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Final Examination
PHY 250
Fall 2005
Instructions:
1. Express your answers in full, grammatically correct sentences using unambiguous, technical
language. Avoid pronouns.
2. Do not place unexplained symbols or acronyms in the text.
3. The acceptable answers must be complete and, if applicable, with a convincing justification..
4. Do not include irrelevant information.
5. Represent physical quantities by conventional symbols.
6. Use proper notation. Explain all symbols in the formulas unless asked otherwise.
7. Indicate vectors with arrows. (I will interpret the symbol without the arrow as the magnitude
of the vector.)
Grading policy:
There are twenty topics on the final. Each topic contains four questions with progressive levels
of difficulty (D, C, B, A). The highest-level question that is answered correctly determines the
grade for the topic.
For 85% of the topics (17) you must answer at least one question properly.
For each grade you need to answer 60% of the questions (12) on the appropriate level.
Topic 1
ELECTRIC CHARGE
D.
The law of charge conservation states that the electric charge of
is always conserved.
C.
If the "charged" particles in a system (substance) cannot move we say that
the system is
B.
Conductors are substances such that
In metals,
in electrolytes, the
allow for transport of charge;
fulfill this function.
A.
In general, charge density relates the charge of an object with its dimensions.
Knowing the distribution of charge density, the charge Q of the object can be
found directly from the definition.
For a linear object with linear charge density λ(x), its charge is
Q=
Topic 2
ELECTRIC FIELD
D. A line along which the electric field vector E is tangent to the line at each point,
is called the
.
C. According to Coulomb’s law, the electrostatic force
F21, that particle 1 exerts on particle 2 depends on the
charges, q1 and q2, of the particles, and the relative
position of the particles.
+
2
+
F21 =
1
(Mark schematically the force and the unit vector on the included figure.)
By definition, the electric field vector E(r) at location r is
a vector such that
B.
,
+
E
-
E
is
F=
(Indicate the force at each particle.)
A. Write an expression for the differential contribution to the electric field vector
at point P due to the indicated fragment of the thin rod. Express the answer in
terms of the variables marked in the figure and the linear charge density of the rod.
y
dE =
P
y
x
Mark this field on the figure.
l
L
Topic 3
GAUSS’S LAW
D. According to Gauss’s law, the net electric flux through any closed (Gaussian)
surface is proportional to the net charge
the surface.
rr
C. By definition, the flux of vector f (r ) over the indicated
differential surface is
rr
r
dΦ = f (r ) ⋅ dA
r
where dA is a vector
v
f
dA
the (differential) surface and its magnitude is equal to
dA
R
Q
B. Using the definition of electric flux, relate the flux over the
indicated differential surface of the sphere to the magnitude E of
the electric field (mark on the figure) produced by the charged
particle located at the center of the sphere, the area of the surface,
and the appropriate angle between these vectors.
dΦE =
A. Using Gauss’s law, relate the flux over the surface
of the cylinder of radius R and height h, to the linear
charge density λ of the uniformly charged rod.
ΦE =
R
h
Topic 4
ELECTRIC POTENTIAL
such that the
D. Electric potential V(r) at position r is a
electric potential energy U(r) of a particle with charge q placed at this location
would be
U(r) =
C. If electrostatic work performed on a charged particle
moving from point A to point B along path 1 is
W1 = 3.2 ⋅10-19 J, the work performed along path 2, which
is twice as long, is
2
VA = 1V
1
VB = 3V
W2 =
B. The charge of the particle discussed in the previous question is
Q=
A. Write an integral expression for the electric potential at point P due to the
charged thin rod. Express the answer in terms of the variables marked in the figure
and the linear charge density of the rod.
V=
L
P
l
x
Topic 5
CAPACITORS
D.
Capacitance relates the charge transferred between the plates of the capacitor
with the voltage
C.
The capacitance C of a
capacitor depends on the geometrical shape of the capacitor and the properties of
the dielectric between the plates
C = κε0
A
d
where A is the area of the plates, d is the distance between the plates, and κ is the
of the
.
B. The electric field between the plates of a
3V
capacitor is approximately uniform. The broken
lines in the figure (draw them) show the 1V and
2V equipotential surfaces and the solid lines are
the electric field lines in this kind of a capacitor.
0V
A. Since, in general, the electric field (vector) is opposite to the
,
of the electric potential, the voltage V between the plates of the capacitor separated
by distance d yields a uniform electric field of strength
A
V
E =
.
κ
d
Topic 6
ELECTRIC CURRENT
D.
The phenomenon of charge transport is referred to as the electric
.
C. The electric current through a
,
dq
is defined by I =
, where dq is the charge
dt
+
+
+
+
+
+
+
+
in time dt.
r
1 r
The average velocity v d ≡ ∑ v i of charge carriers, over a differential
N i
r
vicinity of a given location r , is called the
B.
dA
vd
of the carriers at this location.
θ
vddt
n
A. The charge dq transferred in time
dt, through the differential surface dA indicated in the figure, is
dq =
Hence, the current through the entire surface is equal to the
over that surface
I=
r r
J
∫ ⋅ dA .
surface
Topic 7
ELECTRICAL ELEMENTS
D. A resistor is an electrical element with two sides for which (at any instant) the
passing through this
element (any cross section) is proportional to the potential difference (voltage)
between its terminals. The proportionality coefficient R is called the
of the resistor.
C. A capacitor is an electrical element with two sides, called plates, for which the
potential difference (voltage) between the plates is proportional to the charge Q
Q
Va
-Q
The proportionality coefficient C is called the
Vb
of the capacitor. Consistently with the figure
Va - Vb =
B. Voltage produced by a real source of electromotive
force depends on its
ε
r
Va
ε and
Vb
r.
Consistently with the figure
I
Va - Vb =
A. A transistor is a semiconductor device with three terminals:
, for which the
resistance.
current affects the
Topic 8
ELECTRICAL CIRCUITS
D. According to Kirchhoff’s junction rule, the sum of
is equal to zero.
According to Kirchhoff’s loop rule,
is equal to zero.
C. Consider a circuit containing an ideal seat of electromotive force and three
resistors. Using Kirchhoff’s rules, write the necessary equations to find the currents
indicated in the following figure
I1
I3
R1
ε
I
B.
R
ε
L
I2
R2
R3
Consider a circuit containing an ideal seat of
electromotive force, a resistor and an inductor.
Using Kirchhoff’s loop rule, write the necessary
differential equations to find the current in the
loop
A. For the same loop, write the equations for the complex current at frequency ω
Topic 9
ALTERNATING CURRENT
D. For a sinusoidal alternating current, it value I(t) is a sinusoidal function of time
I (t) = Im sin (ωt + δI)
where Im is called the
of the current,
ω is called the
of the current,
and δI is called the
of the current.
C. By definition, impedance Z of an element relates the
across the element with the
through the element by
B. The number ϕω, relating the
with
is called the phase angle between the current and the voltage.
A. Complex impedance Zω relates the complex voltage with the
by
V(t) =
If the complex impedance of a system has (complex) value Zω, the impedance of
the system is
Zω =
and the phase angle between the voltage across the system and the current through
the system is
ϕω =
Topic 10
RLC CIRCUIT
D.
The average electrical power delivered to an RLC circuit depends on the
through the circuit, the
across the circuit, and the
Pav = Im Vm cosϕ .
C.
In the given RLC circuit, the voltage across the resistor is in phase with the
current in the circuit while the voltage across the inductor
current
by
voltage across the capacitor
by 90°.
90°
and
the
R
L
C
current
B. In a given series RLC circuit the resonance occurs
at the frequency for which the
I
V
assumes its lowest value. Therefore
ω0 =
A. If at resonance the rms value of the voltage across each element is 1V, the rms
value of the voltage across the AC source is
Vrms =
Topic 11
ELECTRICAL CIRCUITS
IN
D. Identify the circuit and explain its purpose.
R2
OUT
R1
C. Identify the circuit and explain its purpose.
IN
OUT
R
C
B. Identify the circuit and explain its purpose.
+
OUT
IN
+5
A. Identify the circuit and explain its purpose.
A
B
X
Topic 12
MAGNETIC INTERACTION
D. According to the definition of the magnetic field vector,
the magnetic force FB (mark on the figure), exerted on a
and
particle, depends on the
B
v
+
of the particle, and the
by
FB =
B
C. Since
the magnetic work (performed on the particle) is zero.
B.
The magnetic potential energy of an object with
magnetic moment μ (mark on the figure) placed
in the magnetic field B is
N
UB =
B
B
S
According to this expression the potential energy
degrees.
has a zero value when the angle between the two vectors is
A. Since the differential magnetic force (mark on the figure) exerted on a
differential fragment of the wire has the value
dFB =
,
B
the
magnetic
force
exerted on a circular loop of radius R is
I
B
FB =
B
.
Topic 13
MAGNETIC FIELD VECTOR
D. According to Gauss’s law, the net magnetic flux ΦB over
B
has a value equal to
.
M
z
N
C. On the included figure, mark the direction of
the magnetic field at points K, M and N.
I
y
K
x
B.
The Biot-Savart law allows one to determine the magnetic field produced by
arbitrarily shaped conducting wire carrying electrical current I. According to this
law, the contribution dB to the magnetic field from a differential fragment “ds” of
the wire is
dB =
where ds is a
A. In the included figure, a uniform magnetic field of known value B, is in the
plane of the rectangle with dimensions given in the figure. Determine the linear
integral of the field along this rectangle (closed loop).
B
b
a
Topic 14
SOURCES OF MAGNETIC FIELD
D. Mark the magnetic field lines for
the field produced by the bar magnet.
N
C.
I
Mark the magnetic field lines for the
field produced by the solenoid. Mark
the location of the magnetic poles of the
solenoid.
I
B. The magnetic field outside the solenoid is approximately zero (vector). Inside
the solenoid the field is uniform and its strength
B0 =
B
where I is the current
.
and the n
A. Permanent magnets are made from ferromagnetic
substances because they exhibit the
of magnetization. As seen in the included figure (complete
the drawing), the magnetization (magnetic field of the
substance) is sustained even after the
.
Topic 15
MAGNETIC INDUCTION
D. Explain the difference between magnetic induction, inductor and inductance.
Inductor is
,
induction is
,
.
and inductance is
C. According to Faraday’s law of induction
ε= −N
the
dΦ B
dt
induced in a coil consisting of N loops
is proportional to the rate of change in the
over the surface of the loop.
v
B. In the figure, the magnet is moving away from the
ring. Mark the direction of the (positive) magnetic flux
over the surface of the ring and the direction of the
(positive) current induced in the ring.
N
A. When a coil, consisting of N loops, rotates in a magnetic field (as in an
electrical generator), the magnetic flux over the surface of this coil is a sinusoidal
function of time Φ(t) = ΦB,maxcos(ωt). Use Faraday’s law of induction to derive an
explicit function for the induced electromotive force.
ε(t) =
.
Topic 16
ELECTROMAGNETIC RADIATION
D. For a certain electromagnetic radiation (a superposition of monochromatic
waves), the function relating the intensity of each sinusoidal wave with the
wavelength is called
of that radiation.
C. Dispersion is an effect in which light is
.
B. Formally defined as the average magnitude of the Poynting vector, the intensity
of the radiation
dU
I = Sav =
dA ⋅ dt
represents the amount of energy passing
of area dA in time dt.
A. The plane wave solutions to the wave equation have the form
E (x , t ) = E m cos(kx − ωt ) .
This wave has a frequency
y
f=
and wavelength
x
λ=
Mark the direction of wave propagation.
z
Topic 17
REFLECTION AND REFRACTION
D. Identify the rays marked in the figure.
Mark the angle of incidence, the angle of
reflection and the angle of refraction.
C.
Snell’s law of refraction states that
the angle of refraction and the angle of
incidence are related by
n1
n2
where n1, and n2 are the refraction indexes of the two media at the refracting
interface.
B.
The critical angle θc is an angle of incidence at which
.
The critical angle depends on the indexes of refraction of the media at the interface
sin θc =
.
A.
The index of refraction n of a medium is defined by
n≡
where
.
Topic 18
MIRRORS
D. The location s of the object and the location s’ of the image both with respect to
the mirror are related by the mirror equation
where f is the
C. Draw a ray diagram to find the
image I produced by a plane mirror
from a real object O. Is the image real
or virtual?
The image is
O
.
B. Draw a ray diagram to find the image I produced by a concave mirror from a
real object O. Is the image real or virtual?
O
The image is
F
A. Draw a ray diagram to find the image I produced by a diverging lens from a
virtual object O. Is the image real or virtual?
The image is
F
F
O
Topic 19
LENSES
D. For a thin lens, the position of an object (s) and the position of the image (s’) is
related by the thin lens equation:
where f is the
C. The magnification of an optical element or instrument is defined as the
A negative value of magnification means that
.
Mark these dimensions on the figure.
B. Identify all rays marked on the figure.
O
Fi
Fo
I
A. Abberation of an element is a
.
Spherical aberration results from
of the lens.
Chromatic aberration results from
of the lens.
Topic 20
THE LAB EXPERIMENTS
D.
An ammeter should be connected in
in which the current is measured.
with the element
with the element
A voltmeter should be connected in
across which the current is measured.
C. When a Weatston Bridge is balanced, the current in the galvanometer has a
value.
The resistance of the investigated resistor is ralated
to the known resistances R1, R2, and R3 by
R2
X
G
R3
R4
X=
B. Hall effect is used to measure magnetic fields. An electric field, transverse to
is produced.
A. According to the law of Malus, the intensity of the light transmitted through a
polarizer, depends on the intensity of the polarized light incident on the polarizer
I = I0 cos2 θ
where θ is the angle between
.