Lecture 42: FRI 05 DECLecture 42: FRI 05 DECFinal Exam ReviewFinal Exam Review

Physics 2113

Jonathan Dowling

The Grading• Midterms  -  100 points each• Final Exam  -  200 points• Homework   -  75 points • In Class PP - 25 points

TOTAL: 600 points • Numerical Grade: Total Points / 6

• A: 90–100      B: 75–89      C: 60–74      D: 50–59      F: <50

Final Exam

• 5:30PM-7:30PM MON 08 DEC

• Cox Auditorium

• ≥ 50 % of Exam From HW01-14

• 100 PTS: CH 13, 21–30

• 100 PTS: CH 31–33

• At least 1 Problem from Exam I, II, or III

Final Exam

• 8 Questions & 6 Problems

• Questions are labeled “Question” are multiple

choice or similar and no partial credit.

• Problems are labeled “Problem” and you must

show all your work to get any partial credit. In

particular an answer in a problem with no

explanation or no work will result in no credit.

What do you need to make on the final to get an A, B, C, etc.?

A: 90–100      B: 75–89      C: 60–74      D: 50–59      F: <50

Solve this simple equation for x:

Where mt1=exam1, mt2=exam2, mt3=exam3, hw=total points on your hws01–14 (out of 450), icppc=checks, icppx=X’s, icpp=number of times you were called on, max is the binary maximum function, and y is your desired cutoff number, y = 90, 75, 60, or 50. Then x is the score out of 200 you need on the final to make that cutoff grade y. This assumes no curve.

Example: John Doe wants to know what he needs to make on the final in order to get an A = 90 in this class.

It is very likely impossible for John to get an A as he’d need better than a perfect score on the final. How good does he need to do to avoid a C = 74?

John is extremely unlikely to get an A, and is unlikely to get a C, so the most probable outcome is that John will get a B in this class.

LC Circuits

The magnetic field on the coil starts to deflux, which will start to recharge the capacitor.

Finally, we reach the same state we started with (withopposite polarity) and the cycle restarts.

PHYS2113 An Electromagnetic LC OscillatorPHYS2113 An Electromagnetic LC Oscillator

Capacitor discharges completely, yet current keeps going. Energy is all in the B-field of the inductor all fluxed up.

Capacitor initially charged. Initially, current is zero, energy is all stored in the E-field of the capacitor.

A current gets going, energy gets split between the capacitor and the inductor.

Electric Oscillators: the MathElectric Oscillators: the Math

Energy as Function of TimeEnergy as Function of Time Voltage as Function of TimeVoltage as Function of Time

Amplitude = ?

ExampleExample• In an LC circuit,

L = 40 mH; C = 4 μF• At t = 0, the current is a

maximum;• When will the capacitor

be fully charged for the first time?

• ω = 2500 rad/s• T = period of one complete cycle •T = πω = 2.5 ms• Capacitor will be charged after T=1/4 cycle i.e at • t = T/4 = 0.6 ms

ExampleExample• In the circuit shown, the switch is in

position “a” for a long time. It is then thrown to position “b.”

• Calculate the amplitude ωq0 of the resulting oscillating current.

• Switch in position “a”: q=CV = (1 μF)(10 V) = 10 μC• Switch in position “b”: maximum charge on C = q0 = 10 μC• So, amplitude of oscillating current =

0.316 A

ba

E=10 V1 mH 1 μF

Damped LCR OscillatorDamped LCR Oscillator

Ideal LC circuit without resistance: oscillations

go on forever; ω = (LC)–1/2

Real circuit has resistance, dissipates energy:

oscillations die out, or are “damped”

Math is complicated! Important points:

– Frequency of oscillator shifts away from

ω = (LC)-1/2

– Peak CHARGE decays with time constant =

τQLCR=2L/R

– For small damping, peak ENERGY decays

with time constant

τULCR= L/R

C

RL

0 4 8 12 16 200.0

0.2

0.4

0.6

0.8

1.0

E

time (s)

U

t(s)

Q(t)

Example, Transformer:

B !

E

i

B

i

B

Displacement Displacement “Current”“Current”

Maxwell proposed it based on symmetry and math — no experiment!

Changing E-field Gives Rise to B-Field!

32.3: Induced Magnetic Fields:

Here B is the magnetic field induced along a closed loop by the changing electric flux E in the region encircled by that loop.

Fig. 32-5 (a) A circular parallel-plate capacitor, shown in side view, is being charged by a constant current i. (b) A view from within the capacitor, looking toward the plate at the right in (a).The electric field is uniform, is directed into the page (toward the plate), and grows in magnitude as the charge on the capacitor increases. The magnetic field induced by this changing electric field is shown at four points on a circle with a radius r less than the plate radius R.

Example, Magnetic Field Induced by Changing Electric Field:

Example, Magnetic Field Induced by Changing Electric Field, cont.:

32.4: Displacement Current:

Comparing the last two terms on the right side of the above equation shows that the term must have the dimension of a current. This product is usually treated as being a fictitious current called the displacement current id:

in which id,enc is the displacement current that is encircled by the integration loop.

The charge q on the plates of a parallel plate capacitor at any time is related to the magnitude E of the field between the plates at that time by in which A is the plate area.

The associated magnetic field are:

AND

Example, Treating a Changing Electric Field as a Displacement Current:

Magnetic Moment vs. Magnetization

32.10: Paramagnetism:

The ratio of its magnetic dipole moment to its volume V. is the magnetization M of the sample, and its magnitude is

In 1895 Pierre Curie discovered experimentally that the magnetization of a paramagnetic sample is directly proportional to the magnitude of the external magnetic field and inversely proportional to the temperature T.

is known as Curie’s law, and C is called the Curie constant.

Fig. 33-5

Mathematical Description of Traveling EM Waves

Electric Field:

Magnetic Field:

Wave Speed:

Wavenumber:

Angular frequency:

Vacuum Permittivity:

Vacuum Permeability:

All EM waves travel a c in vacuum

Amplitude Ratio: Magnitude Ratio:

EM Wave Simulation

(33-5)

Electromagnetic waves are able to transport energy from transmitterto receiver (example: from the Sun to our skin).

The power transported by the wave and itsdirection is quantified by the Poynting vector.

John Henry Poynting (1852-1914)

The Poynting Vector: The Poynting Vector: Points in Direction of Power FlowPoints in Direction of Power Flow

Units: Watt/m2

For a wave, sinceE is perpendicular to B:

In a wave, the fields change with time. Therefore the Poynting vector changes too!!

The direction is constant, but the magnitude changes from 0 to a maximum value.

The average of sin2 overone cycle is ½:

Both fields have the same energy density.

EM Wave Intensity, Energy DensityEM Wave Intensity, Energy DensityA better measure of the amount of energy in an EM wave is obtained by averaging the Poynting vector over one wave cycle. The resulting quantity is called intensity. Units are also Watts/m2.

The total EM energy density is then

The intensity of a wave is power per unit area. If one has a source that emits isotropically (equally in all directions) the power emitted by the source pierces a larger and larger sphere as the wave travels outwards: 1/r2 Law!

So the power per unit area decreases as the inverse of distance squared.

So the power per unit area decreases as the inverse of distance squared.

EM Spherical WavesEM Spherical Waves

ExampleExampleA radio station transmits a 10 kW signal at a frequency of 100 MHz. Assume a spherical wave. At a distance of 1km from the antenna, find the amplitude of the electric and magnetic field strengths, and the energy incident normally on a square plate of side 10cm in 5 minutes.

Radiation PressureRadiation PressureWaves not only carry energy but also momentum. The effect is very small (we don’t ordinarily feel pressure from light). If lightis completely absorbed during an interval Δt, the momentum Transferred Δp is given by

Newton’s law:

Now, supposing one has a wave that hits a surfaceof area A (perpendicularly), the amount of energy transferred to that surface in time Δt will be

therefore

I

A

Radiation pressure:

and twice as much if reflected.

[Pa=N/m2]

F

Radio transmitter:If the dipole antennais vertical, so will bethe electric fields. Themagnetic field will behorizontal.

The radio wave generated is said to be “polarized”.

In general light sources produce “unpolarized waves”emitted by atomic motions in random directions.

EM waves: polarizationEM waves: polarization

When polarized light hits a polarizing sheet,only the component of the field aligned with thesheet will get through.

And therefore:

Completely unpolarized light will have equal components in horizontal and vertical directions. Therefore running the light through a polarizer will cut the intensity in half: I=I0/2

Completely unpolarized light will have equal components in horizontal and vertical directions. Therefore running the light through a polarizer will cut the intensity in half: I=I0/2

EM Waves: PolarizationEM Waves: Polarization

ExampleExampleInitially unpolarized light of

intensity I0 is sent into a system of

three polarizers as shown. What

fraction of the initial intensity

emerges from the system? What

is the polarization of the exiting

light?

• Through the first polarizer: unpolarized to polarized, so I1=½I0.

• Into the second polarizer, the light is now vertically polarized. Then, I2 =

I1cos2(60o)= 1/4 I1 = 1/8 I0.

• Now the light is again polarized, but at 60o. The last polarizer is

horizontal, so I3 = I2cos2(30o) = 3/4 I2 =3 /32 I0 = 0.094 I0.

• The exiting light is horizontally polarized, and has 9% of the original

amplitude.

Completely unpolarized light will have equal components in horizontal and vertical directions. Therefore running the light through first polarizer will cut the intensity in half: I=I0/2

Completely unpolarized light will have equal components in horizontal and vertical directions. Therefore running the light through first polarizer will cut the intensity in half: I=I0/2

When the now polarized light hits second polarizing sheet, only the component of the field aligned with the sheet will get through.