hz signal with a peak voltage of 5. 0 m. V. The tuning circuit consists of a 60 m. H inductor in series with a variable capacitor. The inductor coil has a resistance of 0. 25 W, and the resistance of the rest of the circuit is negligible. (a) To what capacitance should the capacitor be tuned to listen to this radio station. (b) What is the peak current through the circuit at resonance? (c) A stronger station at 1050 k. Hz produces a 10 m. V antenna signal. What is the current in the radio at this frequency when the station is tuned to 1000 k. Hz. 12/17/2021 Physics 122 B - Lecture 25 8

Electromagnetic Fields and Forces 12/17/2021 Physics 122 B - Lecture 25 9

Field Lines Field lines start and stop on charges (if any). Q -Q Field lines never cross. Field line spacing indicates field strength. weak strong Field lines form closed loops only when there is a current or a flux change in the other field (i. e. , energy flow). 12/17/2021 Physics 122 B - Lecture 25 10

Gauss’s Law Revisited ò ò (magnetic monopoles go here) 12/17/2021 Physics 122 B - Lecture 25 11

The Lorentz Force Coulomb’s electric force law Magnetic force on a moving charge Lorentz Force Law The most general statement of electromagnetic forces on a charge. E and B may be frame-dependent (see the later part of this lecture), but the Lorentz Force does not change with frame. 12/17/2021 Physics 122 B - Lecture 25 12

Example: The Motion of a Proton A proton is launched with velocity v 0^j into a region of space where an electric field E 0^i and a magnetic field B 0^i are parallel. How many cyclotron orbits will the proton make while traveling a distance L along the x axis? Find an algebraic expression and evaluate your answer for E 0 = 10 k. V/m, B 0 = 0. 1 T, v 0 = 1. 0 x 10^5 m/s, and L = 10 cm. 12/17/2021 Physics 122 B - Lecture 25 13

Question In what direction is the net force on the moving charge? (a) Left; (b) Right; (d) Up and left at 450; 12/17/2021 (c) Into page; (e) Down and left at 450 Physics 122 B - Lecture 25 14

The Amperian Surface ò r r B. dl = m 0 I Ampere’s Law Question: What restricts the shape and extent of the surface bounded by the integration path? Answer: The shape of the surface does not matter. Any surface should be valid. If the surface intersects no current, the line integral is zero. Otherwise, it has a non -zero value. 12/17/2021 Physics 122 B - Lecture 25 15

ò r r B. dl = m 0 I Something is Missing !!!! Maxwell’s Paradox: Consider a capacitor that is being charged by a battery, with a current flow to the positive plate and from the negative plate. If the Ampere’s Law surface goes through the wire, a current passes through it. If the Ampere’s Law surface goes through the capacitor gap, no current passes through it. Thus there is a paradox. The line integral of Ampere’s Law appears to depend on which surface is used, bringing its validity into question. Maxwell’s Solution: Add a “displacement current” term that depends on the changing electric field in the gap. r r ò B. dl = m 0 (Ithro + Idisp) 12/17/2021 Physics 122 B - Lecture 25 16

Displacement Current ò ò 12/17/2021 Physics 122 B - Lecture 25 17

Induced Magnetic Field Thus, the situation is symmetric: a changing magnetic field induces an electric field, and a changing electric field induces a magnetic field. In both cases, the induced field lines are in closed loops, and represent potential sources of energy. Note, however, that there is a sign difference. The loops are in opposite directions. 12/17/2021 Physics 122 B - Lecture 25 18

Example: Fields in a Charging Capacitor A 2. 0 cm diameter parallel plate capacitor with a 1. 0 mm gap is being charged at the rate of 0. 50 C/s. What is the magnetic field strength in the gap at a radius of 0. 5 cm? ò 12/17/2021 Physics 122 B - Lecture 25 19

A Prelude to Maxwell’s Equations Suppose you come across a vector field that looks something like this. What are the identifiable structures in this field? 1. An “outflow” structure: 2. An “inflow” structure: 3. An “clockwise circulation” structure: 4. An “counterclockwise circulation” structure: Maxwell’s Equations will tell us that the “flow” structures are charges (+ and -) and the “circulation” structures are energy flows in the field. 12/17/2021 Physics 122 B - Lecture 25 20

Maxwell’s Equations ò ò Gauss’s Law ò Faraday’s Law (magnetic monopole charge goes here) Gauss’s Law for magnetism ò Ampère-Maxwell Law (magnetic monopole current goes here) Lorentz Force Law 12/17/2021 Physics 122 B - Lecture 25 21

A Prelude to Waves Maxwell’s formulation of electricity and magnetism has an interesting consequence. The equations can be manipulated to give a wave equations for E and B of the form: This can be recognized as describing an electromagnetic wave traveling through space with a velocity of: 12/17/2021 Physics 122 B - Lecture 25 22

E or B? It’s frame dependent. Sharon runs past Bill carrying a positive charge. From Bill’s perspective the charge is moving, but from Sharon’s perspective the charge is at rest. 12/17/2021 Now turn on a magnetic field into the diagram. From Bill’s perspective the charge experiences a upward vx. B force. But from Sharon’s perspective, the charge is not moving and should experience no magnetic force. Do we have a paradox? Physics 122 B - Lecture 25 23

Galilean Relativity Consider a reference frame S that is at rest, and another reference frame S’ that is moving at a constant velocity V with respect to S. Therefore, a force F as observed in S must have the same magnitude and direction when observed in S’. 12/17/2021 Physics 122 B - Lecture 25 24

Transformation of E and B Consequently, in the reference frames of Bill and Sharon, it wasn’t the force that changes with the motion. Therefore, it must have been the fields. In Sharon’s frame, if there was no magnetic force, there must have been an electric force. In other words, in her moving frame there must have been an induced electric field that produced a force in the upward direction. More generally, if an electric field E is present in S, then in S’: 12/17/2021 Physics 122 B - Lecture 25 25

Example: Transforming the Electric Field In a laboratory at rest there are fields of E = 10 k. V/m and B = 0. 10 T , both in the +x direction in the laboratory frame. What is the electric field in a reference frame moving with velocity V = 1. 0 x 105 m/s in the +y direction. 12/17/2021 Physics 122 B - Lecture 25 26

Producing B from Moving E Now consider Sharon and Bill again. Now the charge is at rest in Bill’s reference frame. From Bill’s perspective B=0, and there is only an electric field E: From Sharon’s perspective there is the same electric field E’, since q and r are the same as in Bill’s frame: However, Sharon also sees a magnetic field B’ produced by the charge moving at -V: 12/17/2021 Physics 122 B - Lecture 25 27

Example: Two Views of a Magnetic Field A 1. 0 T magnetic field points upward. A rocket flies by the laboratory, parallel to the ground, with a velocity of 1000 m/s. What are the fields between the magnet’s pole tips, as viewed from a scientist aboard the rocket? . 12/17/2021 Physics 122 B - Lecture 25 28

Question Reference frame S observes E and B fields as shown. Which diagram shows the fields in reference frame S’? 12/17/2021 Physics 122 B - Lecture 25 29

Lecture 25 Announcements l Midterm 3 is graded and can be picked up at the end of lecture l. The Final Exam is Tuesday June 5 at 2. 30 – 4. 20 pm 12/17/2021 Physics 122 B - Lecture 25 30