21111
21111
Instructor: Profs. Paul Avery, Selman Hershfield
PHYSICS DEPARTMENT
PHY 2049
Exam 3
April 6, 2011
Name (print, last first):
Signature:
On my honor, I have neither given nor received unauthorized aid on this examination.
YOUR TEST NUMBER IS THE 5-DIGIT NUMBER AT THE TOP OF EACH PAGE.
DIRECTIONS
(1) Code your test number on your answer sheet (use 76–80 for the 5-digit number). Code your name on your
answer sheet. DARKEN CIRCLES COMPLETELY. Code your student number on your answer sheet.
(2) Print your name on this sheet and sign it also.
(3) Do all scratch work anywhere on this exam that you like. At the end of the test, this exam printout is to be turned in.
No credit will be given without both answer sheet and printout with scratch work most questions demand.
(4) Blacken the circle of your intended answer completely, using a #2 pencil or blue or black ink. Do not
make any stray marks or the answer sheet may not read properly.
(5) The answers are rounded off. Choose the closest to exact. There is no penalty for guessing.
>>>>>>>>WHEN YOU FINISH <<<<<<<<
Hand in the answer sheet separately.
Constants: e = 1.6 × 10−19 C mp = 1.67 × 10−27 kg
²o = 8.85 × 10−12 C 2 /N · m2
me = 9.1 × 10−31 kg
k = 1/(4π²o ) = 9 × 109 N · m2 /C 2
c = 3 × 108 m/s
micro = 10−6
µo = 4π × 10−7 T · m/A
nano = 10−9
|q1 ||q2 |
Coulomb’s Law: |F~ | =
(point charge)
4π²o r2
~
~ =F
Electric field: E
q
~ =
E
H
~A=
Gauss’ law: Φ = n̂ · E
q
r̂ (point charge)
4π²o r2
~ dA =
n̂ · E
~ =
E
R
dq
r̂ (general)
4π²o r2
E=
σ
(plane)
2²o
qenc
²o
R
1
1
F~ · d~s = mvf2 − mvi2 = Kf − Ki
2
2
R
For conservative forces Uf − Ui = − F~ · d~s → Ki + Ui = Kf + Uf
Energy: W =
Electric potential: V =
Vb − Va = −
Rb
a
Ex dx = −
U
q
Rb
Capacitors: q = CV
~ · d~s
E
C=
U=
Resistors: i =
a
V =
q2
2C
dq
= jA
dt
V =
∂V
,
∂x
∂V
,
∂y
Ex = −
Ey = −
R
dq
(general)
4π²o r
Ez = −
∂V
∂z
K²o A
(parallel-plate)
C = C1 + C2 (parallel)
d
1
1
1
1
u = ²o E 2
=
+
(series)
2
C
C1
C2
R=
q = CV (1 − e−t/RC ) (charging)
q
(point charge)
4π²o r
V
i
R=
ρL
(wire)
A
P = iV
q = qo e−t/RC (discharging)
R = R1 + R2 (series)
1
1
1
=
+
(parallel)
R
R1
R2
~
~ ×B
~
~
~
Magnetism: F~ = q~v × B
F~ = iL
µ = N iA
~τ = µ
~ ×B
U = −~
µ·B
H
µo i
µo i
µo iN
~ = µo id~s × r̂
~ · d~s = µo ienc
dB
B
B=
, (wire)
(loop center),
(solenoid)
2
4π r
2πR
2R
L
L = N ΦB /i (definition)
UB =
1
Li2
uB =
B2
H
H
~ · d~s = − dΦB
E
dt
di
L = µo n2 Al (solenoid)
E = −L
dt
E
i = (1 − e−t/τL )
i = io e−t/τL
~A=
Induction: ΦB = n̂ · B
~ dA
n̂ · B
E=
E1 = −M
τL =
L
di2
dt
Vs
L = L1 + L2 (series)
=
Ns
1
=
1
+
1
(parallel)
21111
21111
AC Circuits: ω = √
tan φ =
1
(LC circuit)
LC
XL − XC
R
I=
Em
Z
q = Qo e−Rt/(2L) cos(ω 0 t + φ)
First 2 Maxwell’s Eqs.:
Last 2 Maxwell’s Eqs.:
EM Waves: c =
I = Io cos2 θ
H
i = I sin(ωt − φ) (driven RLC)
R2 + (XL − XC )2
XL = ωL,
¡
¢1/2
ω 0 = ω 2 − (R/(2L))2
~ · n̂dA = 0
B
H
H
θc = sin−1
~ =B
~ m sin(~k · ~r − ωt)
B
pr =
XC =
Pavg =
1
IEm cos φ
2
1
di
, vL = L ,
ωC
dt
vC =
cos φ =
q
C
~ · d~s = −N dΦB
E
dt
~ · d~s = µo ²o dΦE + µo ienc
B
dt
~= 1E
~ ×B
~
S
µo
n1 sin θ1 = n2 sin θ2
I
(total absorption)
c
p
~ · n̂dA = qenc
E
²0
E
1
=√
B
µo ²o
~ =E
~ m sin(~k · ~r − ωt)
E
pr =
H
Z=
E = Em sin(ωt)
1 2
E
cµo rms
n2
θB = tan−1
n1
I = Savg =
n2
n1
~m ⊥ B
~ m ⊥ ~k
E
2I
(total reflection)
c
p=
id = ²o
dΦE
dt
Em
Erms = √
2
I=
Ps
4πr2
c = ω/k = f λ
U
(mom. carried by EM radiation of energy U )
c
R
Z
21111
21111
1. If the electric field in a plane electromagnetic wave is given by E(x, t) = 8500 sin(33x − ωt), in SI units, the frequency
of the wave is:
(1) 2.1 × 103 Hz
(2) 5.3 × 106 Hz
(3) 9.9 × 109 Hz
(4) 6.2 × 1011 Hz
(5) 1.6 × 109 Hz
2. Which of the following equations, along with a symmetry argument, can be used to calculate the magnetic field between
the plates of a charging parallel plate capacitor with circular plates?
(1)
(2)
(3)
(4)
(5)
H
H
H
H
~ · d~s = µ0 i + µ0 ²0 dΦE /dt
B
~ · dA
~ = qenc /²0
B
~
B · d~s = −dΦB /dt
~ · dA
~=0
B
none of these
3. A cylindrical region contains a uniform electric field that is along the cylinder axis and is changing with time. If r is the
distance from the cylinder axis the magnitude of the magnetic field within the region is:
(1) proportional to r
(2) proportional to 1/r2
(3) uniform
(4) proportional to r2
(5) proportional to 1/r
4. In the figure at right E = 120V , R1 = 10Ω, R2 = 20Ω, R3 = 30Ω, and L = 4H.
Immediately after switch S is closed, what is the current through R2 ?
(1)
(2)
(3)
(4)
(5)
4.5
5.5
6.0
4.0
5.0
A
A
A
A
A
5. What energy in eV is required to flip the spin of an electron in a magnetic field of 12.9 T from antiparallel alignment to
parallel alignment? The magnitude of the magnetic moment of an electron is 9.27×10−24 J/T, and an eV is 1.6×10−19 J.
(1) 1.5 × 10−3
(2) 3.7 × 10−4
(3) 4.5 × 10−3
(4) 7.5 × 10−4
(5) 3.0 × 10−3
6. A 30 W light bulb is placed 1.8m below the surface of a pool of water (n = 1.333). What is the diameter (in meters)
of the circle on the surface of the water through which light is transmitted? (Some of the light is totally internally
reflected.)
(1) 2.4
(2) 4.1
(3) 2.7
(4) 4.8
7. In an LC circuit the charge across the capacitor, q(t), is given by graph 2.
Which graph represents the derivative of the current, di/dt?
(1)
(2)
(3)
(4)
(5)
graph 1
graph 4
graph 3
none of the graphs
graph 2
(5) 1.8
21111
21111
8. A small rocket of mass 10 kg. emits an intense light beam as a rocket exhaust (because electromagnetic energy U carries
momentum U/c). What must be the minimum total power of the beam in gigawatts (= 109 W) needed to lift the rocket
against the force of gravity at the earth’s surface? (g = 9.8 m/s2 .)
(1) 59
(2) 0
(3) 50
(4) 15
(5) 29
9. Of the three chief kinds of magnetic materials (diamagnetic, paramagnetic, and ferromagnetic) which are used to make
permanent magnets?
(1)
(2)
(3)
(4)
(5)
only paramagnetic
only diamagnetic
all three
only ferromagnetic
only paramagnetic and ferromagnetic
10. A transformer is to take an AC voltage of amplitude 120 V down to an AC voltage of amplitude 6 V. If the number of
turns in the primary coil is 1000, how many turns are in the secondary coil?
(1) 50
(2) 500
(3) 20
(4) 20,000
(5) 1,000
11. A metal rod is forced to move with constant velocity along two parallel metal
rails, which are connected with a strip of metal (see figure). A magnetic field
of 0.5 T points out of the page. The separation between the rails is L = 30 cm,
the rod has resistance 40Ω, and velocity 70 cm/s. What force must be applied
to the rod to keep it moving at the constant velocity?
(1) 1 × 10−4 N
(2) 5 × 10−4 N
(3) 3 × 10−4 N
(4) 4 × 10−4 N
(5) 2 × 10−4 N
12. For a driven LCR circuit the current and voltage as a function of time are given
by the graphs in the figure. For this circuit is XL > XC or XC > XL ? Also,
does one need to increase or decrease the driving frequency, ωd , in order to
obtain resonance?
(1)
(2)
(3)
(4)
(5)
XL > XC , decrease ωd
XL < XC , decrease ωd
XL > XC , increase ωd
XL < XC , increase ωd
none of the other answers are correct
13. Gauss’ law for magnetism tells us:
(1)
(2)
(3)
(4)
(5)
the net charge in any given volume
the magnetic field of a current element
that the line integral of a magnetic field around any closed loop must vanish
that magnetic monopoles do not exist
that charges must be moving to produce magnetic fields
14. A circular loop of wire has radius 20 cm. If the magnetic field inside the loop is uniform and has the time dependence,
B(t) = 1 + 2t + 3t2 , where t is measured in seconds and B is measured in Tesla. What is the magnitude of the induced
emf in the loop at time t = 3s?
(1) 20 V
(2) 2.5 V
(3) 1.7 V
(4) 4.3 V
(5) 3.0 V
21111
21111
15. The current i in a long wire is going up as shown in the figure, but decreasing
in magnitude. What is the direction of the induced current in the left loop and
the right loop. (List the direction of the induced current in the left loop first.)
(1)
(2)
(3)
(4)
(5)
counterclockwise, clockwise
clockwise, counterclockwise
counterclockwise, counterclockwise
clockwise, clockwise
There is no induced current.
16. The speed of light in vacuum depends on which of the following wave properties?
(1) Wavelength
(2) None of these
(3) Intensity
(4) Frequency
(5) Amplitude
17. An unpolarized beam of light with intensity 72 W/m2 is incident on a stack of 3 polarizing sheets with each sheet’s
polarization axis rotated an angle θ relative to that of the previous sheet. If the beam emerging from the stack has
intensity 4 W/m2 , what is its intensity after it has passed through the first 2 sheets?
(1) 12 W/m2
(2) 8 W/m2
(3) 18 W/m2
(4) 6 W/m2
(5) 24 W/m2
18. An LC circuit has C = 8µF and L = 5 mH. If the maximum voltage on the capacitor is 3 V, what is the maximum
current through the inductor?
(1) 0.24 A
(2) 0.48 A
(3) 0.12 A
(4) 0.96 A
(5) 0.06 A
19. The magnetic field is 2 T into the page and 1 T out of the page in regions 1
and 2 respectively (see figure). The magnitude of both fields is decreasing at a
rate of −0.005 T/s. If region 1 has area 0.3 m2 and region 2 has area 0.6 m2 ,
H
~ · d~s for the path of integration shown in the figure?
what is E
(1) 4.5 × 10−3 V
(2) −4.5 × 10−3 V
(3) −2.0 × 10−3 V
(4) 1.5 × 10−3 V
(5) −1.5 × 10−3 V
20. Consider an RLC circuit with driving emf amplitude Em = 12V , resistance R = 6Ω, inductance L = 1H, and capacitance
C = 2.0µF . Find the amplitude of the voltage across the resistor at resonance.
(1) 1400 V
(2) 120 V
(3) 2 V
(4) 12 V
(5) 700 V