Final Exam
PHYS 1220
F14
12/16/2014
__________
name
Nine 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 xaxis 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
4𝜋𝜀0
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 -2uniform 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 𝑖̂ − 𝐵0 𝑘̂
Problem 6 – Fields and Forces due to Current Carrying Wires
The long straight wire AB, shown in the figure below, carries a current of
10[A]. The trapezoidal loop, whose long edges are parallel to the wire,
carries a current of 2[A].
a) What does the magnetic field line pattern of the long wire look
like?
b) Draw the directions of the magnetic force on each segment of the
loop (call to the right, upward, and into the plane positive. Include
in your vector drawings whether these forces are constant or
whether and how they vary as function of position in the wire
segment.
c) Do any of these forces completely cancel out with forces on other
segments? Why or why not?
d) Find the magnitude and direction of the net force exerted on the
loop by the magnetic field of the wire or explain why there is no
net force.
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 – Capacitor Networks and Energy in Capacitors
a) Derive the equations for series and for parallel network equivalent
capacitance.
For parts b) and c), consider the capacitor network in the figure below (all
capacitances are in farad):
b) What is the equivalent capacitance of the circuit?
How much energy is stored in the network, if Vab= 220[V]?
How much charge is on the 1[F] capacitor in that case?
c) How does the energy stored in the network change when the source is
disconnected while the capacitors remain charged and the branches are rearranged as shown below? Or does the energy not change?
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:
b) Find the equivalent resistance of the circuit.
c) 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 = 𝛼𝐿0 ∆𝑇 𝑎𝑛𝑑 ∆𝑉 = 𝛽𝑉0 ∆𝑇
𝑄 = 𝑚𝑐∆𝑇 𝑎𝑛𝑑 𝑄 = ±𝑚𝐿
𝑑𝑄
𝑇 −𝑇
𝑑𝑄
 ( 𝑑𝑡 )
= 𝑘𝐴 𝐻𝐿 𝐶 𝑎𝑛𝑑 ( 𝑑𝑡 )
𝑐𝑜𝑛𝑑
𝑟𝑎𝑑
= 𝐴 𝑒 𝜎 ∙ (𝑇 4 − 𝑇𝑆4 )
 𝑝 ∙ 𝑉 = 𝑛 𝑅𝑇 𝑎𝑛𝑑 𝑚𝑡𝑜𝑡 = 𝑛𝑀, 𝑀 = 𝑁𝐴 𝑚
3
 𝐾𝑡𝑟𝑎𝑛𝑠 =
3𝑘𝐵 𝑇
𝑘𝐵 𝑇 𝑎𝑛𝑑 𝑣𝑟𝑚𝑠 = √
2
𝑚

𝑉
 𝑊 = ∫𝑉 2 𝑝 𝑑𝑉 𝑎𝑛𝑑 ∆𝑈 = 𝑄 − 𝑊
1
 𝑒=
𝑊
2 𝑑𝑄
 ∆𝑆 = ∫1
 𝐹=
𝑇
1
4𝜋𝜀0
∙
𝐶 ≡
1
4𝜋𝜀0
𝑄
𝑉𝑎𝑏
∑i
𝑇𝐻

𝑎𝑛𝑑 𝑆 = 𝑘 ∙ ln(𝑤)
|𝑞1 |∙|𝑞2 |
𝑟2
 Φ𝐸 = ∫ 𝐸⃗ ∙ 𝑑𝐴 =
𝑈=
𝑇𝐶
𝑎𝑛𝑑 𝑒𝐶𝑎𝑟𝑛𝑜𝑡 = 1 −
𝑄𝐻
qi
ri
𝑎𝑛𝑑 𝐸⃗ =
𝐹0
𝑞0

𝑄𝑒𝑛𝑐𝑙.
𝜀0
and V =
𝑎𝑛𝑑 𝐶𝑝𝑙𝑎𝑡𝑒 = 𝜀0
𝐴
𝑑
U
q0
𝑎𝑛𝑑 𝐸⃗ = − (𝑖̂
𝑎𝑛𝑑, 𝑖𝑛 𝑠𝑒𝑟𝑖𝑒𝑠,
1
𝐶𝑒𝑞
𝜕𝑉
𝜕𝑥
+ 𝑗̂
= ∑𝑖
1
𝐶𝑖
𝜕𝑉
𝜕𝑦
+ 𝑘̂
𝜕𝑉
)
𝜕𝑧
, 𝑎𝑛𝑑, 𝑖𝑛 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙, 𝐶𝑒𝑞 = ∑𝑖 𝐶𝑖
𝑄2
1
1
1
= 2 𝐶 𝑉 2 = 2 𝑄 𝑉 𝑎𝑛𝑑 𝑢 = 2 𝜀 𝐸 2
2𝐶
𝑑𝑄
𝐼 = 𝑑𝑡 = 𝑛 |𝑞| 𝑣𝑑 𝐴 𝑎𝑛𝑑 𝐽 = 𝑛 𝑞 𝑣𝑑
𝐸
𝐿
𝜌=
𝑎𝑛𝑑 𝑉 = 𝐼𝑅 𝑤𝑖𝑡ℎ 𝑅 = 𝜌
𝐽
𝐴
𝑉2
𝑃 = 𝑉𝑎𝑏 ∙ 𝐼 = 𝐼 2 𝑅 = 𝑎𝑏
𝑅
1
1
𝑅𝑒𝑞 = ∑𝑖 𝑅𝑖 (𝑠𝑒𝑟𝑖𝑒𝑠) 𝑅 = ∑𝑖 𝑅 (𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙)
𝑒𝑞
𝑖
𝑈=
𝐾𝑖𝑟𝑐ℎℎ𝑜𝑓𝑓 𝑅𝑢𝑙𝑒𝑠 ∑ 𝐼 = 0 (𝑗𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑟𝑢𝑙𝑒), ∑ 𝑉 = 0 (𝑙𝑜𝑜𝑝 𝑟𝑢𝑙𝑒)
𝑡
𝜀
𝑡
Capacitor charging 𝑞 = 𝑄𝑓 ∙ (1 − 𝑒 −𝑅𝐶 ) 𝑎𝑛𝑑 𝑖 = 𝑅 ∙ 𝑒 −𝑅𝐶
⃗ and Φ𝐵 = ∮ 𝐵
⃗ ∙ 𝑑𝐴 = 0 and 𝐹 = 𝐼 𝑙 × 𝐵
⃗ (𝑓𝑜𝑟 𝑠𝑡𝑟𝑎𝑖𝑔ℎ𝑡 𝑤𝑖𝑟𝑒𝑠)
𝐹 = 𝑞𝑣 × 𝐵
⃗ ×𝑟
𝜇0 𝑞𝑣
⃗ =
𝐵
∙ 𝑟2 𝑚𝑎𝑔𝑛𝑒𝑡𝑖𝑐 𝑓𝑖𝑒𝑙𝑑 𝑐𝑟𝑒𝑎𝑡𝑒𝑑 𝑏𝑦 𝑚𝑜𝑣𝑖𝑛𝑔 𝑐ℎ𝑎𝑟𝑔𝑒
4𝜋
𝜇0 𝐼 𝑑𝑙 ×𝑟
∙ 𝑟2 𝐵𝑖𝑜𝑡 − 𝑆𝑎𝑣𝑎𝑟𝑡 𝑓𝑖𝑒𝑙𝑑 𝑐𝑟𝑒𝑎𝑡𝑒𝑑
4𝜋
𝜇0 𝐼
𝐹
𝜇0 𝐼∙𝐼′
𝐵 = 2𝜋𝑟
and 𝐿 = 2𝜋𝑟
𝑓𝑜𝑟 2 𝑤𝑖𝑟𝑒𝑠
⃗ ∙ 𝑑𝑙 = 𝜇0 𝐼𝑒𝑛𝑐𝑙 𝐴𝑚𝑝𝑒𝑟𝑒 ′ 𝑠𝐿𝑎𝑤
∮𝐵
𝑑Φ
𝜀 = − 𝑑𝑡𝐵 = ∮ 𝐸⃗ ∙ 𝑑𝑙 𝐹𝑎𝑟𝑎𝑑𝑎𝑦′𝑠 𝐿𝑎𝑤
𝑑𝑖
𝑁Φ𝐵
𝜀 = −𝐿
𝑤ℎ𝑒𝑟𝑒 𝐿 =
𝑑𝑡
𝑖
⃗ =
𝐵
𝑏𝑦 𝑠𝑡𝑟𝑎𝑖𝑔ℎ𝑡 𝑤𝑖𝑟𝑒 𝑐𝑎𝑟𝑟𝑦𝑖𝑛𝑔 𝑐𝑢𝑟𝑟𝑒𝑛𝑡
1
2
𝑈=
𝐿 𝐼 2 𝑎𝑛𝑑 𝑢 =
1
𝐿𝐶
𝜔= √
𝐼𝑟𝑚𝑠 =
𝐼
√2
𝐵2
2𝜇
1
𝐿𝐶
𝑓𝑜𝑟 𝐿𝐶 𝑐𝑖𝑟𝑐𝑢𝑖𝑡𝑠 𝑎𝑛𝑑 𝜔′ = √
−
𝑅2
4𝐿2
𝑎𝑛𝑑 𝑖 = 𝐼 ∙ cos(𝜔𝑡) 𝑉 𝑎𝑐𝑐𝑜𝑟𝑑𝑖𝑛𝑔𝑙𝑦 𝐴𝐶 𝑐𝑢𝑟𝑟𝑒𝑛𝑡
𝑉𝑅 = 𝐼𝑅, 𝑉𝐿 = 𝐼𝑋𝐿 , 𝑉𝐶 = 𝐼𝑋𝐶 , 𝑉 = 𝐼𝑍
1
𝑍 = √𝑅 2 + (𝑋𝐿 − 𝑋𝐶 )2 = √𝑅 2 + (𝜔𝐿 − 𝜔𝐶 )
𝑃𝑎𝑣𝑔 =
1
2
𝑉𝐼 cos(𝜙) 𝑎𝑛𝑑 tan(𝜙) =
𝐵 = 𝜇0 𝜀0 𝑐𝐸 𝑎𝑛𝑑 𝑐 =
𝑆=
1
𝜇0
1
√𝜇0 𝜀0
𝜔𝐿−
𝑎𝑛𝑑 𝑣 =
2
1
𝜔𝐶
𝑅
1
√𝜇𝜀
⃗ 𝑎𝑛𝑑 𝐼 = 𝑆𝑎𝑣𝑔
𝐸⃗ × 𝐵
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)