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Principi e Metodi della Fisica
Elettromagnetismo: dalla carica elettrica, ai campi, alle equazioni di
Maxwell ed alle onde elettromagnetiche
Relatività: dal mistero della velocità della luce, allo spazio-tempo, alla
equivalenza di massa ed energia. La gravitazione come geometria
Meccanica Quantistica: la crisi della Fisica Classica, comportamenti
corpuscolari ed ondulatori di materia e radiazione, i quanti di
energia. Principio di indeterminazione. Statistiche quantistiche e
struttura atomica della materia
Bibliografia: Robert H. March, Fisica per Poeti, Ed. Dedalo, 1994
Website: http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
LT in Comunicazione Scientifica, a.a. 2004-05
Mauro Anselmino
Electric Charge
The unit of electric charge is the coulomb. Ordinary matter is made up of atoms which have positively charged nuclei and
negatively charged electrons surrounding them. Charge is quantized as a multiple of the electron or proton charge:
The influence of charges is characterized in terms of the forces between them (Coulomb's law) and the electric field and voltage
produced by them. One coulomb of charge is the charge which would flow through a 120 watt lightbulb (120 volts AC) in one
second. Two charges of one coulomb each separated by a meter would repel each other with a force of about a million tons!
The rate of flow of electric charge is called electric current and is measured in ampères.
In introducing one of the fundamental properties of matter, it is perhaps appropriate to point out that we use simplified sketches
and constructs to introduce concepts, and there is inevitably much more to the story. No significance should be attached to the
circles representing the proton and electron, in the sense of implying a relative size, or even that they are hard sphere objects,
although that's a useful first construct. The most important opening idea, electrically, is that they have a property called
"charge" which is the same size, but opposite in polarity for the proton and electron. The proton has 1836 times the mass of
the electron, but exactly the same size charge, only positive rather than negative. Even the terms "positive" and "negative" are
arbitrary, but well-entrenched historical labels. The essential implication of that is that the proton and electron will strongly
attract each other, the historical archtype of the cliche "opposites attract". Two protons or two electrons would strongly repel
each other. Once you have established those basic ideas about electricity, "like charges repel and unlike charges attract", then
you have the foundation for electricity and can build from there.
From the precise electrical neutrality of bulk matter as well as from detailed microscopic experiments, we know that the proton
and electron have the same magnitude of charge. All charges observed in nature are multiples of these fundamental charges.
Although the standard model of the proton depicts it as being made up of fractionally charged particles called quarks, those
fractional charges are not observed in isolation -- always in combinations which produce +/- the electron charge.
One of the fundamental symmetries of nature is the conservation of electric
charge. No known physical process produces a net change in electric charge.
Coulomb's Law
Like charges repel, unlike charges attract.
The electric force acting on a point charge q1 as a result of the presence of a second point charge q2 is given by Coulomb's Law:
Where ε0 = permittivity of space
Note that this satisfies Newton's third law because it implies that exactly the same magnitude of force acts on q2 .
Coulomb's law is a vector equation and includes the fact that the force acts along the line joining the charges. Like
charges repel and unlike charges attract. Coulomb's law describes a force of infinite range which obeys the inverse
square law, and is of the same form as the gravity force.
Electric Force Example
(1 Kg-peso = 9.8 N)
How many electrons in matter?
If such enormous forces would result from our hypothetical charge arrangement, then why don't we
see more dramatic displays of electrical force? The general answer is that at a given point in a wire,
there is never very much departure from electrical neutrality. Nature never collects a Coulomb of
charge at one point. It might be instructive to examine the amount of charge in a sphere of copper
of volume one cubic centimeter. Copper has one valence electron outside of closed shells in its
atom, and that electron is fairly free to move about in solid copper material (that's what makes
copper a good electrical conductor). The density of metallic copper is about 9 grams/cm3 and one
mole of copper is 63.5 grams so the cubic centimeter of copper contains about 1/7th of a mole or
about 8.5 x 1022 copper atoms. With one mobile electron per atom, and with the electron charge of
1.6 x 10-19 Coulombs, this means there are about 13,600 Coulombs of potentially mobile charge in
one cm3 of copper.
A mole (abbreviated mol) of a pure substance is a mass of the material in grams that is numerically
equal to the molecular mass in atomic mass units (amu). A mole of any material will contain
Avogadro's number of molecules.
1 mole contains ≈ 6 1023 molecules,
Avogadro’s number
Fundamental Forces
Electromagnetic forces
bind atoms in molecules
Strong forces bind protons and
neutrons (nucleons) in nuclei
u,d are quarks
Nucleus contains protons with
charge +e and uncharged neutrons
-10
10 m
10
-14
m
“Empty” atomic matter
Nuclear matter
Composite nucleons
No evidence yet for
further structure of
electrons and quarks
Electric field is defined as the electric force per unit charge. The direction of the field is taken to
be the direction of the force it would exert on a positive test charge. The electric field is radially
outward from a positive charge and radially in toward a negative point charge.
The electric field of a point charge can be obtained from Coulomb's law:
The electric field from any number of point
charges can be obtained from a vector sum
of the individual fields. A positive number is
taken to be an outward field; the field of a
negative charge is toward it.
Electric Potential Energy and Electric Potential
Potential energy can be defined as the capacity for doing work which arises from position or configuration. In the
electrical case, a charge will exert a force on any other charge and potential energy arises from any collection of
charges. For example, if a positive charge Q is fixed at some point in space, any other positive charge which is
brought close to it will experience a repulsive force and will therefore have potential energy. The potential energy
of a test charge q in the vicinity of this source charge will be:
where k is Coulomb's constant
The potential energy of a point charge can be evaluated by calculating the work
necessary to bring a test charge q in from an infinite distance to some distance r.
The zero of potential is chosen at infinity. From the knowledge of the electric
potential energy one can compute the electric force.
In electricity, it is usually more convenient to use the electric potential
energy per unit charge, just called electric potential or voltage.
Magnetic Fields
Magnetic fields are produced by electric currents, which can be macroscopic currents in wires, or
microscopic currents associated with electrons in atomic orbits. The magnetic field B is defined in terms
of force on moving charge in the Lorentz force law. The interaction of magnetic field with charge leads to
many practical applications. Magnetic field sources are essentially dipolar in nature, having a north and
south magnetic pole.
Electric and Magnetic Monopoles
The electric field of a point charge is
radially outward from a positive charge
Electric sources are inherently "monopole"
or point charge sources.
The magnetic field of a bar magnet
Magnetic sources are inherently dipole sources you can't isolate North or South "monopoles".
Magnetic Force
The magnetic field B is defined from the Lorentz Force Law, and specifically from the
magnetic force on a moving charge:
The implications of this expression include:
1. The force is perpendicular to both the velocity v of the charge q and the magnetic field B.
2. The magnitude of the force is F = qvB sinθ where θ is the angle < 180 degrees between the velocity and the
magnetic field. This implies that the magnetic force on a stationary charge or a charge moving parallel to the
magnetic field is zero.
3. The direction of the force is given by the right hand rule. The force relationship above is in the form of a vector
product.
From the force relationship above it can be deduced that the units of magnetic field are Newton seconds /(Coulomb
meter) or Newtons per Ampere meter. This unit is named the Tesla. It is a large unit, and the smaller unit Gauss is
used for small fields like the Earth's magnetic field. A Tesla is 10,000 Gauss. The Earth's magnetic field is on the
order of half a Gauss.
Lorentz Force Law
Both the electric field and magnetic field can be defined from the Lorentz force law:
The electric force is straightforward, being in the direction of the electric field if the
charge q is positive, but the direction of the magnetic part of the force is given by the
right hand rule.
Magnetic Field of Current
The magnetic field lines around a long wire which carries an electric current form
concentric circles around the wire. The direction of the magnetic field is perpendicular to the wire and is in the direction the
fingers of your right hand would curl if you wrapped them around the wire with your thumb in the direction of the current.
An electric current acts with a force on a test small magnet (Oersted)
The magnetic field of an infinitely long straight wire can be obtained by applying Ampere's law. The expression for the
magnetic field is
r = distance from wire
µ0 = permeability of free space
Magnetic Field of (Atomic) Current Loop
Electric current in a circular loop
creates a magnetic field which is
more concentrated in the center
of the loop than outside the loop.
at center of loop
B = (μoi) / (2r)
Magnetic Dipole Moment
(Notice: µ of dipole is not the
same as the permeability µ0 )
Magnetic Force on a Current (Faraday)
F=iLxB
Magnetic Force Between Wires
Energy in Electric and Magnetic Fields
Both electric fields and magnetic fields store energy. For the electric field
the energy density is
This energy density can be used to calculate the energy stored in a capacitor.
For the magnetic field the energy density is
Electric Field Energy in Capacitor
The energy stored on a capacitor is in the form of
energy density of an electric field and is given by
This can be shown to be consistent with the energy stored
on a charged parallel plate capacitor
E = V/d
Capacitance of Parallel Plates
The electric field between two large parallel plates is given by
The voltage difference between the two plates can be expressed in terms of the work
done on a positive test charge q when it moves from the positive to the negative plate.
it then follows from the definition of capacitance C = Q/V
The Farad, F, is the SI unit for capacitance, equal to a Coulomb/Volt.
Storing Energy in a Capacitor
The energy stored on a capacitor can
be expressed in terms of the work
done by the battery. Voltage
represents energy per unit charge, so
the work to move a charge element dq
from the negative plate to the positive
plate is equal to V dq, where V is the
voltage on the capacitor. The voltage
V is proportional to the amount of
charge which is already on the
capacitor.
Element of energy stored:
If Q is the amount of charge stored when the whole battery
voltage appears across the capacitor, then the stored
energy is obtained from the integral:
This energy expression can be put in three equivalent forms by just
permutations based on the definition of capacitance C = Q/V:
Magnetic Interactions with Moving Charge
Applications - Voltage Generated in a Moving Wire
Motional EMF
The magnetic force exerted on the charges in a moving conductor will generate a
voltage (a motional emf). This motional emf is one of many settings in which the
generated emf is described by the Faraday’s law.
The motional emf expression is an application of Faraday's Law, as can be seen from:
Φ = BA is the flux of magnetic field: it is given by B times the perpendicular
area that it penetrates. ΔΦ/Δt gives the variation with time of the flux.
Faraday's Law
Any change in the magnetic environment of a coil of wire will cause a voltage (emf) to be "induced" in the coil. No matter
how the change is produced, the voltage will be generated. The change could be produced by changing the magnetic
field strength, moving a magnet toward or away from the coil, moving the coil into or out of the magnetic field, rotating
the coil relative to the magnet, etc.
AC Generator
The turning of a coil in a magnetic field produces motional emfs in both sides of the coil which
add. Since the component of the velocity perpendicular to the magnetic field changes
sinusoidally with the rotation, the generated voltage is sinusoidal or AC. This process can be
described in terms of Faraday's law when you see that the rotation of the coil continually
changes the magnetic flux through the coil and therefore generates a voltage.
Magnetic Force on Moving Charge
Circular Path from Magnetic Field
If a charge moves into a magnetic field with direction perpendicular to the field, it will follow a circular
path. The magnetic force, being perpendicular to the velocity, provides the centripetal force.
DC Motor Operation
Electric Shock
The primary variable for determining the severity of electric shock is the
electric current which passes through the body. This current is of course
dependent upon the voltage and the resistance of the path it follows through
the body. An approximate general framework for shock effects is as follows:
Electric Current
(1 second contact)
Physiological Effect
1 mA
Threshold of feeling, tingling sensation.
10-20 mA
"Can't let go!" current - onset of sustained
muscular contraction.
100-300 mA
Ventricular fibrillation, fatal if continued.
One instructive example of the nature of voltage is the fact that a bird can sit on a highvoltage wire without harm, since both of its feet are at the same voltage. You can also see
that the bird is not "grounded" -- you will not be shocked by touching a high voltage if there
is no path for the current to reach the Earth or a different voltage point. Typically if you
touch a 120 volt circuit with one hand, you can escape serious shock if you have insulating
shoes which prevent a low-resistance path to ground. This fact has led to the common
"hand-in-the-pocket" practice for engineers and electrical workers. If you keep one hand in
your pocket when touching a circuit which might provide a shock, you are less likely to
have the kind of path to ground which will result in a serious shock.
Will the bird on the high voltage wire be shocked?
Electric current flow is proportional to voltage
difference according to Ohm's law, and both
the bird's feet are at the same voltage. Since
current flow is necessary for electric shock, the
bird is quite safe unless it simultaneously
touches another wire with a different voltage.
Want a scary job? Maintenance on high
voltage transmission lines is sometimes done
with the voltage "live" by working from a
platform on a helicopter, sitting on a metal
platform! The helicopter must make sure it
doesn't touch neighboring wires which are at a
different voltage.
Power:
P = VI = RI2
Maxwell's Equations
Maxwell's equations represent one of the most elegant and concise ways to state the fundamentals of electricity and
magnetism. From them one can develop most of the working relationships in the field. Because of their concise statement,
they embody a high level of mathematical sophistication and are therefore not generally introduced in an introductory
treatment of the subject, except perhaps as summary relationships.
I. Gauss' law for electricity
II. Gauss' law for magnetism
III. Faraday's law of induction
IV. Ampere's law
ρ, J = charge, current density
The Wave Equation
Maxwell's Equations contain the wave equation for electromagnetic waves:
In 1-dimension
with the same form applying to the magnetic field wave in a plane perpendicular the electric
field. The wave equation for electromagnetic waves arises from Maxwell's equations. The
forms of a plane wave solution for the electric and the magnetic fields are
To be consistent with Maxwell's equations, these solutions must be related by
Electromagnetic waves carry energy as they travel through empty space. There is an energy density associated
with both the electric and magnetic fields. The rate of energy transport per unit area is described by the vector
which is called the Poynting vector. This expression is a vector product, and since the magnetic field is
perpendicular to the electric field, the magnitude can be written
(B = E/c)
Traveling Wave Relationships
A single frequency traveling wave will take the form of a sine wave. A snapshot of the wave in space at an instant of time
can be used to show the relationship of the wave properties: frequency, wavelength and propagation velocity.
The motion relationship "distance = velocity x time" is the key to the basic wave relationship. With the wavelength as
distance, this relationship becomes λ = vT. Then using f = 1/T gives the standard wave relationship v = f λ
Waves change periodically in space and time
k = 2π/λ
ω = 2π/T
Transverse Waves
For transverse waves the displacement of the medium is perpendicular to the direction of propagation of the wave.
A ripple on a pond and a wave on a string are easily visualized transverse waves. Electromagnetic waves are
transverse waves.
Longitudinal Waves
In longitudinal waves the displacement of the medium is parallel to the propagation of the wave.
Sound waves in air are longitudinal waves.
Sound Waves in Air
A single-frequency sound wave traveling through air will cause a sinusoidal pressure variation in the air. The
air motion which accompanies the passage of the sound wave will be back and forth in the direction of the
propagation of the sound, a charateristic of longitudinal waves.
Interference and Phase
If a mass on a rod is rotated at constant
speed and the resulting circular path
illuminated from the edge, its shadow will
trace out simple harmonic motion. If the
shadow vertical position is traced as a
function of time, it will trace out a sine wave.
A full period of the sine wave will
correspond to a complete circle or 360
degrees. The idea of phase follows this
parallel, with any fraction of a period related
to the corresponding fraction of a circle in
degrees.
waves add with their phases
Diffraction
This is an attempt to more clearly visualize the nature of single slit diffraction. The phenomenon of diffraction
involves the spreading out of waves past openings which are on the order of the wavelength of the wave. The
spreading of the waves into the area of the geometrical shadow can be modeled by considering small elements of
the wavefront in the slit and treating them like point sources.
If light from symmetric elements near each edge of
the slit travels to the centerline of the slit, as
indicated by rays 1 and 2 above, their light arrives
in phase and experiences constructive
interference. Light from other element pairs
symmetric to the centerline also arrive in phase.
Although there is a progressive change in phase
as you choose element pairs closer to the
centerline, this center position is nevertheless the
most favorable location for constructive
interference of light from the entire slit and has the
highest light intensity
The first minimum in intensity for the light through a single slit can be visualized in terms of rays 3 and 4. An element at
one edge of the slit and one just past the centerline are chosen, and the condition for minimum light intensity is that light
from these two elements arrive 180° out of phase, or a half wavelength different in pathlength. If those two elements suffer
destructive interference, then choosing additional pairs of identical spacing which progress downward across the slit will
give destructive interference for all those pairs and therefore an overal minimum in light intensity.
Double Slit Interference
Double Slit Diffraction
Speed of light
AM Radio Band
The Amplitude Modulated (AM) radio carrier frequencies are in the frequency range 535-1605
kHz. The frequencies 30-535 kHz are used for maritime communication and navigation and for
aircraft navigation. Carrier frequencies of 540 to 1600 kHz are assigned at 10 kHz intervals.
Frequencies: 500-1500 kHz
f = 500 kHz = 5 105 s-1
T = 1/f = 2 10-6 s
Wavelengths: 600 - 200 m
λ = cT = 3 108 m/s 2 10-6 s = 600 m
Quantum energies: 2 - 6 x 10-9 eV
E = h f = 6.6 10-34 J s 5 105 s-1 = 33 10-29 J =
33 10-29 6.25 1018 eV = 2 10-9 eV
TV and FM Radio Band
Frequencies: 54-1600 MHz
Wavelengths: 5.55 m - 0.187 m
Quantum energies: 0.22 x 10-6 - 0.66 x 10-5 eV
h = Planck constant = 6.6 10-34 J s
1 eV = 1.6 10-19 J
Rays with quantum energy of 1 eV correspond to λ ≈ 12 10-7 m, E = hc/λ
Visible Light
The narrow visible part of the electromagnetic spectrum corresponds to the wavelengths near the maximum of
the Sun's radiation curve. White light may be separated into its spectral colors by dispersion in a prism.
Frequencies: 4 - 7.5 x 1014 Hz
Wavelengths: 750 - 400 nm
Quantum energies: 1.65 - 3.1 eV
1 nm = 10 Å
X-rays and gamma-rays
X-ray was the name given to the highly penetrating rays which emanated when high energy electrons struck a
metal target. Within a short time of their discovery, they were being used in medical facilities to image broken
bones. We now know that they are high frequency electromagnetic rays which are produced when the electrons
are suddenly decelerated - these rays are called bremsstrahlung radiation, or "braking radiation". X-rays are also
produced when electrons make transitions between lower atomic energy levels in heavy elements. X-rays
produced in this way have definite energies just like other line spectra from atomic electrons. They are called
characteristic x-rays since they have energies determined by the atomic energy levels
The term gamma ray is used to denote electromagnetic radiation from the nucleus as a part of a radioactive
process. The energy of nuclear radiation is extremely high because such radiation is born in the intense conflict
between the nuclear strong force and the electromagnetic force, the two strongest basic forces. The gamma ray
photon may in fact be identical to an x-ray, since both are electromagnetic rays; the terms x-ray and gamma rays
are statements about origin rather than implying different kinds of radiation.
In interactions with matter, both X and gamma rays are ionizing radiation and produce physiological effects which
are not observed with any exposure of non-ionizing radiation, such as the risk of mutations or cancer in tissue.
Frequencies: 7.5 x 1014 - 3 x 1016 Hz
Frequencies: typically >1020 Hz
Wavelengths: 400 nm - 10 nm
Wavelengths: typically < 10-12 m
Quantum energies: 3.1 - 124 eV
Quantum energies: typically >1 MeV
X-rays
gamma-rays
Gamma radioactivity is
composed of
electromagnetic rays. It is
distinguished from x-rays
only by the fact that it
comes from the nucleus.
Most gamma rays are
somewhat higher in energy
than x-rays and therefore
are very penetrating. It is
the most useful type of
radiation for medical
purposes, but at the same
time it is the most
dangerous because of its
ability to penetrate large
thicknesses of material.
Radiation and the Human Body
Microwave Interactions
The quantum energy of microwave photons is in the range 0.00001 to 0.001 eV which is in the range of
energies separating the quantum states of molecular rotation and torsion. The interaction of microwaves with
matter other than metallic conductors will be to rotate molecules and produce heat as result of that molecular
motion. Conductors will strongly absorb microwaves and any lower frequencies because they will cause
electric currents which will heat the material. Most matter, including the human body, is largely transparent to
microwaves. High intensity microwaves, as in a microwave oven where they pass back and forth through the
food millions of times, will heat the material by producing molecular rotations and torsions. Since the
quantum energies are a million times lower than those of x-rays, they cannot produce ionization and the
characteristic types of radiation damage associated with ionizing radiation.
E = (10-5 – 10-3) eV
→
λ = (12 – 0.12 ) cm
Visible Light Interactions
The primary mechanism for the absorption of visible light photons is the elevation of
electrons to higher energy levels. There are many available states, so visible light is
absorbed strongly. With a strong light source, red light can be transmitted through the
hand or a fold of skin, showing that the red end of the spectrum is not absorbed as
strongly as the violet end.
While exposure to visible light causes heating, it does not cause ionization with its risks. You may be
heated by the sun through a car windshield, but you will not be sunburned - that is an effect of the higher
frequency UV part of sunlight which is blocked by the glass of the windshield.
Ultraviolet Interactions
The near ultraviolet is absorbed very strongly in the surface layer of the skin by electron transitions. As you go to higher
energies, the ionization energies for many molecules are reached and the more dangerous photoionization processes take
place. Sunburn is primarily an effect of UV, and ionization produces the risk of skin cancer.
The ozone layer in the upper atmosphere is important for human health because it absorbs most of the harmful ultraviolet
radiation from the sun before it reaches the surface. The higher frequencies in the ultraviolet are ionizing radiation and can
produce harmful physiological effects ranging from sunburn to skin cancer.
Health concerns for UV exposure are mostly for the range 290-330 nm in wavelength, the range called UVB. According to
Scotto, et al, the most effective biological wavelength for producing skin burns is 297 nm. Their research indicates that the
biological effects increase logarithmically within the UVB range, with 330 nm being only 0.1% as effective as 297 nm for
biological effects. So it is clearly important to control exposure to UVB.
Since the quantum energies of x-ray photons are much
too high to be absorbed in electron transitions between
states for most atoms, they can interact with an electron
only by knocking it completely out of the atom. That is, all
x-rays are classified as ionizing radiation. This can occur
by giving all of the energy to an electron (photoionization)
or by giving part of the energy to the photon and the
remainder to a lower energy photon (Compton scattering).
At sufficiently high energies, the x-ray photon can create
an electron positron pair.
Heart Electrical Phenomena
The rhythmic contractions of the heart which pump the life-giving blood occur in response to periodic electrical control
pulse sequences. The natural pacemaker is a specialized bundle of nerve fibers called the sinoatrial node (SA node).
Nerve cells are capable of producing electrical impulses called action potentials. The bundle of active cells in the SA
node trigger a sequence of electrical events in the heart which controls the orderly pattern of muscle contractions that
pumps the blood out of the heart.
The electrical potentials (voltages) that are generated in
the body have their origin in membrane potentials where
differences in the concentrations of positive and
negative ions give a localized separation of charges.
This charge separation is called polarization. Changes
in voltage occur when some event triggers a
depolarization of a membrane, and also upon the
repolarization of the membrane. The depolarization and
repolarization of the SA node and the other elements of
the heart's electical system produce a strong pattern of
voltage change which can be measured with electrodes
on the skin. Voltage measurements on the skin of the
chest are caled an electrocradiogram or ECG.
The heart's electrical control system must properly
synchronize the pumping functions illustrated above.