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Instructor: Profs. Selman Hershfield, Aneta Petkova
PHYSICS DEPARTMENT
PHY 2049
Exam 3
November 17, 2010
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>>>>>>>>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
Coulomb’s Law: |F~ | =
me = 9.1 × 10−31 kg
k = 1/(4π²o ) = 9 × 109 N · m2 /C 2
c = 3 × 108 m/s
µo = 4π × 10−7 T · m/A
micro = 10−6
nano = 10−9
|q1 ||q2 |
(point charge)
4π²o r2
~
~ =F
Electric field: E
q
~ =
E
H
~A=
Gauss’ law: Φ = n̂ · E
q
r̂ (point charge)
4π²o r2
~ =
E
R
dq
r̂ (general)
4π²o r2
E=
σ
(plane)
2²o
~ dA = qenc
n̂ · E
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
V =
∂V
,
∂x
∂V
,
∂y
Ex = −
Ey = −
K²o A
(parallel-plate)
d
u=
dq
= jA
dt
q
(point charge)
4π²o r
R=
q = CV (1 − e−t/RC ) (charging)
1
²o E 2
2
V
i
R
dq
(general)
4π²o r
Ez = −
∂V
∂z
C = C1 + C2 (parallel)
1
1
1
=
+
(series)
C
C1
C2
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
~ · d~s = µo ienc
~ = µo id~s × r̂
B
B=
, (wire)
(loop center),
(solenoid)
dB
2
4π r
2πR
2R
L
L = N ΦB /i (definition)
UB =
1 2
Li
2
uB =
B2
2µ
H
H
~ · d~s = − dΦB
E
dt
di
L = µo n2 Al (solenoid)
E = −L
dt
~A=
Induction: ΦB = n̂ · B
~ dA
n̂ · B
i=
E=
E
(1 − e−t/τL )
R
i = io e−t/τL
E1 = −M
τL =
L
R
di2
dt
L = L1 + L2 (series)
Vs
Ns
=
V
N
1
1
1
=
+
(parallel)
L
L
L
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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
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1. In a series driven RLC circuit the current leads the voltage. Using the convention V (t) = Em sin(ω
√ d t) and I(t) =
Im sin(ωd t − φ), what is the sign of φ, and the relation between ωd and the resonance frequency, 1/ LC?
(1)
(2)
(3)
(4)
(5)
none of the other
√ answers is correct
φ > 0, ωd < 1/√LC
φ < 0, ωd > 1/√LC
φ < 0, ωd < 1/√LC
φ > 0, ωd > 1/ LC
2. If graph 2 at right is the current as a function of time in an RLC series circuit,
which graph represents voltage across a capacitor?
(1)
(2)
(3)
(4)
(5)
none of the graphs
graph 3
graph 4
graph 2
graph 1
3. A cube has six faces, which we label as 1,2,. . .,6. If the flux through faces i = 1, . . . , 5 is (i − 2) in Webers, what is the
flux through face 6?
(1) −7 Wb
(2) +1 Wb
(3) −5 Wb
(4) −3 Wb
(5) −1 Wb
4. In an LC circuit (see figure) the amplitude of the current oscillations is 2A
and the amplitude of the voltage oscillations across the capacitor is 3V . If the
capacitance is 5µF , what is the inductance?
(1) 2.3 µH
(2) 3.7 µH
(3) 11 µH
(4) 5.4 µH
(5) 7.5 µH
5. The current i increases in the loop labeled in the figure. What is the direction
of the induced current in the smaller inner loop and the loop at the right?
(List the direction of the induced current for the smaller loop first.)
(1)
(2)
(3)
(4)
(5)
clockwise, clockwise
There is no induced current.
counterclockwise, clockwise
counterclockwise, counterclockwise
clockwise, counterclockwise
6. A generator consists of a circular coil of wire with 500 turns and radius 2 cm. It rotates in a uniform magnetic field of
magnitude 3 T with frequency f = 60 Hz. What is the voltage or emf amplitude produced by the generator?
(1) 113 V
(2) 492 V
(3) 711 V
(4) 581 V
(5) 237 V
7. The magnetic field is 1 T into the page and 2 T out of the page in regions 1
and 2 respectively (see figure). The magnitude of both fields is increasing at
a rate of 0.005 T /s. If region 1 has area 0.4 m2 and region 2 has area 0.8 m2 ,
H
~ · d~s for the path of integration shown in the figure?
what is E
(1) −6 × 10−3 V
(2) 6 × 10−3 V
(3) −2 × 10−3 V
(4) 4 × 10−3 V
(5) 2 × 10−3 V
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8. Unpolarized light of intensity Io is sent through three polarizers. If the polarization axes of the polarizers is are θ1 = 0◦ , θ2 = 30◦ , and θ3 = −30◦ , what is
the intensity of the light emerging from all three polarizers?
(1)
(2)
(3)
(4)
(5)
0.188
0.562
0.281
0.094
0.375
Io
Io
Io
Io
Io
9. A series RLC circuit is driven by a voltage of amplitude Em = 5V . If L = 5H, C = 3µC, R = 2kΩ, and the angular
frequency is ω = 1000s−1 , what is the amplitude of the voltage oscillations across the capacitor?
(1) 0.33 V
(2) 2.27 V
(3) 4.92 V
(4) 3.49 V
(5) 1.97 V
10. Which of the Maxwell equations explains why a voltage is created when a loop of wire turns in a magnetic field?
H
(1) H B · dA = 0
(2) E · ds = −dΦB /dt
(3) F
H = q(E + v × B)
(4) H E · ds = µo ienc + µo ²o dΦE /dt
(5) E · dA = qenc /²o
11. The electric field in a plane electromagnetic wave is Ez = Em sin(ky + ωt). An electric field of 3.0 kV/m in the (-z)
direction is measured at some point and time along the travel path of the wave. What is the magnetic field at the same
point and time?
(1) 1.0 × 10−5 T ĵ
(2) 1.3 × 10−5 T î
(3) 1.0 × 10−5 T î
(4) −1.0 × 10−5 T ĵ
(5) −1.0 × 10−5 T î
12. A current of 2 A flows into a capacitor with circular plates of radius R = 7 mm.
What is the magnitude of the magnetic field at point P in between the plates
at a radial distance r = 4 mm from the center of the plates?
(1) 1.0 × 10−5 T
(2) 3.1 × 10−4 T
(3) 3.3 × 10−5 T
(4) 4.0 × 10−5 T
(5) 1.0 × 10−4 T
13. For the same geometry as in the previous problem, what is the magnitude of the magnetic field at a radius r = 10 mm
from the center of the plates?
(1) 11.4 × 10−5 T
(2) 3.3 × 10−5 T
(3) 1.0 × 10−5 T
(4) 4.0 × 10−5 T
(5) 3.1 × 10−4 T
14. At t = 0 the switch is closed in the RL circuit at right. If L = 4mH, R = 3Ω,
and E = 6V , what is the voltage across the inductor at time t = 2ms?
(1) 1.3 V
(2) 2.1 V
(3) 0.7 V
(4) 4.7 V
(5) 3.9 V
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15. Light is incident on a triangular piece of glass with index of refraction 1.4.
What is the angle θ in the figure at which the light emerges? Assume the
index of refraction of air is 1. (The figure is not drawn to scale so do not try
to read the angle directly from it.)
(1)
(2)
(3)
(4)
(5)
10◦
19◦
7◦
13◦
16◦
16. An optical fiber has a plastic core (n1 = 1.4) surrounded by a plastic sheath
(n2 = 1.5). A light ray is incident on one end of the fiber at angle θ. The ray
is to undergo total internal reflection at point A, where it encounters the coresheath boundary. What is the maximum value of θ that allows total internal
reflection at A?
(1) 26◦
(2) 24◦
(3) 33◦
(4) 28◦
(5) 30◦
17. A common refrigerator magnet is an example of a
(1) induced magnet
(2) diamagnet
(3) none of the other answers is correct
(4) paramagnet
(5) ferromagnet
18. A square loop of side 2 cm moves with velocity 3 m/s at t = 0 as it exits a
region of uniform magnetic field 5 T into the page (see figure). If the loop has
resistance 0.4 Ω, what is the magnitude of the magnetic force on it at t = 0?
(1) 3.7 × 101 N
(2) 7.5 × 10−2 N
(3) 3.0 × 10−3 N
(4) 7.5 × 10−1 N
(5) 4.2 N
19. In a transformer used to take an AC voltage source of amplitude 120 V down to a voltage amplitude of 5 V, which can
be used to run an electronic device, what is the ratio of the number of turns in the primary coil to the number of turns
in the secondary coil, Np /Ns ?
(1) 0.04
(2) 4.9
(3) 1.0
(4) 0.20
(5) 24
20. A space craft has a reflecting sail of area 5000 m2 . Electromagnetic radiation of intensity 1200 W/m2 is incident on the
sail. What is the net force on the sail?
(1) 0.05 N
(2) 0.04 N
(3) 0.01 N
(4) 0.03 N
(5) 0.02 N