About half has past…
What have we learned:
1. Kinematics – description of motion
2. Dynamics – reasons for motion
3. Applications to some physical phenomena
Now: Electricity – just one more force
1. What property is necessary for this force to
2. Description of the force…
3. What does this force depend on?
Example – force of gravity:
1. Any massive object (that has a property of
gravitational mass) creates the gravitational field
around it. (The mass changes the space.) If another
massive object appears in this “changed” space, a
gravitational force acts on it.
2. The force of gravity between two point-like objects
is central, attractive, and proportional to both
3. The force depends on the distance between the
objects, ~ 1/d2.
What have we learned ?
• When we rub objects they acquire a ‘charge’.
• Like charges repel, unlike charges attract.
• Force between charges depends on the
What sort of property is “charge” ?
Charge is a property of some of the subatomic
particles that “make up” matter.
Charge comes in two varieties, by
By convention: electrons carry negative charge
protons carry positive charge
neutrons carry no charge – they are neutral
What happens when you rub a piece of glass/rubber ?
Charges can neither be created nor destroyed.
They can be “separated” – in this case
negatively charged electrons leave the glass
charging the silk negatively and the glass
positively (fur positively, rubber negatively).
The net charge remains zero as it was in the
Total charge can neither be created nor destroyed.
net charge = positive charge – negative charge
Example: you connect three charges together
+3nC, +5nC, and -4nC, the net charge is: 3 + 5 - 4 = 4nC
When I rub a piece of rubber, some charges (electrons)
get removed from the fur and transferred to the
IONS are formed:
ION = charged atom
Negative ION
Positive ION
Electric force vs. Gravitation
ƒ Electric forces can be repelling or attractive: e.g.
two positive charges repel each other, a positive and
a negative charge attract.
ƒ Electric forces are much stronger than gravity.
The pith ball did not move unless is was charged.
ƒ If the charges of two particles is zero, the electric
force between them is zero. Gravity acts on
everything that is massive (no negative mass).
Why are electrical forces important ?
All atomic and molecular interactions,
and therefore structure of all atomic
matter is guided by electric forces
Most of (macroscopic) forces that
surround us in everyday are electric in
nature: string tension, elastic forces of
springs, any solid or liquid, friction in
solids and liquids, etc.
Most of the devices use electric power,
or electrically guided…
Charge is measured in Coulombs (C)
The charge of one electron is only 1.6 x 10-19 C
1 Coulomb is a very large unit – not really convenient
for electrostatics
The reason: 1 C/s = 1 Ampere
Two ‘point’ charges q1 and q2 act on each other with the
electrical force proportional to these charges and inversely
proportional to the distance squared between them.
q1 q 2
F = ke 2 ,
k e = 9 × 10
The force is acting along the straight line connecting the
charges and is attractive if negative and repulsive if
What is the force acting
between two +1 µC charges
1 cm apart?
F = 9 × 10
= 90 N
−2 2
(10 )
F =k 2
Match the picture the with force acting on the left charge in
each case:
What is the force on the charge on the right ?
4. None of the above
Answer 2 – because of the Newton’s 3rd law
In which of the two situations shown is the electrical
force larger ? Or are they the same ?
F = ke 2 = ke 2
1 q
= ke
= ke 2
F = ke
2 d
( 2d )
What does a charge do to make other charges feel
a force ?
An electric charge alters space around it by creating the
electric field. Another charge in this field is acted upon
by the electric force.
Electric (static) force:
1. Any charged object (that has a property of electric
charge) creates the electric field around it. (The
charge changes the space.) If another charged
object appears in this “changed” space, an electric
force acts on it.
2. The electric force between two point-like charged
objects is central, attractive or repulsive, and
proportional to both charges. Attractive between
opposite charges and repulsive between like ones.
3. The force depends on the distance between the
objects, ~ 1/d2.
Electric field (strength)
Force acting on a charged object
Electric field strength =
charge on this object
Example : Find the force acting on the
F electron in an electric field of 100 N/C.
Electric field E =
q E = F ⇒ F = q × E = −1.6 × 10 −19 × 100
Unit :
= − 1.6 × 10 −17 N
1.6 × 10 −17
Find acceleration : a = = −
9.1 × 10 −31
13 m
= 1.8 × 10 2
The electric potential energy
ƒ Imagine you have two opposite charges some distance apart
from each other. Let’s fix one of the charges. One can apply
an external force to a movable charge and move it to a larger
distance from the fixed charge. The work done by the
external force will be positive because it is directed in the
same direction as the displacement. This force will “work”
against the electric force and the latter will do negative
ƒ After that, one can release the charge, and it will be
attracted back to the fixed charge by the electric force.
The electric force will then do positive work.
ƒ We can introduce the potential energy in the same fashion
we did for the gravitational field
ƒ The external force does positive work against the electric
force, the potential electric energy increases by exactly this
amount. Then, when the charge is allowed to move, the
electric force does positive work, and the potential energy
Potential difference
ƒ This potential energy (and the work of the external force) is
proportional to the product of these charges. Then (similarly to the
introduction of the field) we can divide the potential energy by the
value of the moving charge and introduce a new scalar quantity, the
electric potential. This quantity is independent of the moving
charge and characterizes the electric field of the fixed charge.
ƒ The potential as well as the potential energy is a relative term – you
have to choose the point of zero potential. Then the electric
potential at any other point will be a potential difference between
that point and the chosen point of zero potential.
ƒ This is similar to what we did with gravitational potential energy
∆PE = mgd
d – is the vertical distance between two positions – a
relative term
ƒ Sometimes it is useful to introduce an “absolute” value for potential
(and therefore PE), namely – zero at infinity where the field is
ƒ The sign convention is such that if an object is charged positively,
the potential around it is positive, negatively – negative.
ƒ One can think of it as a potential to attract (if negative) or repel
(if positive). The same sign convention is valid for PE.
Total energy and bound states
If we ascribe zero potential to
infinity, and there is a positively
charged object (a nucleus), then
negatively charged objects will be
attracted to it.
If this negatively charged object
rotates around the positively charged
one on a stationary orbit – such that
the centripetal force is supplied by
the electric attraction. This forms a
bound state.
The total energy (KE+PE) of the bound
particle is less than zero. Therefore
the charge cannot escape (as the
planet cannot), because otherwise at
infinity where PE=0, the KE would have
to be negative which is impossible.
states E>0
One of the
states E<0
Conductors and Dielectrics
Materials can be divided into two groups - conductors
and dielectrics.
In dielectrics, electrons are tightly bound to their hosts,
the atoms or molecules keeping them neutral.
If such material is exposed to the electric field, the
molecules or atoms may rearrange themselves a little
(the shape of electron clouds changes) to “make the
negative charges closer to the positive charges”.
However, free motion of charges are not possible.
In conductors, some electrons are “free” from their
hosts and can readily move when the field is applied.
Typical conductors – metals, salty water…
Typical insulators – plastics, ceramics, etc.
Can there be electrical forces between a charge object
and a neutral object ?
1. Yes
2. No
Let’s do the experiment…
Why ?
“Neutral” objects also contain charges.
They just have the same amount of positive &
negative charge, balancing each other.
If charges can move, objects can become
Three different ways of charging objects:
1. Charging by contact – in this case as you touch the
object, the electrons rearrange:
a. If you have a negatively charged object – there is
an “excess” of electrons – some of them (but not
all) move to the other object and charge it
b. If you have a positively charged object – there is a
lack of electrons – some of the other object’s
electrons partially recover this deficit and charge
the other object positively.
In both cases the charge of the object that you
charge is the same as the charge of the originally
charged object
2. Charging by induction: Only for conductors –
a. Bring a charged object close to the conducting
object, but not touching it.
b. Connect the conducting object with the ground
c. Remove connection with the ground
d. Remove the originally charged object
a. The opposite charges are attracted to be closer to
the charged object, and the like charges as far as
b. The like charges can now escape
c. Now they cannot return
d. The object “lost” like charges and is therefore
charged with charge opposite to the original.
3. Charging by rubbing – in this case the charges are
separated as a result of mechanical work.
• Charges of two objects are equal but opposite
• What object acquires negative charge? – The one
whose affinity to electrons is larger – balloon and hair,
plastic and fur, silk and glass, etc.
Charges can move.
The motion of charges is called the electric
CURRENT = amount of charge that flows by a location in
space per time.
I = q/t
Unit: Ampere = 1 A = 1 C/s
Direction of the electric current
ƒ By convention, the positive direction of the current
is the direction of motion of a positive charge which
is repelled by positive charges and is attracted by
negative charges. If a loose positive charge is
located at a position with a higher potential, it will
move in the direction of the lower potential, so that
the electric force does positive work and the
charge loses the potential energy as it is moving.
ƒ Since the negatively charged electrons are the
carriers of charge in any metal conductor, they
move in the opposite direction to the conventional
direction of the current; e.g. if the (positive)
current is flowing from A to B in a copper wire, it
means that the electrons are drifting from B to A.
If charges move – electric current = charge passing by per
unit time
I = q/t ,
Ampere = Coulomb/second
Conventional direction of current from higher potential
to lower potential regardless of the charge carrier –
compare the result of two events
1. Positive q moved from left to right
2. Negative q moved from right to left
For the charge counter that counts charges to
determine the current, q/t, there is no difference
whether positive moved to the right or negative
moved to the left
Small drift – large current
ƒ Let’s say we have a current of 1A in copper wire of
cross section area of 1 mm2 at a room temperature.
ƒ What is the thermal average speed of the
electrons? It’s 1.2×105m/s. If all electrons were
moving with this speed along the wire, the current
would have been 3.2×105A !!
ƒ Instead they rapidly move back and forth with this
speed and very slowly drift with a drift velocity
which is about 4 mm/s in this case.
ƒ Therefore, the electrons do not have to move much
on the average to cause a substantial current.
Constant force – constant current?
1. If a charged particle is placed in a uniform field E, a
force F = qE is acting on it, and its acceleration = F/m
2. What happens if a conductive wire is placed in the
electric field that is “maintained” constant by some
external means? – A constant (direct) electric current is
the result.
3. Where is the acceleration? Electrons accelerate, but
collide with the lattice or other electrons, stop, then
accelerate again, and so on. On the average, they move at
a constant speed making the current constant.
4. This drag – is the reason for resistance – “the
opposition” to the current flow.
5. What does the resistance depend on?
RESISTANCE – a measure of the opposition to current flow
In regular conductors like metals it depends on 4 factors:
1. Properties of the material – conductivity the higher the
conductivity, the lower the resistance.
2. The length of a conductor – the longer the wire, the
larger the resistance
3. The cross section area of the conductor – the larger the
area, the lower the resistance
4. The temperature – the higher the temperature, the
larger the resistance
1 l
σ A
(1 + β∆T )
1. Can use the
formula and
2. Totally different picture – electrons
accelerate in the field and barely
collide with anything – cannot use
the formula…
3. More difficult
dependence than
in (1) because of
ion motion. The
formula is not
Voltage in a circuit
ƒ The current is flowing through the circuit because the
carriers of charge can move and because there is a
potential difference between the terminals of the circuit.
ƒ This potential difference is due to a work done by the
external forces in a battery or a power supply. This work
is done against the electric force and it results in
separation of charges, or in other words, charges gaining
potential energy. This potential energy is equal to the
amount of separated charges multiplied by the potential
difference across the power supply or voltage across the
power supply: W external = PE = qV .
ƒ You will see that it is much more convenient to discuss this
voltage across the power supply rather than potential
energy because voltage is independent of charge and is an
adequate characteristics of a power supply.
Analogy – if the current that you observe is similar to a
waterfall, the battery is similar to the water pump that
pumps the water upwards.
V = Work/q = PE/q
UNIT: 1 Volt = 1 V = 1 J/C
Now, after being separated the positive and negative
charges can happily reunite, but not through the battery –
through the circuit – and by doing that they will spend the
acquired energy.
Electro – mechanical analogy
How an electron
spends its energy
When a charge moves along a wire moved by the electric
force (field), the electric force does work – the charge
loses its potential energy in the electric field. Where
does it go? to going through resistance! – to internal
energy, light, etc.
1. On each part of the circuit, some energy is spent,
therefore some voltage is reduced as well.
2. The voltage drop across a portion of the circuit is
proportional to the current and the resistance of this
portion – Ohm’s law
V = I×R,
I= ,
3. Alternatively: The current in the circuit is equal to the
voltage across it divided by its resistance.
4. Or: The resistance of a circuit element is equal to the
voltage drop on it divided by the current in the circuit.
A 100 Ω light bulb is connected to a 9 V battery. How
high is the current ?
I = V / R = (9V) / (100 Ω) = 0.09 A = 90 mA
NOTE: 1 Ω = 1 V/A
Example: You are making a circuit from a 12V battery and
a resistor (an element that has a calibrated
resistance). If you want a current to be less than 1mA,
what resistance should you use?
V = I×R,
I= ,
= 12,000Ω = 12kΩ
Example: What voltage is needed to get a 10A current
through the 20Ω resistance?
V = I×R,
I= ,
V = 10A × 20Ω = 200V
A 10 Ω light bulb is connected to a 12 V battery. Compare this
situation to a circuit where three 10 W bulbs are connected to a 12
V battery in series. What can you say about the series circuit?
1. The resistance of the circuit.
2. The voltage from the battery.
1. Is increased
3. The current.
2. Is the same
4. The voltage on each bulb.
3. Is decreased
5. The brightness of the bulb.
A 10 Ω light bulb is connected to a 12 V battery. Compare this
situation to a circuit where three 10 W bulbs are connected to a 12
V battery in parallel. What can you say about the series circuit?
The resistance of the circuit.
2. The voltage across the battery.
1. Is increased
3. The current.
2. Is the same
4. The voltage drop on each bulb.
3. Is decreased
5. The brightness of the bulb.
6. The current through each bulb
ƒ Now we know that charges spend their energy
traveling along the circuit. Apparently we use some of
this energy.
ƒ Consider a light bulb – let’s call the energy used by
this bulb useful (regardless of the efficiency)
ƒ When a charge q passes through the bulb, the charge
loses energy E=qV, where V is a voltage drop across
the bulb
ƒ There are many charges pass the bulb over time t.
How many? – The rate is q/t = I, therefore q = It
ƒ Then E = VIt – the energy dissipated on a bulb over
time t.
ƒ Can also introduce the power dissipated on the bulb
P = E/t = VI
V = Energy per unit charge = E/q
I = Charge per unit time = q/t
Power = Energy per unit time = E/t
= (Energy per unit charge) x (Total charge per unit time)
Units of power = Volt x Ampere = J/C x C/s = J/s = Watt
A 100 W light bulb is connected to 100 V power supply.
What is the current through the bulb ?
P = V I ⇒ I = P/V = (100W)/(100V) = 1 A
Power and Ohm’s law
I= ,
V = I×R,
P = VI = I R =
If you want to increase the power dissipated on, say,
light bulb, what should you do?
Increase its resistance?
Decrease its resistance?
An automobile headlight and a dashboard light use the
same voltage, but the power input to the headlight is much
larger because
1. Its resistance is lower
2. The current in it is larger
3. It uses energy faster
4. All of the above
9P=V2/R, I=V/R, P
A searchlight installed on a truck requires 60 watts
of power when connected to 12 volts.
a) What is the current that flows in the searchlight?
b) What is its resistance?
P = VI
60W = 12 V x I
60W/12 V = I
V = IR
12 V/5 A = R
R = 2.4 Ohms
Example: A light bulb is rated at 60W when connected to
120 V.
What current flows through the bulb in this case?
What is the bulb’s resistance?
What would be the current if it were connected to
60V (if the resistance is the same)?
What would be the power consumption in this case?
1. I = P / V = 60 / 120 = 0.5A
2. R = V/I = 120 / 0.5 = 240Ω
3. I = V/R = 60 / 240 = 0.25A
4. P = VI = 60 ⋅ 0.25 = 15W
kW-h – kilo Watt hours = unit for the consumed energy
Example: The toaster is rated 1,500 W. It takes 6 min to
make a toast. How much does it cost if the energy
price is $.1/kW-h ?
1. Energy consumed = 1.5kW ⋅ hour = 0.15kW ⋅ h
2. Cost = .1$/kW ⋅ h ⋅ 0.15kW ⋅ h = $0.015
The resistance of an electric heater is 10Ω when
connected to 120 V. How much energy does it use
during 30 min of operation?
E = Pt =
W ⋅ 0.5h = 0.72kW ⋅ h
AC = Alternating Current
• Household outlets
• Used for lights, heaters etc. where direction of
current doesn’t matter.
• Can be stepped up or down by using transformers.
DC = Direct Current
•Can be produced from AC by “rectification”
• Needed for electronics etc.
• Batteries produce DC voltages
Some more things that I want to mention
Semiconductors –
are close to insulators – have a complete band of
electrons, but if something changes – temperature or
impurities are added, they may acquire free carriers
of electric charge
p – type
n - type
the fourth state of matter – all (or most) atoms or
molecules are ionized
the temperature is high enough so the kinetic energy
of electrons is larger than ionization energy
dense and rarefied plasmas
gas with two temperatures – ion and electron
Electron–positron plasma, fusion, etc.
Piezoelectric Effect
Crystals which are capable of separating charges when
compressed, twisted or distorted are said to be
Mechanical work is used to separate charges.
The voltage due to these separated charges can be used
for the external loads such as light bulbs in shoes or
the speakers in the vinyl disc players.
Many applications in technology…
ƒ First we approach to the magnetic force as just
another force
ƒ Then we will notice that the magnetic force is
related to the electric force, and that the electric
and magnetic fields are related as well
ƒ We will discuss the features of the electromagnetic
field and will notice problems that cannot be
resolved in frames of the Newtonian physics and
require the new physics, the theory of relativity.
First notion: Magnets
ƒ Have two poles (North & South)
ƒ Poles cannot be separated
ƒ Like repel, unlike attract
ƒ Magnets create a field around them:
The Earth’s Magnetic field
Does the compass needle always point towards the
north pole of the Earth?
No, the magnetic north pole is not the same as the
geographic north pole.
Which of the following statements is true:
1. The north pole of a compass needle points toward the
north pole of earth.
2. The south pole of a compass needle points towards
the north pole of earth.
The first statement is
correct, but it’s all about
conventions: The south
magnetic pole of the
Earth is near the north
pole of the Earth
What is magnetism ?
What we know so far:
• Electric charges produce electric fields around
•An electric field causes a force on any charged
object placed in it.
•Magnets produce magnetic fields.
•A magnetic field causes a force on the poles of any
magnet placed in it.
It turns out that electricity and magnetism are related
Experiment with moving charges – electric
current and a compass
A moving electric charge
(current) produces a magnetic
field around it.
Thus magnetic fields is caused by the electric current…
The magnetic field around a wire
with a current is proportional to the
current and inversely proportional
to the distance from the wire.
Earth’s magnetic field
Why are some materials “magnetic” and others are not?
Electrons move around atoms = Moving charge!
Electrons produce their own tiny magnetic field.
In some materials these fields can all
line up, producing a big field. These
materials are called “Ferromagnetic”.
Moving charge (current) in a magnetic field:
A magnetic field exerts a force on a
moving electric charge (current)
The force on the moving electric charge (current) is
proportional to the charge, its velocity, and the
strength of the field
F = km q B
F = IB∆l
The forces in both cases are perpendicular to the both
magnetic field and velocity of the charge
The first problem with Newtonian mechanics
ƒ Have you noticed that the force is proportional to
the velocity of charge? There is a problem here:
think of different observers – recall that two
observers in different inertial frames (moving with
respect to each other uniformly) do not agree about
the velocity of the moving particle but agree about
its acceleration and the force acting on it!
ƒ So if the force explicitly depends on the velocity, it
may be zero in one inertial frame and not zero in
another! The observers disagree about the Force!
ƒ This problem cannot be resolved within Newtonian
mechanics. We will remember this (and everything
else I’ve ever said) and continue our study of the
magnetic field.
Side note
ƒ Some of you may have noticed that we have already
seen a force that depended on the velocity – the
force of air resistance. It did not cause any
agitation. Why?
ƒ Because the air resistance is due to collisions with
molecules, similar to the pressure force on the walls
of a vessel containing gas. The larger the velocity,
the more the momentum transfer, the larger the
force is.
ƒ However, no problem arises in this case since this
resistance force is not proportional to the velocity
in general, but to the relative velocity with respect
to the air; there is a special frame of reference in
this case (the air), but not in the case of the
magnetic force.
The second problem with Newtonian mechanics
ƒ The second striking difference of the magnetic
force is its direction.
ƒ The Lorentz force is directed perpendicularly to
both magnetic field and the velocity of the charge.
If the charge is moving in the same direction as the
field, there is no magnetic force acting on it.
ƒ Problems with this facts intersect with problems
related to the force depending on the velocity, but
the mere fact that no mechanical phenomenon has
such features seriously questions the possibility of
a mechanistic description of magnetism.
Since a moving charge creates a magnetic field and the
latter acts on the moving charge, the moving charges may
interact magnetically:
q1q2 v1v2
F~ 2
d c
Parallel currents act on each other with magnetic forces
– opposite currents repel each other while the currents
directed in the same direction attract…
I1 I 2
– similar to the Coulomb law
Electric motors
Moving charges create a magnetic field.
How about moving magnets ?
A moving magnet induces an
electric field – is a source of an
electric current.
If a closed loop is moving in the
vicinity of a magnet, an electric
current is induced in it
Regenerative braking
Eddy currents
How does it work ?
Voltage is the same
in each loop and
adds up: V ~ N
Vout N out
N in
INPUT: AC current
Produces a changing magnetic field
Induces a current in the
output coil
You connect a stereo to the 110 V outlet. The stereo
needs 11 V voltage. You have a transformer with 100 turns
in the input coil. How many do you need in the output coil ?
Vout N out
N in
(11 V)/(110 V) = Nout / 100
(11 V)/(110 V) = 1/10 = Nout/100
1/10 x 100 = 10 = Nout
ƒ In 1911, Kamerlingh Onnes
discovered that at very low
temperatures, the resistance in
metal conductors vanishes
ƒ Meissner discovered that the
superconductors also do not
allow the magnetic field to go
through them
ƒ It turned out that the regular
resistance does not vanish at
all, but the super current is
flowing along the surface. This
current does not experience
resistance and it also
compensates the magnetic field
inside the superconductor
Superconductivity continued
ƒ The phenomenon of superconductivity was only
understood in 1950s
ƒ Different types of superconductors were discovered
including High TC superconductors whose critical
temperature can be as high as 150K.
1. A moving charge, an electric current, or changing
electric field induce a magnetic field
2. A changing magnetic field induces an electric field.
3. The electric charge is the source of electric and
magnetic fields
4. Electric and magnetic fields are related
with each other
Maxwell equations – complete picture of
Predicted Electro-Magnetic waves
Light – EM wave
James Clerk Maxwell (1831 – 1879)
The new way of description of physical
ƒ Maxwell’s description of the electromagnetism is
especially important because it is a field description
rather than force description.
ƒ Maxwell’s equations are equations for the electric and
magnetic fields that adequately describe the
properties of these fields and completely cover all
electromagnetic phenomena.
ƒ They have a strong predictive power. They include
possible sources of the fields such as charges or
currents, and predict and describe the
electromagnetic waves without any charges in the
ƒ Since then, all new phenomena on the fundamental
level has been described in the same fashion – in the
field language.
Electromagnetic Waves
Wave: A traveling disturbance consisting of
coordinated vibrations that transmits energy but
not matter.
EM waves:
1. Source of EM waves
2. What oscillates?
3. EM waves propagation
Heinrich R. Hertz
1857 – 1894
If the electric field is disturbed by moving charges back
and forth… what happens?
• The moving charge creates an oscillating electric field and
an oscillating magnetic field.
• The oscillating magnetic
field produces a new
changing electric field
opposing the original
electric field.
• The oscillating electric
field will produce a new
opposing magnetic field.
An oscillating (accelerating)
charge is a source of EM
• EM wave is transverse – E
and B fields are mutually
perpendicular and both
perpendicular to the
direction of propagation
• Both E and B fields
oscillate in phase
• EM wave propagates with
speed of light
• does not need a medium…
What kind of wave is an electromagnetic wave ?
1. Transverse Wave
2. Longitudinal wave
3. Neither
Differences between EM waves and mechanical waves:
• EM waves are really two (coupled) waves: an electric
field and a magnetic field wave.
• EM waves do not require a medium and can travel
through vacuum.
SPEED of EM waves:
In vacuum: c = 299,792,458 m/s
c ≈ 300,000 km/s = 186,000 miles/s
In a medium they are slower
c medium = c/n
n – index of refraction
A radio station transmits EM waves with frequency 100 MHz.
What is the wavelength of the EM waves ?
c = fλ
f =
f = 100 MHz = 100 x 106 Hz = 100,000,000 Hz
λ= c/f = (300,000,000 m/s )/(100,000,000 Hz) = 3 m
Or alternatively:
λ=c/f = 3∗108m/s/108Hz = 3m
EM Waves with
f = 100 … 109 = 1,000,000,000 Hz
λ = 3000 km … 0.3 m
Can be generated directly by moving charges back and
forth (changing electric field) in an antenna.
• Radio: AM = 700 – 1400 kHz, FM = 88 – 108 MHz
• TV
• Emissions from Planets and stars
Very Large Array, New Mexico
EM waves with
f = 109 … 1012 Hz
l = 30 cm … 0.3 mm
Can be generated by very sophisticated electronics
and antennas.
• Communication: Satellites
• Radar
• Cooking food
Microwaves bounce off metallic objects (Reflection).
Measure time it takes to reach object and return.
If it takes 20 microseconds for the
signal to return, how far away is the
airplane ?
20 µs = 20 x 10-6 s = 0.00002 s
d = c t/2 = (300,000 km/s)(0.00002 s)/2
= 3 x 105 km/s x 2 x10-5 s /2 = 3 km
Venus’ surface viewed by radar:
Microwave oven:
Water molecules have are electric dipoles.
Water molecules rotate. When heated, they rotate more
If we apply a changing electric field to the water, it will
be forced to rotate, increasing its kinetic energy.
… and its temperature!
EM waves with
f = 1012 … 4 x 1014 Hz
λ = 0.3 mm … 0.75 µm (750 nm)
Emitted by warm objects, lasers, LEDs
• Heat radiation
• Remote controls
• Some Wireless devices
• Lasers
• Fiber-optic communication
Infrared Photography:
EM waves with
f = 4 x 1014Hz … 7.5x 1014Hz
λ = 750 nm … 400 nm
Very narrow range of EM waves which happens to
be detectable by human eyes.
• Seeing
• Optics
• TV
• Photography
• Telescopes
• Microscopes
Why do we see visible light and not other EM waves ?
EM waves with
f = 7.5 x 1014 … 1018 Hz
λ = 400 nm … 0.3 nm
Emitted by the sun, very hot objects
• Tanning
• Lithography to make computer chips
EM waves with
f = 1016 … 1020 Hz
λ = 30 nm … 0.3 pm
Made by bombarding a target with electrons.
Can travel through matter almost unhindered.
• X-ray imaging
• X-ray diffraction
Materials with “heavier” atoms in them stop x-rays more
efficiently, for example, calcium in bones.
Since X-rays have
wavelength the size of
atoms, they can reveal
atomic structure of
crystals and molecules.
For example:
Structure of DNA
How are X-rays generated ?
Electron gun
EM waves with
f = 3 x 1019 … > 1023 Hz
λ = 10 pm … < 3 fm
In nuclear processes, such as γ radioactivity
In the process of electron-positron annihilation
Supernovae explosions
Neutron star collisions, etc.
Passage of the EM waves through the atmosphere
EM waves passage through the Earth’s atmosphere:
Ions high up (~ 90 km) in atmosphere can reflect
certain radio waves (shortwave).
Greenhouse effect:
Certain gases (H2O, CO2, CH4) reflect or absorb infrared
radiation. Keep heat from escaping earth into space.
•Keeps it about 35o
higher than without…
Important for life
•Regulates temperature
on Earth
•Responsible for high
temperature on Venus
(460 C) – runaway
greenhouse effect
•Global warming
Ozone layer
At about 20-40 km above sea level: High concentration
of ozone (O3)
Stops UV light, protects life on earth.
Ozone hole: Certain pollutants can reduce amount of ozone
in ozone layer (CFC’s)
Do not confuse ozone
problem with the greenhouse
Light and
• Light is an electromagnetic wave with frequencies
in the range of 4 x 1014 to 7.5 x 1014 Hz
• In air & vacuum, wavelengths range from 450 nm to
750 nm, where 1 nm = 1 billionth of a meter.
• Color is determined by frequency (wavelength)
• White light is a mixture of all colors
• In vacuum, the speed of light is 3 x 108 m/s.
Light passing through a very narrow slit will spread out.
Every point reached by the wave (including those in the
slit) becomes a source of waves
The resultant signal at any point is a result of
interference from all directly arriving waves
INTERFERENCE: summation of waves – superposition
Constructive Interference
Destructive Interference
Waves in opposite phases
Waves in the same phase
Two slit interference
Summation of waves with different paths
– for a maximum the difference in paths has to be the
integer number of wavelengths
- for a minimum – half integer
Interference effects: thin films
Some wavelengths interfere
destructively, some constructively
In most light, the E-vector points in random directions, but
the direction of the magnetic field is always determined by
the direction of the electric field
light, e.g.
laser light:
Unpolarized light can be polarized by
• Reflection
• Passing through a polarization filter (polaroid)
Note: B field direction and magnitude is always related to
the E field’s direction and magnitude
Polarized plated filter out
different polarizations leaving
only one – along its axis
Liquid Crystal Displays
No signal –
transparency because
of correct rotation of
Electric signal
changes polarization
in the crystal – no
transparency - image
The Law of Reflection:
The angle of incidence equals the angle of reflection.
Reflection: Specular reflection
Specular reflection – depends on the reflecting
surface only – all rays are reflected similarly
– the surface is flat enough
- Depends on the wavelength –the shorter the
wavelength, the better quality mirror is required for
to obtain the specular reflection
Diffuse reflection:
Reflection off rough surfaces – most common.
Most light we see is diffuse reflected light.
Colored Objects:
• Reflect only some frequencies of light and absorb others.
If an object appears red, it
Reflects red and absorbs other colors such as blue,
yellow, green, etc.
REFLECTION, plane mirror
When you look into a mirror, what is reversed ?
1. Nothing is reversed.
2. Left and Right are reversed.
3. Up and Down are reversed.
4. Front and Back are reversed.
One-way mirrors
Normally, part of light
is reflected and the
rest is transmitted
Curved mirrors
Convex mirror
Van Eyck: “The Arnolfini couple”
An image from a convex mirror – always virtual, always
smaller than the object
Concave mirror – virtual image
if the object is closer than the
Virtual image
Enlarged & upright
Concave mirror – real image if the
object is farther than focus
Real image
reduced & inverted
Plane mirror
Concave mirror
Convex mirror
Curved Mirrors Application: TELESCOPES
Hubble space telescope
Tiny error in mirror, repaired in 1993
2.4 m mirror, too flat on one edge by 1 / 50th of the width of a single
human hair
Why ?
1. The pencil actually bends
when in contact with water.
2. It’s some kind of
interference effect.
3. It’s a magic trick.
When light enters a medium it slows down.
Now assume light hits a boundary (= interface)
between two media under an angle:
What happens ?
1. Some of it reflects off the interface.
2. Some gets transmitted, but how ?
Analogy: Car leaving road and entering mud
Because the right wheel slows down first, the
car rotates.
Light does the same thing when it crosses the
interface between two different media:
A light ray bends towards the normal when it
enters a transparent medium in which light travels
It bends away from the normal if it enters a
medium in which light travels faster.
Which way is faster for the light, 1 or 2 ?
Glass into air
Air into glass
How to explain the pencil in water ?
From more optically dense medium to less
optically dense medium:
What happens if the incident angle is increased?
It increases to some
maximum angle (critical
angle), at which
something strange
happens: the light does
not come out from the
more dense medium
If this angle is
exceeded: it is
completely reflected –
total internal
Happens after exceeding the
Optical fiber
Recall: Light rays can be focused by a curved mirror.
They can also be focused by using refraction:
Convex surface
Opposite case:
Concave surface
= Combinations of concave and convex surfaces,
utilizing refraction to manipulate light
Two convex surfaces =
biconvex lens
How does it work ?
Converging Lens
Diverging Lens
Biconcave Lens
Image formation
Lenses are used to form images of objects.
How they do that can be determined by “ray tracing”.
Optical Axis
Focal points
In this case light re-converges and projects a real image
that is inverted.
Another example:
What’s different ??
The light rays are not converging, the lens is not projecting
an image. Looking through the lens a virtual image appears.
= Magnifier
Is the image in a camera real or virtual ?
1. Real
2. Virtual
Lens formula
s− f
In a slide projector a slide is located 11 cm from a lens
with a 10 cm focal length. Where should the screen be
located to get a sharp image ?
s− f
p = (10 x 11)/(11-10) = 110/1 = 110 cm = 1.1 m
A magnifying glass has a focal length of 10 cm. You place
a coin at 5 cm from the lens. Where is the image?
s− f
p = (5 x 10)/(5-10) = 50/(-5) = -10 cm
What does that mean ??
1. There is no image
2. There is a real image
3. There is a virtual image
Example: Magnifier, s = 5 cm, p = -10 cm
M = -(-10)/5 = 10/5 = 2
Virtual upright image
Example : slide projector, s = 11 cm, p = 110 cm
M = -(110)/11 = -10
What does that mean ?
Real, inverted image
Telescope, microscope, etc.
The object – first lens – first image – second lens –
second image and so on.
The human eye
Refraction depends on wavelength of light (color)
In glass, shorter wavelengths travel slower than
longer wavelengths.
Rainbow and halo are results of
collective refraction and
Primary rainbow
Secondary rainbow
Halo effect
Blue sky and red sunsets – Scattering of
Where are we?
Historically – at the end of the 19th century
ƒ Mechanics is seemingly complete – Newton’s laws and
ƒ Electricity and magnetism – Maxwell’s equations and
applications, waves
ƒThe end? End of the XIX century – a time of decadence !!
What about relation between them – mechanics and E&M?
There are two approaches to this question – theoretical
and experimental
Since they were independent (historically) we will start
with the theoretical
Principle of relativity – Galileo – Newton:
1. All inertial systems of reference (those that are
moving at constant velocities) are equivalent – there
is no difference who is moving and who is at rest.
Example: if you are driving a car at a constant
velocity, you are moving with respect to trees, but
they are moving with respect to you.
2. The same laws of physics are valid in any inertial
frame of reference.
3. The rules of changing the frame of reference:
t = t'
time is absolute
d = d '−vrel t distance is corrected by the distance passed by the origin
v = v'−vrel velocity is corrected by the velocity of the origin
Example: You are in a train that is moving at 100 mph.
You define your frame – the frame of the train – nonprimed, the frame related to the ground - primed
You are walking along the car in the direction of motion
of the train at 3 mph
v = 100 mph
t = t'
time is absolute
d = d '−vrel t distance is corrected by the distance passed by the origin
v = v'−vrel velocity is corrected by the velocity of the origin
You start your watch at t=0, d=d’=0, then
d ' = d + vrel ⋅ t = v ⋅ t + vrel ⋅ t = 3t + 100t = (3 + 100)t = 103t
v' = v + vrel = 3 + 100 = 103
Time is absolute, but distance is relative
Theoretical problems:
If one tries to rewrite beautiful Maxwell equations in a
frame moving at a velocity v, i.e. study E&M in a moving
frame, using Galilean transformations, the Maxwell
equations become very very ugly.
This conspicuous ugliness suggests that there is only ONE
frame of reference – the absolute frame of reference
where they are not ugly.
Lorentz force (magnetic force) manifestly dependent on
the velocity is another problem. The inertial frames of
reference are good for mechanical phenomena, but not
for the electromagnetic.
Experimental reasons
ƒ Most physicists of the XIX century agreed that EM
waves light included propagate in a special medium
called ether. The ether was believed to be massless
and not interacting with matter.
ƒ To verify this hypothesis, the speed of ether was
measured in different directions by Michelson and
Morley with a surprising but consistently confirmed
result – the speed of light is the same in all directions.
ƒ Another, Trouton – Noble experiment also confirmed
that there is no “ether wind”.
ƒ Dilemma: we live in the “best of the worlds” where the
ether is at rest, elsewhere the physics (E&M) is ugly
or there is no ether at all and the Galilean
transformations are not valid for EM waves.
The dilemma
The problem can be stated in the following form
there are three statements
ƒ Galilean transformations of coordinates and
absolute time.
ƒ The speed of light is constant in all frames of
ƒ All laws of Nature are the same in all inertial
frames of reference.
ƒ These three statements contradict each other. One
has to be dropped to resolve the problem.
The special theory of relativity
ƒ Einstein’s solution – special theory of
relativity: two postulates, 1905:
ƒ All inertial frames of reference are
equally suitable for the description of
physical phenomena.
ƒ The speed of light in vacuum is the
same for all observers and is
independent of the motion of the
The toll: Lorentz transformations for time and coordinates
replace Galileo transformations
t '− 2 d '
time is relative
1 − rel2
d '−vrel t '
distance measured in a moving frame
1 − rel2
new formula for velocity addition
v' vrel
1− 2
v + vrel
v' =
1+ 2
Notice, that
become Galilean
if the ratio v/c is
small, v/c << 1, or
v << c
1. Relativity of simultaneity: If
two events occur at
different points in the rest
frame simultaneously – at
the same time, the time
interval between these
events in the moving frame
will not be zero, but
proportional to the distance
between these points
2. Time dilation: The time
interval appears to be longer
to the moving observer than
it does to the one at rest
with respect to the clock.
2. Time dilation continued: prime denotes
the frame which is moving with respect to
the clock.
∆t ' =
1− 2
Example: a lifetime of a muon (a particle) is
2.2x10-6s, how long will it live for a
stationary observer if the muon moves at
0.8c ? How far will it move before
∆t = 2.2 ⋅10
2.2 ⋅10
2.2 ⋅10
∆t ' =
= 3.67 ⋅10 −6 s
1 − 0.64
d ' = c ⋅ ∆t ' = 3 ⋅108 ⋅ 3.67 ⋅10 −6 = 1,100 m
3. Lorentz contraction: Found by a moving observer, the
length in the direction of motion is contracted:
L' = L ⋅ 1 − 2
v1 + v2
1+ 2
4. Velocity addition formula
Example: An observer
measures the speed of light
coming from the headlights of
my car.
Extreme case: c+c
v+c v+c v+c
v c+v
1+ 2 1+
2c 2c
cc 1 + 1 2
1+ 2
Dynamical consequences:
E0 = mc 2
Rest energy
mc 2
1− 2
1− 2
Total enrgy (rest and kinetic)
Linear momentum
1. Energy and mass are equivalent
2. Kinetic energy at v << c is equal to mv2/2
3. Energy – momentum conservation in collisions
4. Momentum always conserves!
As a result Einstein has pacified Maxwell Electricity
and magnetism with Newtonian mechanics.
1. No medium is needed for EM wave propagations
2. EM waves propagate in vacuum at speed of light,
same in all directions and frames.
3. EM waves carry both energy and momentum.
Experimental evidence:
1. Michelson – Morley experiment, 1887 – speed of light
is the same in all directions
2. Trouton – Noble experiment – no ether wind.