Slide 1 Physics 122B Electricity and Magnetism

Lecture 25 (Knight: 34.1 to 34.5)

Maxwell’s Equations

Martin Savage

Slide 2 ### Lecture 25 Announcements

- Midterm 3 is graded and can be picked up at the end of lecture
- The Final Exam is Tuesday June 5 at 2.30 – 4.20 pm

Physics 122B - Lecture 25

Slide 3 ### About the Final Examination

- Final is at 2:30 pM on Tuesday (June 5)
- 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.
- There will be assigned seating. Look up your seat assignment on Tycho before coming to the Final.

Physics 122B - Lecture 25

Slide 4 ### 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= vR + vL + vC.

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 = I cos(wt-f) where f is the phase angle between current i and drive voltage E. If vL>vCthen the current ilagsE and f>0. If vC>vLthen i leadsE and f<0.

Physics 122B - Lecture 25

Slide 5 ### Analyzing an LRC Circuit

Draw the current vector I at some arbitrary angle. All elements of the circuit will have this current.

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.

The phasors VR and VL-VC form the sides of a right triangle, with E0 as the hypotenuse. Therefore, E02= VR2+(VL-VC)2.

Draw the emf E0 as the vector sum of VR and VL-VC. The angle of this phasor is wt, where the time-dependent emf is E0 cos wt.

Physics 122B - Lecture 25

Slide 6 ### Impedance and Phase Angle

We can define the impedanceZ of the circuit as:

From the phasor diagram ,we see that the phase anglef of the current is given by:

Physics 122B - Lecture 25

Slide 7 C

### Resonance

The current I will be a maximum when wL=1/wC. This defines the resonant frequency of the system w0:

Physics 122B - Lecture 25

Slide 8 ### Example:Designing a Radio Receiver

- An AM radio antenna picks up a 1000 kHz signal with a peak voltage of 5.0 mV. The tuning circuit consists of a 60 mH 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.
- To what capacitance should the capacitor be tuned to listen to this radio station.
- What is the peak current through the circuit at resonance?
- A stronger station at 1050 kHz produces a 10 mV antenna signal. What is the current in the radio at this frequency when the station is tuned to 1000 kHz.

Physics 122B - Lecture 25

Slide 9 ### Electromagnetic Fields and Forces

Physics 122B - Lecture 25

Slide 10 ### Field Lines

Field lines start and stop on charges (if any).

Q

-Q

Field lines never cross.

weak

Field line spacing indicates field strength.

strong

Field lines form closed loops only when there is a current or a flux change in the other field (i.e., energy flow).

Physics 122B - Lecture 25

Slide 11 ò

ò

### Gauss’s Law Revisited

(magnetic monopoles go here)

Physics 122B - Lecture 25

Slide 12 ### 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.

Physics 122B - Lecture 25

Slide 13 ### Example:The Motion of a Proton

A proton is launched with velocity v0j into a region of space where an electric field E0i and a magnetic field B0i 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 E0 = 10 kV/m, B0 = 0.1 T, v0 = 1.0x10^5 m/s, and L = 10 cm.

^

^

^

Physics 122B - Lecture 25

Slide 14 ### Question

In what direction is the net force on the moving charge?

(a) Left; (b) Right; (c) Into page;

(d) Up and left at 450; (e) Down and left at 450

Physics 122B - Lecture 25

Slide 15 r

r

ò

B.dl = m0 I

### The Amperian Surface

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.

Physics 122B - Lecture 25

Slide 16 r

r

ò

B.dl = m0 I

r

r

ò

B.dl = m0 (Ithro + Idisp)

### 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.

Physics 122B - Lecture 25

Slide 17 ò

ò

### Displacement Current

Physics 122B - Lecture 25

Slide 18 ### 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.

Physics 122B - Lecture 25

Slide 19 ò

### 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?

Physics 122B - Lecture 25

Slide 20 ### A Prelude toMaxwell’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.

Physics 122B - Lecture 25

Slide 21 ò

ò

ò

ò

### Maxwell’s Equations

(magnetic monopole

charge goes here)

Gauss’s Law

Gauss’s Law for magnetism

Ampère-Maxwell Law

Faraday’s Law

(magnetic monopole

current goes here)

Lorentz Force Law

Physics 122B - Lecture 25

Slide 22 ### 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:

Physics 122B - Lecture 25

Slide 23 ### E or B? It’s frame dependent.

Now turn on a magnetic field into the diagram. From Bill’s perspective the charge experiences a upward vxB force. But from Sharon’s perspective, the charge is not moving and should experience no magnetic force. Do we have a paradox?

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.

Physics 122B - Lecture 25

Slide 24 ### 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’.

Physics 122B - Lecture 25

Slide 25 ### 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’:

Physics 122B - Lecture 25

Slide 26 ### Example:Transforming the Electric Field

In a laboratory at rest there are fields of E = 10 kV/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 = 1.0x105 m/s in the +y direction.

Physics 122B - Lecture 25

Slide 27 ### 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:

Physics 122B - Lecture 25

Slide 28 ### 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?

.

Physics 122B - Lecture 25

Slide 29 ### Question

Reference frame S observes E and B fields as shown. Which diagram shows the fields in reference frame S’?

Physics 122B - Lecture 25

Slide 30 ### Lecture 25 Announcements

- Midterm 3 is graded and can be picked up at the end of lecture
- The Final Exam is Tuesday June 5 at 2.30 – 4.20 pm

Physics 122B - Lecture 25