Physics 122B Electricity and Magnetism

Lecture 25 (Knight 34.1 to 34.5) Maxwells

Equations

- Martin Savage

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

About the Final Examination

- Final is at 230 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.

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 Kirchhoffs 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 vLgtvC then the

current i lags E and fgt0. If vCgtvL then i leads

E and flt0.

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.

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

Resonance

The current I will be a maximum when

wL1/wC. This defines the resonant frequency of

the system w0

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.

Electromagnetic Fields and Forces

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).

Gausss Law Revisited

(magnetic monopoles go here)

The Lorentz Force

Coulombs 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.

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 1.0x105 m/s, and

L 10 cm.

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

The Amperian Surface

Amperes 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.

Something is Missing !!!!

Maxwells 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 Amperes Law surface goes

through the wire, a current passes through it.

If the Amperes Law surface goes through the

capacitor gap, no current passes through it.

Thus there is a paradox. The line integral of

Amperes Law appears to depend on which surface

is used, bringing its validity into question.

Maxwells Solution Add a displacement current

term that depends on the changing electric field

in the gap.

Displacement Current

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.

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?

A Prelude to Maxwells 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

Maxwells Equations will tell us that the

flow structures are charges ( and -) and the

circulation structures are energy flows in the

field.

Maxwells Equations

(magnetic monopole charge goes here)

Gausss Law

Gausss Law for magnetism

Ampère-Maxwell Law

Faradays Law

(magnetic monopole current goes here)

Lorentz Force Law

A Prelude to Waves

Maxwells 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

E or B? Its frame dependent.

Now turn on a magnetic field into the

diagram. From Bills perspective the charge

experiences a upward vxB force. But from

Sharons 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 Bills perspective the charge is

moving, but from Sharons perspective the charge

is at rest.

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.

Transformation of E and B

Consequently, in the reference frames of

Bill and Sharon, it wasnt the force that changes

with the motion. Therefore, it must have

been the fields. In Sharons 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

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.

Producing B from Moving E

Now consider Sharon and Bill again. Now the

charge is at rest in Bills reference frame.

From Bills perspective B0, and there is only an

electric field E

From Sharons perspective there is the same

electric field E, since q and r are the same as

in Bills frame

However, Sharon also sees a magnetic field

B produced by the charge moving at -V

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 magnets pole tips, as

viewed from a scientist aboard the rocket?

.

Question

Reference frame S observes E and B fields as

shown. Which diagram shows the fields in

reference frame S?

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