Final Exam PHYS 1220 Fantasy Term 12/16/2044

__________ nameNine problems of equal value.

You can skip four of the problems. You must work at least one of problems 5, 6, and 7 and at least any two of problems 5 through 9. If you work more problems, the best grades that are consistent with the above rules will count toward the exam grade.

By choosing which problems you want to skip you can turn the exam from a cumulative to a non-cumulative one.

Exam Tips:

Show your work for full credit.

All multiple choice problems may have more than one correct answer. Full credit will be awarded when you mark all correct answers and no incorrect answer.

Problem 1 Lecture videos

Recall the following lecture videos:

1. Energy stored in a capacitor

2. Freezing Liquid Nitrogen

For one of these two lecture videos,

a) Describe the experiment and the physical principles tested.Use equations, phase diagrams, etc. as appropriate.

b) What would be a random error in that experiment? Explain what this error would do to your data.

c) What would be a systematic error in that experiment? Explain what this error would do to your data.

Problem 2 Electrostatics: Electric fields and electric potential differences

Three point charges with charge +2q, -q, and -2q are arranged along the x-axis as shown (see figure).

a) Divide the x-axis into suitable ranges and explain for each range with appropriate vector diagrams whether the total electric field could be zero somewhere along the axis.

b) For the region between ‘-q’ and ‘+2q’, where is the point of zero potential difference, if any?

Hint: work in multiples of q

4 π ε0 for any calculation

Problem 3 – Thermodynamics

A heat engine takes 1 mole of an ideal diatomic gas first in an isobaric process at 4 105 [Pa] from point ‘a’ at a volume of 0.005 [m3] to point ‘b’ at 0.01 [m3]. Next the pressure is iso-volumetrically lowered to half its original value to point ‘c’. Finally, the cyclic process goes in a single quasi-static process back to the original point ‘a’ in the pV diagram.

a) Show that the final process is isothermal.

b) Calculate the temperatures at points a, b, and c.

c) Draw the pV and the pT diagram.Hint: Make sure to show all relevant information

d) How much heat is exchanged during one cycle? Is the net heat released or absorbed by the gas?

Problem 4 Electrostatics, Gauss’ Law

Consider an insulating very long solid cylinder of radius R with a charge density of -2uniform throughout.

Use Gauss’s Law to determine expressions for the electric field inside and outside the cylinder. Explain your choice of Gaussian surfaces.

Problem 5 – Magnetic Fields

Consider at each of the points a, b, c, d, and e in the picture a positive charge of magnitude q. All charges move with the same speed v in the directions shown by the arrows. The side length of the cube, which is shown to illustrate directions and distances, is L.

a) Determine the direction of the magnetic force on charges b, c, d, and e due to charge a, if any.

b) Determine the direction of the magnetic force on each charge due to an isotropic field

Bext=B0 i−B0 k

Problem 6 – Thermodynamics

A straight composite rod consists of a 1[m] long section of steel and a section of copper of unknown length. Both parts have the same cross section: 2 [cm] by 2 [cm]. The rod is completely insulated except at the end plates: they are immersed in boiling water (steel) and ice water (copper).

a) How long (in meters) does the copper section have to be so that the steady state temperature at the steel-copper joint is 50[°C]?

We watched a number of videos in lecture and demonstrations regarding thermodynamic processes that occur, which take a substance along a path in a phase diagram. Among these were:

• fast freezing water in a long enclosure of which one end is cooled by liquid nitrogen;• water that starts boiling when ice is added to the outside of its closed container;• water that starts boiling when a vacuum pump evacuates the gas on top of the water;• a cotton ball that ignites when a piston is moved.

b) Explain two of these demonstrations and draw for each the path the water takes in the p-V and in the p-T diagram.

Problem 7- RC circuits

a) Which statements about RC circuits with an initially charged capacitor are correct? Indicate clearly for each whether the statement is correct or false.

A- On discharging the capacitor, the current is largest at first and then the current asymptotically approaches zero.

B- During the discharge process, the capacitor produces less and less voltage drop in the circuit.

C- The time constant for discharging is different from the time constant for charging the capacitor.

D- After the time for the time constant has elapsed, the capacitor is fully discharged.

E- During the discharging of the capacitor the current in the resistor is constant.

b) Derive the equations for q(t) and i(t) for the case of discharging the capacitor.

c) Draw the q(t) and i(t) diagrams.

Problem 8 – First Law of ThermodynamicsAn engine takes 2 moles of an ideal diatomic gas through the cycle (1-4) shown in the figure. C v=

52

R(diatomic ideal gas)

Hint: CP = CV + R. Express your results in multiples of p0V0.

a) Find the Work done in each process

b) Find the heat absorbed or released in each process

c) Compare your results to those one gets from a geometrical analysis of the plot for the total work and total heat for the cyclical process. Explain how Wtot and Qtot relate and why they relate that way.

9. Resistor Networks

a) Derive the equations for series and for parallel network equivalent __resistance.

For part b) and c), consider the resistor network in the figure below:

a) Find the equivalent resistance of the circuit.b) What is the voltage across the 15 resistor? What is the current through

the 3 resistor?Hint: Show your work for full credit.

Master Equations – Physics 1220

∆ L=α L0 ∆ T∧∆ V =βV 0 ∆ T

Q=mc ∆ T∧Q=± mL

( dQdt )

cond=kA

T H −TC

L∧( dQ

dt )rad

=A e σ ∙ (T 4−T S4 )

p ∙V =n RT∧mtot=nM , M=N A m

K trans=32

kB T∧vrms=√ 3 kB Tm

W =∫V 1

V 2

p dV∧∆ U =Q−W

e= WQH

∧eCarnot=1−T C

T H

∆ S=∫1

2 dQT

∧S=k ∙ ln (w )

F= 14 π ε0

∙|q1|∙|q2|

r2 ∧E=F0

q0

ΦE=∫ E ∙d A=Qencl .

ε0

U = 14 π ε0

∑i

q i

ri∧V =U

q0∧E=−(i ∂V

∂ x+ j ∂V

∂ y+k ∂ V

∂ z ) C ≡ Q

V ab∧C plate=ε0

Ad

∧,∈series , 1Ceq

=∑i

1C i

,∧,∈¿ ,C eq=∑i

Ci

U= Q2

2 C=1

2C V 2=1

2QV ∧u=1

2ε E2

I=dQdt

=n|q|vd A∧J=n q vd

ρ=EJ∧V =IR with R=ρ L

A

P=V ab ∙ I=I2 R=V ab

2

R

Req=∑i

Ri (series ) 1Req

=∑i

1R i

(¿)

Kirchhoff Rules∑ I=0 ( junctionrule ) ,∑ V=0 (looprule)

Capacitor charging q=Qf ∙ (1−e−tRC )∧i= ε

R∙ e

−tRC

F=q v× B and ΦB=∮ B ∙ d A=0 and F=I l × B( for straight wires)

B= μ0

4 π∙ q v ×r

r2 magnetic field created by moving charge

B=μ0

4 π∙ I d l× r

r2 Biot−Savart field created by straight wire carrying current

B=μ0 I2 πr

and FL

=μ0 I ∙ I '

2 πrfor 2wires

∮ B ∙ d l=μ0 I encl Amper e' sLaw

ε=−dΦB

dt=∮ E ∙ d l Faraday ' s Law

ε=−L didt

where L=N ΦB

i

U=12

L I 2∧u= B2

2 μ

ω=√ 1LC

for LC circuits∧ω '=√ 1LC

− R2

4 L2

I rms=I

√2∧i=I ∙ cos ( ωt ) V accordingly AC current

V R=IR , V L=I X L ,V C=I X C ,V=IZ

Z=√R2+ ( X L−XC )2=√ R2+(ωL− 1ωC )

2

Pavg=12

VI cos (ϕ )∧tan (ϕ )=ωL− 1

ωCR

B=μ0ε 0 cE∧c= 1√μ0 ε0

∧v= 1√με

S=1μ0

E × B∧I=Savg

Need to take your mind off the exam for a minute? Check this out:

There once was an old man from Esser,Who's knowledge grew lesser and lesser. It at last grew so small,He knew nothing at all,And now he's a College Professor.

Who said it? (Solution at bottom)

1. I don't know what's the matter with people: they don't learn by understanding, they learn by some other way — by rote or something. Their knowledge is so fragile!

2. Most teachers waste their time by asking questions which are intended to discover what a pupil does not know, whereas the true art of questioning has for its purpose to discover what the pupil knows or is capable of knowing.3. To know that we know what we know, and to know that we do not know what we do not know, that is true knowledge.4. Mathematics is written for mathematicians.5. The true measure of a man is what he would do if he knew he would never be caught.6. Well, it’s no secret that the best thing about a secret is secretly telling someone your secret, thereby adding another secret to their secret collection of secrets, secretly.7. It is not enough to have a good mind. The main thing is to use it well.8. We cannot ... prove geometrical truths by arithmetic.9. Physics is like sex: sure, it may give some practical results, but that's not why we do it.10. An expert is a person who avoids the small errors while sweeping on to the grand fallacy.

(Richard Feynman, Albert Einstein, Copernicus, Copernicus, Lord Kelvin, Spongebob, Rene Descartes, Aristotle, Richard Feynman, Steven Weinberg)