Name__________________
Final Exam
Physics 248
May 14, 2007
Problems will be graded on reasoning and intermediate steps as well as on
the final answer. You must show all your work to receive full credit. If
you only provide your final answer, and do not show your work, you will
receive very few points. Be sure to include units and direction of vectors
when needed.
8
9
Total
2
Constants:
c ≈3 x 10 m/s, k ≈9x10 Nm /C , me=9.1x10-31 kg = 0.51 MeV/c2, ε0=8.85x10-12 C2/(Nm2)
mproton = 1.67 x 10-27 kg, h= 6.62 x 10-34 m2 kg/s, hc = 197.3eV " nm , e = 1.6 x 10-19 C
Problem
1
2
3
4
5
2
!
Score
/20
/20
/20
/20
/20
1. [20 points, 5 each]
Short problems, multiple choice answers and true/false questions. Provide an
explanation that leads to the answer.
A) Of the 5 masses shown in the figure in orbit around the central mass M, indicate
which are the ones that would require the most energy to escape from their orbits:
a) 1: mass 2m at distance 4R;
b) 2: mass 0.5m at distance R;
c) 3: mass 3m at distance 3R;
d) 4: mass m at distance 5R;
e) 5: mass 2m at distance 2R.
Explain.
B) Is the following statement True or False? If true which is the phenomenon
that would cause this effect?
A clock 1 is placed on top of the Sears tower (so it is at a higher gravitational
potential) than clock 2 that is at the street level, hence clock 1 runs faster than 2.
C) Chose one of the following answers and explain which phenomenon causes
this effect. The demonstration of the formula is not requested.
The angle by which light bends in the presence of a massive object of mass M and
radius R is proportional to:
i)
ii)
iii)
iv)
v)
GMR2/c2
GM/(c2R)
GM c2/R
c2R/(GM)
R/(GM c2)
D) Chose one of the following answers. Explain.
In a double slit experiment the distance between the slits is increased by a factor of 4.
If the distance between the bright fringes is small compared with the distance from
the slits to the screen, the distance between adjacent fringes near the center of the
interference pattern:
a) increases by a factor of 2;
b) does not change;
c) decreases by a factor of 2;
d) decreases by a factor of 4;
e) increases by a factor of 4.
2. [20 points] In a region of space, a particle has a wave function given by
ψ (x) = A exp[-x2/(2L2)]
and energy E= h 2/(2mL2) where L is some length.
(a) Find the potential energy as a function of x.
!
(b) Draw the potential energy versus x.
(c) What is the classical potential that has this dependence?
3. [20 points, 5 each]
A) An electric field of 1 V/m is applied at the 2 ends of a cylindrical copper conductor
(resistivity ρ = 17.2 nΩ m) of L=1 m length and a cross-section of A = 1 cm2.
a) Find the resistance of the conductor.
b) Find the current flowing in the conductor.
c) The same electric field is applied at the 2 ends of a similar conductor with same
section A and double length 2L. What is the resistance of this second conductor?
d) What is the resistance if the two conductors are placed in parallel?
B) A coaxial cable consists of 2 very thin-walled conducting coaxial cylinders of radii
r1 and r2, as shown in the figure. The current I flows in one direction down the inner
cylinder and in the opposite direction in the outer cylinder. Calculate the magnetic
field in all regions of space, inside the inner cylinder, outside the outer one and
between the two cylinders. Indicate the direction of the field lines in the drawing
below.
C) The electric field of an electromagnetic wave oscillates in the y direction and the
Poynting vector is given by:
S(x,t) = (10W/m2) sin2[kx- ωt] ux = (10W/m2) sin2[2.4 x 106 π (x- ct)] ux,
where x is in meters, t in seconds, and c=3 x 108 m/s. is the speed of light in vacuum.
a) What is the direction of propagation of the wave? What is the direction of B?
b) Find the wavelength of the wave and the frequency.
c) Find the intensity of the electromagnetic wave.
[Hint: use the relationship between the intensity of the wave and the average of the
magnitude of the Poynting vector: I = |Sav|.]
d) Find the expression of the electric field E(x,t) and of the magnetic field B(x,t) and
calculate the amplitude of the plane waves.
[Hint: use the relationship between the intensity and the amplitude of the electric field and
of the magnetic field waves.]
D) Consider the circuit in the figure with ε = 10V, R1 = 10 Ω, R2 = 100 Ω and L = 2 H.
Neglect the resistance in the connecting wires, in the inductor and of the battery.
R1 =
ε=
R2 =
=L
a) Calculate the current in the 100 Ω resistor at t=0 when the switch is closed.
b) Calculate the current in the 10 Ω resistor and in the 100 Ω resistor at a very long
time after the switch has been closed.
4.[20 pts, 4 pts each]
An isolated parallel plate capacitor is made of two conducting plates of area A=4m2
separated by d=2 cm. The charge on the 2 plates has absolute value of 2 µC.
Consider the field uniform and entirely contained inside the plates and the medium
between plates vacuum.
a) Calculate the electric field magnitude and direction by applying Gauss’ law in the
regions of space I,II,III. Indicate in the figure below the direction of the electric field
and describe which kind of Gaussian surfaces are suitable for this calculation.
d
2 µC
-2 µC
III
I
II
b) Calculate the electric potential difference between the left and the right plate,
Vleft-Vright.
c) How much work W must you do to pull the plates apart to twice their original
separation?
d) If we want to discharge this capacitor by 1/2 of its initial charge value in 0.1s, which
is the resistor value we have to choose to connect its plates?
e) How much energy will be dissipated in total in the resistor?
5. [20 pts, 5 pts each]
A conducting rod MN of length b = 5 cm slides with constant velocity of v=5 cm/s in
the direction indicated in the figure along 2 metal parallel rails separated by a
distance b and connected at one end by another metallic rod OP as show in the
figure. The circuit formed by the rails, the connecting rod OP and the sliding rod MN
has a total resistance R = 5 Ω. The circuit is in a region of uniform magnetic field,
perpendicular to this page and entering it, of magnitude B = 0.4 T. If the rod is kept in
motion with constant speed in the direction indicated in the figure, calculate
(neglecting the self-induction of the circuit):
M
P
v
B
O
N
1) the induced emf in the circuit;
2) the induced current in the circuit and draw an arrow along the circuit to show its
direction;
3) the mechanical power that is needed to maintain the constant speed of the sliding
rod at the value 5 cm/s;
4) the heating power dissipated in the circuit due to the Joule effect.