ECE Written Qualifying Examination, Basic Physics - Spring 2014
1. An incompressible, inviscid (no viscosity) fluid of mass density ρ drains from a
cylindrical tank of cross-sectional area A1 in the presence of gravity of acceleration
constant g. The fluid exits the tank through a tube with varying cross sectional area A2
and A3 , assume A 1 >>A2 , A3 . Also assume the atmospheric pressure on the top surface
of the fluid and at the exit of the tube are the same. The instantaneous height above the
tube of the upper surface of the fluid is h(t) . Give your answers to the following in
terms of
these parameters.
g
A1
h(t)
A2 A3
v
A) What is the speed of the fluid, v , as it
flows out of the tube? (3 pts)
B) In terms of this speed, what is the rate of
change of the height of the fluid? (1 pts)
C) What is the speed of the fluid in the
portion of the tube with area A2 (2 pt)
D) What is the difference in pressures
between the cross-section of the tube
with area A2 and the cross-section of the
tube with area A3 ? (2 pts)
ECE Written Qualifying Examination, Basic Physics - Spring 2014
2. A Michelson interferometer with a gas
cell in one arm is pictured at left. Initially
the gas cell of diameter d is evacuated.
Assume the interferometer is immersed in
air. Monochromatic light from a source
enters the interferometer, is split by the
beam splitter, travels along the indicated
paths, and interferes on a screen at the
bottom producing an image.
A. A micrometer on mirror M2 is turned
moving the mirror a distance ∆L2 away
from the beam splitter. This results in the
image changing from light to dark to light
m1 times. What is the wavelength of the
source in air? (3 pts)
B. The gas cell is now slowly filled with a gas of unknown index of refraction. As it is
filled the image changes from light to dark to light m2 times. What is the index of
refraction of the gas? (3 pts)
ECE Written Qualifying Examination, Basic Physics - Spring 2014
A quantum particle of mass m is confined in a one-dimensional potential well of length L
and depth V.
A. If the potential V is very large, the ground state wave function can be approximated
by that of an infinitely deep well. Write down that wave function including its
normalization. Take 0< x < L to be the domain of the well. (2 pts)
B. Estimate how large V must be for the above approximation to be valid. (2 pts)
C. For a given value of V estimate how many bound states will exist in the well? (2 pt)
ECE Written Qualifying Examination, Basic Physics - Spring 2014
1. An incompressible, inviscid (no viscosity) fluid of mass density ρ drains from a
cylindrical tank of cross-sectional area A1 in the presence of gravity of acceleration
constant g. The fluid exits the tank through a tube with varying cross sectional area A2
and A3 , assume 
A1 ? A2 , A3 . Also assume the atmospheric pressure on the top surface of
the fluid and at the exit of the tube are the same. The instantaneous height above the tube
of the upper surface of the fluid is h(t) . Give your answers to the following in terms of
these parameters.
A1
g
h(t)
A2 A3
v
A) What is the speed of the fluid, v , as it
flows out of the tube? (3 pts)
B) In terms of this speed, what is the rate of
change of the height of the fluid? (1 pts)
C) What is the speed of the fluid in the
portion of the tube with area A2 (2 pt)
D) What is the difference in pressures
between the cross-section of the tube
with area A2 and the cross-section of the
tube with area A3 ? (2 pts)
Solution:
1 2
ρv = ρ gh , thus v = 2gh , same as free fall.
2
dh
A
B)
=− 3v
dt
A1
A
C) v2 = 3 v
A2
D) Pressure at the top and pressure at exit ( A3 ) are equal. According to Bernoulli,
A)
p2 +
1
A2
1
1 2
ρv2 = p3 + ρv 2 . Thus, p2 − p3 = ρv 2 (1 − 32 )
2
A2
2
2
ECE Written Qualifying Examination, Basic Physics - Spring 2014
2. A Michelson interferometer with a gas
cell in one arm is pictured at left. Initially
the gas cell of diameter d is evacuated.
Assume the interferometer is immersed in
air. Monochromatic light from a source
enters the interferometer, is split by the
beam splitter, travels along the indicated
paths, and interferes on a screen at the
bottom producing an image.
A. A micrometer on mirror M 2 is turned
moving the mirror a distance ∆L2 away
from the beam splitter. This results in the
image changing from light to dark to light
m 1 times. What is the wavelength of the
source in air? (3 pts)
B. The gas cell is now slowly filled with a gas of unknown index of refraction. As it is
filled the image changes from light to dark to light m 2 times. What is the index of
refraction of the gas? (3 pts)
Solution
A. Each time mirror M2 moves 1/2 a wavelength the “bulls eye” changes from light to
λair
or λair = 2∆L2 / m1 .
2
B. Light travels through the cell twice. When evacuated this corresponds to n 1
wavelengths where N1 = 2d / λvac . When the cell is filled with gas the number of
wavelengths changes to N 2 = 2dn / λvac , where n is the index of refraction of the gas.
The difference between N 1 and N 2 is the number of fringe shifts,
m2 = N 2 − N1 = 2d(n − 1) / λvac . Thus, (n − 1) = m2 λvac / (2d) .
dark to light. Thus, ∆L2 = m1
ECE Written Qualifying Examination, Basic Physics - Spring 2014
A quantum particle of mass m is confined in a one-dimensional potential well of length L
and depth V.
A. If the potential V is very large, the ground state wave function can be approximated
by that of an infinitely deep well. Write down that wave function including its
normalization. Take 0< x < L to be the domain of the well. (2 pts)
B. Estimate how large V must be for the above approximation to be valid. (2 pts)
C. For a given value of V estimate how many bound states will exist in the well? (2 pt)
Solution
πx
2
sin( )
L
L
2
p2
1  hπ 
=
E0 =
  << V
 2m 2m L
A, ψ 0 (x) =
1  nhπ 
L
En ==
2mV
 ; V , n ;

2m L
 πh

2