Download Exam 3

Name:_______________________ ___
PHY2061
11-18-04
Exam 3
Closed book exam. A calculator is allowed, as is one 8.511” sheet of paper with your own
written notes. Please show all work leading to your answer to receive full credit. Answers
should be calculated to 2 significant digits. Exam is worth 100 points, 25% of your total
grade.
UF Honor Code: “On my honor, I have neither given nor received unauthorized aid in doing
this exam.”
4
V   r2
3
Sphere: S  4 r 2
1
K
  3.1415927
me  9.111031 kg
 0  8.8542 1012 C2 / N m 2
 9  109 N m 2 / C 2
4 0
q
qq

F  K 1 2 2 rˆ12
 E   Ε  dΑ  enc   E 
S
r
0
0
U
F
V
E
W  U   F  ds
C
q0
q0



  xˆ  yˆ  zˆ
F F F
x
y
z
  F  div  F   x  y  z
x
y
z
F  grad  f   f
Q  C V
1
Q2
2
U  C  V  
2
2C
V  iR
P  Vi  i 2 R 
R
L
A
i
F  q(E  v  B)
dq
dt
V2
R
k
0  4 k  1.257 106 T  m /A
i ds × r
dB  k
C B  ds  0 ienc
r3
τ  r ×F
μ  iA
B   B  dΑ
S
U
1 2
Li
2
 RC  RC
dB
dt
2
0E2
U
B
u 

V 2 0
2
L
 LR 
LC 
R
a  b  axbx  a y by  az bz
C
  F dV  
V
Reff  R1  R2
1
1
1
 
Reff R1 R2
1 eV  16022
.
 1019 J
K 0

 107 T  m / A
2
c
4
0i
2ki
rˆ 
rˆ
2 r
r
U  μ  B
  N
V   E  ds
1
1
1
 
Ceff C1 C2
B wire 
τ  μ×B
E  V
Ceff  C1  C2
c  3.0  108 m/s
F  i L×B
e  16022
.
 1019 C
VL  L
1
LC
di
dt
VS 
dBz
dz
NB
L
i
Fz   z
NS
VP
NP
a  b   a y bz  by az  i   axbz  bx az  j   axby  bx a y  k
Page 1 of 10
S
F  dΑ
PHY2061
11-18-04
Name:_______________________ ___
1. Consider the toroid shown with an inner radius a = 5 cm and an outer radius
b = 6 cm. The total number of turns of wire wound around the toroid is N = 400.
The core of the toroid has a rectangular cross-section with a thickness h = 0.5 cm.
(a) [6 points] If the wire carries a current of 2A, what is the magnitude of the
magnetic field inside the toroid at a radius midway between the inner and
outer radii?
Page 2 of 10
PHY2061
11-18-04
Name:_______________________ ___
1. Continued
(b) [8 points] Calculate the inductance of the toroid.
2. [8 points] A long straight wire with a cylindrical cross-section has a radius of 0.3 mm
and carries a current of 100 mA uniformly spread across its cross-section. What is the
ratio of the magnetic field at a radius of 0.1 mm to that at the surface of the wire?
Page 3 of 10
PHY2061
11-18-04
Name:_______________________ ___
3. A square loop of wire with edge length 7 cm is located in the plane of the paper inside
a homogeneous magnetic field of 0.35 T pointing into the paper. It is connected in
series with a resistor of 210 . The magnetic field is now increased at a constant rate
by a factor of 4.5 in 18 s.
(a) [6 points] Calculate the magnitude of the induced EMF in the loop (in V)
during that time.
(b) [6 points] Calculate the current induced in the loop in Amps during that time.
Page 4 of 10
PHY2061
11-18-04
Name:_______________________ ___
4. A rectangular loop of wire is located in the plane of the paper inside a homogeneous
magnetic field of 6 mT pointing into the paper. One side length is fixed at 50 cm, the
other side grows with time according to the formula  0 1  at  , where
0
 50 cm and a  2.5 s -1 .
(a) [6 points] Calculate the magnitude of the induced EMF in the loop.
(b) [6 points] If the resistance of the wire increases as R  R0 1  t  , where
R0  3 and  = 0.5 s-1, what would the acceleration of the moving wire need
to be in order to maintain a constant current in the wire? Hint: Consider
adding a quadratic time dependence to the length of the growing side.
Page 5 of 10
PHY2061
11-18-04
Name:_______________________ ___
5. A solenoid 42 cm long has a circular cross-section of 8.5 cm2. There are 420 turns of
wire carrying a current of 3.25A.
(a) [6 points] Find the magnitude of the magnetic field inside the solenoid
assuming that it is essentially infinite in length (neglect end effects).
(b) [6 points] Find the total energy stored in the magnetic field inside the volume
of the solenoid. Neglect end effects.
Page 6 of 10
PHY2061
11-18-04
Name:_______________________ ___
6. An oscillating LC circuit consists of a 2 mH inductive coil and a 3 nF capacitor. The
capacitor has a voltage drop of 2.8 V when the current through the coil is 1.2 mA.
(a) [6 points] Find the period of oscillation.
(b) [6 points] Find the maximum charge on the capacitor.
(c) [6 points] Find the maximum current through the coil.
Page 7 of 10
PHY2061
11-18-04
Name:_______________________ ___
7. A frog can be considered a magnetic dipole when immersed in strong magnetic field
(a property known as diamagnetism).
(a) [3 points] Does the frog have greater potential energy when its magnetic
dipole moment is aligned with the direction of the magnetic field or opposite
to it?
(b) [3 points] If the field is strong enough, it is possible to levitate the frog.
Would this occur near the center of a solenoid where the field is uniform, or
near the end of a solenoid where the field is diverging?
Page 8 of 10
Name:_______________________ ___
PHY2061
11-18-04
R
(t)
C
L
8. Consider an RLC circuit connected to an alternating-current power supply. The
power supply delivers an electric potential of   t    0 sin t .
(a) [6 points] Write down a differential equation for the charge q  t  on the
capacitor.
(b) [6 points] Estimate the current in the circuit for the following three limiting
1
cases (give your reasons!): 1)   0 , 2)   , and 3)  
LC
Page 9 of 10
PHY2061
11-18-04
Name:_______________________ ___
(c) [6 points] How does adding a transformer to the output of the power supply
affect the behavior of the circuit, if the number of secondary windings on the
transformer (which are connected to the circuit) is half of the number of
primary windings (which are connected to the power supply)?
Page 10 of 10