COLLEGE PHYSICS
2005 SPRING SEMESTER REVIEW
THERMAL PHYSICS AND THERMODYNAMICS
• Heat Capacity and Specific Heat
If heat Q is absorbed by an object, the object’s temperature change ∆T = Tf − Ti is
related to Q by
Q = C ∆T,
in which C is the heat capacity of the object. Using the object’s mass m, we also define
the specific heat capacity of the material making up the object as
c =
C
and C = c m.
m
• Heat of Transformation
Heat absorbed by a material may change the material’s physical state or phase – for
example, from solid to liquid or from liquid to gas. The amount of energy required per unit
mass to change phase (not temperature) of a particular material is its heat of tranformation
L. Thus,
Q = L m.
The heat of vaporization LV is the amount of energy per unit mass that must be added
to vaporize a liquid or that must be removed to condense a gas. The heat of fusion LF is
the amount of energy per unit mass that must be added to melt a solid or that must be
removed to freeze a liquid.
• Work associated with Volume Change
A gas may exchange energy with its surroundings through work. The amount of work
W done by a gas as it expands or contracts from its initial volume Vi to a final volume Vf
is given by
Z
Vf
p dV,
W =
Vi
1
where the pressure p(V ) may vary during the volume change. When pressure is constant
during the volume change, we find W = p ∆V .
• First Law of Thermodynamics
The principle of conservation of energy for a thermodynamic process is expressed in the
first law of thermodynamics
∆E = Ef − Ei = Q − W,
P
where E = (kinetic energy) represents the internal energy of the material, which depends
on temperature T for an ideal gas in a closed system, Q represents the energy exchanged
as heat between the system and its surroundings (Q is positive if the system absorbs heat
and negative if it loses heat), and W is the work done by the system. Both W and Q are
path-dependent while ∆E is path-independent.
• Applications of the First Law for an Ideal Gas
p V = NkB T and E =
3
NkB T
2
where N denotes the number of particles in the gas and kB denotes Boltzmann’s constant.
Adiabatic P rocess :
Isovolumetric P rocess :
Isothermal P rocess :
Isobaric P rocess :
Cyclic P rocess :
Q = 0 and ∆E = − W =
W = 0 and ∆E = Q =
3
NkB ∆T
2
3
NkB ∆T
2
∆E = 0 and Q = W = NkB T ln
W = p ∆V, ∆E =
Vf
Vi
3
p ∆V, and Q = ∆E + W
2
∆E = 0 and Q = W
• Entropy Change and the Second Law of Thermodynamics
The entropy change ∆S for an irreversible process that takes a system from an initial
state i to a final state f is equal to
∆S =
Z
f
i
2
dQ
,
T
where Q is the energy transferred as heat to or from the system during the process and T
is the temperature of the system during the process.
The Second law of Thermodynamics states that if a process occurs in a closed system,
the entropy of the system increases for irreversible processes and remains constant for
reversible processes:
∆S ≥ 0.
• Carnot Engine and Ideal Efficiency
An engine is a device that, operating in a cycle, (1) extracts energy QH as heat from a
high-temperature reservoir, (2) does a certain amount of work W , and (3) expells energy
QL as heat to a low-temperature reservoir, where
QH = W + QL .
An engine is said to be ideal if all processes involved are reversible. The ideal efficiency of an ideal engine is defined as
=
W
QL
= 1 −
.
QH
QH
A Carnot engine is associated with a reversible cycle involving two isothermal processes and two adiabatic processes between a low-temperature (TL) isothermal and a hightemperature (TH ) isothermal. Its efficiency is
C = 1 −
TL
.
TH
ELECTRICITY
• Coulomb’s Law
The electric force between two point (discrete)
by a distance r:
q1 q2
F1→2 =
4π0
3
charges q1 and q2 at rest and separated
b
r
r2
,
where q1 is located at the origin while q2 is located at r ≡ r br. Here, 0 = 8.85 × 10−12
C2 /N · m2 and (4π0 )−1 = 8.99 × 109 N · m2/C2 . The notation F1→2 is used to denote the
electrostatic force on charge q2 due to charge q1. From Newton’s Third Law, we find
F2→1 = − F1→2.
If q1 q2 > 0, the force is repulsive and is radially outward whereas if q1q2 < 0, the force is
attractive and is radially inward.
• Electric Charge Quantization
Any charge can be written as n e, where n = ±1, ±2, ..., and e = 1.602 × 10−19 C is a
constant known as the fundamental charge.
• Electric Charge Conservation
The net (algebraic) charge of any isolated system cannot change.
• Electric Field
The electric field E at any point r is defined in terms of the electrostatic force F that
would be exerted on a positive test charge q0 placed at r:
E ≡
F
.
q0
The units of electric field are therefore N/C.
• Electric Field Lines
Electric field lines provide a means for visualizing the direction and magnitude of electric
fields. The electric field vector at any point is tangent to a field line through that point.
The density of lines in any region is proportional to the magnitude of the electric field
in that region. Field lines are directed outwardly from positive charges and are directed
inwardly toward negative charges.
• Electric Field due to a Single Point Charge
The electric field E at a point r due to a single point charge q located at r0 is
E =
q
r − r0
.
4π0 |r − r0|3
4
• Electric Field due to a Distribution of Point Charges
The electric field E at a point r due to a distribution of point charges (q1, r1 ; q2, r2, ...)
is
E =
X
qi
r − ri
.
4π0 |r − ri |3
i
• Electric Field due to a Continuous Charge Distribution
The electric field at an observation point r due to a continuous charge distribution is
found by treating an infinitesimal charge element dq at point r0 as a point charges
dq r − r0
dE =
4π0 |r − r0 |3
and then summing, via integration E =
charge elements.
R
dE, the electric field vectors produced by all the
• Force on a Point Charge in an Electric Field
When a point charge q is located in an electric field E set up by other (continuous or
discrete) charges, the electrostatic force F on the point charge q is
F = q E.
Hence, charge q moves either in the same direction as E if q > 0 or in the opposite direction
if q < 0.
• Electric Flux
The electric flux ΦE is defined as the number of field lines which come out of a finite
region V minus the number of field lines which go into V . If we denote ∂V as the closed
surface which defines the boundary of V , then the electric flux is defined as
ΦE ≡
I
E · dA,
∂V
where dA is an infinitesimal surface element vector (which is everywhere perpendicular to
the surface ∂V and points away from V ).
• Gauss’ Law
The net charge qV enclosed within region V is expressed as
qV = 0
I
E · dA.
∂V
5
• Electric Potential Energy
The electric potential energy UE of a charge q at a particular point r is defined as in
terms of the work done to bring this charge from infinity to point r:
UE ≡ − W∞r = + Wr∞ ,
where Wr∞ is the work done to bring the charge from r to infinity.
• Electric Potential
The potential difference ∆Vab ≡ Vb − Va between two points a and b is defined in terms
of the work W performed in bringing a point charge q from point a to point b:
∆Vab ≡ −
Wab
.
q
Here, since 0 = W∞a + Wab + Wb∞ , we find
Wab ≡ W∞b − W∞a ≡ − ∆abUE ,
and hence
∆Vab ≡
∆ab UE
.
q
We may therefore define the electric potential V at point r as
V (r) =
UE (r)
.
q
The units of electric potential are volts (V): 1 volt = 1 J/C = N·m/C.
• Equipotential Surfaces
Points on an equipotential surface all have the same electric potential. The electric field
E is always perpendicular to equipotential surfaces. Hence, the electric field E is expressed
in terms of the potential V as
E(r) = − ∇V (r),
so that the electric field is always directed away from regions of high potential toward
regions of low potential.
6
• Finding V from E
Since work is defined in terms of a path integral
Wab ≡
Z
b
F · ds,
a
where the electrostatic force F on charge q in an electric field E is F ≡ q E, we find
Z
∆Vab = −
b
E · ds.
a
Other units of electric field are therefore V/m. Since the electric potential vanishes at
infinity, we also find
Z ∞
E · ds.
V (r) =
r
• Potential due to a Point Charge
The electric potential at r due to a single point charge q (located at the origin) is
V (r) =
Z
∞
r
q
E(r) dr =
4π0
Z
∞
r
q
dr
,
=
r2
4π0 r
where E = E(r) rb, with E(r) = (q/4π0) r−2 , and ds = rb dr. Note that
E(r) ≡ −
dV (r)
.
dr
The electric potential at an observation point r due to a distribution of point charges
(q1, r1 ; q2, r2, ...) is
X
X qi
1
.
Vi (r) =
V (r) ≡
4π0 |r − ri |
i
i
• Potential due to a Continuous Charge Distribution
The infinitesimal electric potential dV (r) at r due to an infinitesimal charge element dq
at r0 is defined as
dq
1
.
dV (r) =
4π0 |r − r0 |
• Equipotential Surfaces of a Conductor
The surface of a conductor is an equipotential surface.
7
ELECTRIC CIRCUITS
• Electric Current
An electric current I in a conductor is defined as I ≡ dq/dt, where dq is the amount
of positive charge that crosses a surface S that cuts across the conductor in time dt. The
unit of current is the ampere (A): 1 A = 1 C/s.
• Electrical Resistance of a Conductor
The resistance R of a conductor is defined as
R ≡
V
I
(Ohm0s Law),
where V is the potential difference along the conductor and I is the current which flows
through it.
• Power and Ohmic Dissipation
The power P in an electrical device maintained at a potential difference V is P = I V .
If the device is a resitor, with resistance R, we find
P = I2 R =
V2
.
R
In a resistor, electric potential energy is converted into internal thermal energy via collisions
between charge carriers and the conductor lattice.
• Electromotive (Emf) Force
An emf device does work on charges to maintain a potential difference between its
output terminals; the emf (work per unit charge) is defined as
dW
.
dq
E =
An ideal emf device has no internal resistance; a real emf device has an internal resistance
(denoted r). The potential difference between its terminals is E only if there is no current
flowing through the device.
8
• Loop Rule
The algebraic sum of the changes in potential encountered in a complete traversal of
any closed loop of a circuit must be zero. Potential drops by a factor − IR in the direction
of the current flow through a resitor R whereas it increases by a factor E as we cross an
ideal emf device from the negative-terminal to the positive-terminal.
• Junction Rule
The sum of the currents entering any junction must be equal to the sum of the currents
leaving that junction.
• Resistors in Series and in Parallel
When resistors (R1 , R2 , ...) are placed in series, the potential drop across each resistor
P
is Vn = I Rn . Since n Vn = V (according to the Loop Rule), we find
V = I
X
Rn ≡ I RSer
eq
n
and thus the equivalent resistance of individual resistors connected in series is
X
=
RSer
eq
Rn .
n
When resistors (R1, R2 , ...) are placed in parallel, each resistor is exposed to the same
potential difference V and thus the current flowing through each resistor is In = V/Rn .
According to the Junction Rule, the total current flowing through the network is therefore
I =
X
n
V
V
≡ P ar
Rn
Req
and thus the equivalent resistance of individual resistors connected in parallel is
RPeqar
=
X
n
9
1
Rn
!−1
.
MAGNETISM
• Magnetic Field
A magnetic field is defined in terms of the force FB acting on a test particle with charge
q moving through the field with velocity v:
FB = q v × B.
The unit of magnetic field is the tesla (T): 1 T = 1 N/(A· m) = 104 gauss.
• Charged Particle Circulating in a Magnetic Field
A charged particle with mass m and charge q moving with velocity v perpendicular to a
magnetic field B will travel in a circle of radius r defined by the force-equilibrium relation
|q| vB =
mv 2
r
→
r =
mv
.
|q|B
The frequency of revolution f , the angular frequency ω, and the period T are given by
ω = 2π f =
|q|B
2π
=
.
T
m
• Magnetic Force on a Current-Carrying Wire
A straight wire carrying current I in a uniform magnetic field experiences a force
FB = I L × B,
where L is a displacement vector in the direction of the current I.
• Biot-Savart Law
The infinitesimal magnetic field dB at an observation point r set up by a currentcarrying element I d~`0 located at point r0 is
µ0 I ~0
dB =
d` ×
4π
10
!
r − r0
,
|r − r0 |3
where µ0 ≡ 4π × 10−7 T·m/A is the vacuum permeability.
• Forces between Parallel Current-Carrying Wires
Parallel straight wires carrying currents in the same direction attract each other, whereas
parallel straight wires carrying currents in opposite directions repel each other. The magnitude of the force per unit length of wire is
F
µ0 Ia Ib
=
,
L
2π d
where d is the wire separation and Ia and Ib are the currents in the wires.
• Ampere’s Law
The path integral of the magnetic field along a closed loop is proportional to the net
current IS flowing through the loop:
I
∂S
B · ds = µ0 IS ,
where the closed loop ∂S is the boundary of the surface S.
• Magnetic Flux
The magnetic flux ΦB through a surface S in a magnetic field B is defined as
ΦB =
Z
B · dA.
S
The unit of magnetic flux is the weber (Wb): 1 Wb = 1 T·m2 = 1 V·s.
• Faraday’s Law of Induction
Consider a closed conducting wire representing the closed loop ∂S, which bounds a
surface S. If the magnetic flux through the surface S changes with time, a current i and
an emf E are induced in the loop:
E = −N
dΦB
,
dt
where N is the number of coils in the conducting loop. Note: The magnetic flux can change
either with a change in the surface S or a change in the orientation or magnitude of the
magnetic field.
11
• Lenz’s Law
An induced current has a direction such that its induced magnetic field produces an
induced magnetic flux which opposes the original change in magnetic flux.
• Induced Electric Field
Consider any closed loop ∂S bounding a surface S. A changing magnetic flux through
the surface S produces an emf along the boundary ∂S. This induced emf in turn is related
to an induced electric field along the closed loop ∂S:
E =
I
E · ds.
∂S
Faraday’s Law therefore becomes
d
dt
Z
B · dA
= −
I
S
E · ds.
∂S
RAY OPTICS
• Ray Picture
The speed of light in matter is defined as
c
v =
≤ c,
n
where n ≥ 1 is called the index of refraction, which is known to be a function of wavelength
λ and the condition dn(λ)/dλ 6= 0 leads to the phenomena of light dispersion.
A light wave can either be represented in terms of the ray picture or the wave front
picture. In the ray picture, we focus our attention on the wave vector k = k bk, where
b
k = be × bb is the unit vector pointing in the direction of propagation. In the wave front
picture, on the other hand, we focus our attention on the wave crests associated with
consecutive electric and magnetic maxima (separated by one wavelength λ).
• Ray Picture of Light & Law of Reflection
In a simple sense, the ray picture of light makes use of the experimental fact that light
travels in a straight line when it propagates in a uniform medium. We use the ray picture
12
to demonstrate the Law of Reflection, whereby an incident ray hitting a smooth reflecting
surface at an angle of incidence θi (measured from the normal to the surface) is reflected
at the surface and a reflected ray is seen departing the surface at an angle of reflection θr
(again measured from the normal to the surface). According to the Law of Reflection, the
angle of reflection θr is equal to the angle of incidence θi :
θr = θi.
Law of Reflection :
• Law of Refraction
The process of light refraction involves an incident ray propagating in medium 1 (with
index of refraction n1 ) crossing the boundary between medium 1 and medium 2 (with index
of refraction n2 ) at an angle of incidence θi (measured from the normal to the boundary).
At the boundary, a reflected ray propagates in medium 1 at an angle of reflection θr = θi
(according to the Law of Reflection) and a transmitted ray is propagating at an angle of
transmission θt (measured from the normal to the boundary).
The Law of Refraction (Snell’s Law) states that
n1 sin θi = n2 sin θt
→
n1
θt = arcsin
sin θi
n2
→





θt < θi (if n1 < n2)
θt > θi (if n1 > n2)
The process of refraction corresponds to the fact that the transmitted ray is propagating
along a direction different from the incident direction.
• Image Formation by a Thin Lens & Thin-Lens Equation
When an object (of height ho ) is placed at a distance do from a lens of focal length f ,
an image (of height hi ) is formed at a distance di from the lens (measured positively on the
opposite side of the lens from where the object is located) defined by the thin-lens formula
1
1
1
+
=
do
di
f
→
di =
do f
.
do − f
By basic trigonometry, the magnification m of the lens is defined as
m =
hi
di
f
.
= −
= −
ho
do
do − f
For a converging lens (f > 0), the image is real (di > 0) and inverted (hi < 0) when
do > f or the image is virtual (di < 0) and upright (hi > 0) when do < f . For a diverging
lens (f < 0), on the other hand, the image is always upright (hi > 0) and virtual (di < 0).
13