Graphs:
Average velocity: slope of the straight line joining the
initial and final position on the position -time graph.
(Instantaneous) velocity at a given point: slope of
the tangent line at given time on the position -time
graph.
Average Acceleration: slope of the straight line
joining the initial and final position on the velocity time graph
(Instantaneous) acceleration at a given point: slope
of the tangent line at given time on the velocity time graph
Displacement is the area under velocity – time graph
Change in velocity is the area under acceleration –
time graph
Air resistance provides a drag force to objects in free fall.
▪ The drag force increases as the speed of the falling object
increases resulting in decreasing downward acceleration
▪ When the drag force reaches the magnitude of the gravitational force,
the falling object will stop accelerating and fall at a constant velocity.
▪ This is called the terminal velocity/speed.
In vacuum
In air
Inertia is resistance an object has to a change of velocity
Mass is numerical measure of the inertia of a body (kg)
Weight is the gravitational force acting on an object . W = mg
Force is an influence on an object that causes the object to accelerate
• 1 N is the force that causes a 1-kg object to accelerate 1 m/s2.
Fnet (resultant force) is the vector sum of all forces acting on an object
Free Body Diagram is a sketch of a body and all forces acting on it.
Newton’s first law: An object continues in motion with constant speed in a straight line (constant velocity)
or stays at rest unless acted upon by a net external force.
Object is in translational equilibrium if
Newton’s second law:
F=
Fnet = 0, a = 0  no change in velocity
Δp
Δt
• Δp is the change in momentum produced by the net force F in time Δt.
• Δp = Δ(mv)
1. velocity changes, mass doesn’t change:
Δp = mΔv → F = ma
If a net force is acting on an object of mass m, object will acquire acceleration
Direction of acceleration is direction of the net force.
2. mass changes, velocity doesn’t change: Δp = vΔm → F =
∆m
v
∆t
Δm/Δt in (kg/s)
Newton’s third law: Whenever object A exerts force on object B, object B exerts an equal in magnitude
but opposite in direction force on object A
Normal/Reaction force (Fn or R) is the force which is preventing an object from falling through
the surface of another body .
That’s why normal force is always perpendicular (normal) to the surfaces in contact.
when you have to draw FREE BODY DIAGRAM (object and all forces acting on
it), there is a requirement to draw as many normal/reaction forces as there are
points of contacts .
For example if you have a 2-D car (with two wheels) you have to draw two
normal /reaction forces. Each on one wheel.
Table with two legs – the same thing.
Emu with two legs – the same thing
Friction force is the force that opposes slipping (relative motion ) between two
surfaces in contact; it acts parallel to surface in direction opposed to slipping. Friction
depends on type and roughness of surfaces and normal force.
vertical direction :
Horizontal direction:
F sin θ + Fn = mg
F cos θ – Ffr = ma
Fn
Fn
Ffr
Ffr
=
direction perpendicular to the incline:
Fnet = ma = 0
Fn = mg cos θ
Along the incline direction:
Fnet = ma
mg sin θ – Ffr = ma
Linear momentum is defined as the product of an object’s mass and its velocity:
p = mv
vector! (kg m/s)
Impulse is defined as the product of the net force acting on an object and time interval of action:
FΔt
vector! (Ns)
Impulse F∆t acting on an object will produce the change of its momentum Δp:
F∆t = ∆p
Δp = mv - mu
Ns = kg m/s
Achieving the same change in momentum over a longer time
requires smaller force, and over a shorter time requires greater force.
WHEN YOU TRY TO FIND CHANGE IN MOMENTUM
REMEMER TO LABEL VELOCITIES AS POSITIVE OR NEGATIVE
The impulse of a time-varying force is represented
by the area under the force-time graph.
Law of Conservation of Momentum: The total momentum of a system of interacting particles is
conserved - remains constant, provided there is no resultant external force. Such a system is called
an “isolated system”.
momentum of the system after collision = momentum of the system before
collision for isolated system (p = pi ) vector sum
REMEMBER TO DRAW A SKETCH OF THE MASSES AND VELOCITIES BEFORE AND AFTER COLLISION.
LABEL VELOCITIES AS POSITIVE OR NEGATIVE.
Elastic collision: both momentum and kinetic/mechanical energy are conserved. That means no
energy is converted into thermal energy
Inelastic collision: momentum is conserved but KE is not conserved.
Perfectly inelastic collision: the most of KE is converted into other forms of energy when objects
after collision stick together.
To find how much of KE is lost in the system, subtract KE of the system after collision from KE of the
system before the collision.
If explosion happens in an isolated system momentum is conserved but KE increases (input of
energy from a fuel or explosive material.)
Work is the product of the component of the force in the direction of displacement and
the magnitude of the displacement. (scalar)
W = Fd cos Ѳ
(Fd = F cos Ѳ)
(Joules)
Work done by a varying force F along the whole
distance travelled is the area under the graph FcosѲ
versus distance travelled.
Energy is the ability to do work. Work changes energy.
Potential energy is stored energy.
(Change in) Gravitational potential energy ∆EP = mg ∆h
Elastic potential energy = work is done by external force in stretching/compressing
the spring by extension x.
EPE = W = ½ kx2
Kinetic energy is the energy an object possesses due to motion EK = ½ mv2
Work done by applied/external force is converted into (changes) potential energy
(when net force is zero, so there is no acceleration).
Work – Kinetic energy relationship: work done by net force changes kinetic energy:
W = ∆KE = final EK – initial EK = ½ mv2 – ½ mu2
the work done by centripetal force is zero:
Wnet = 0 → Wnet = ∆KE → no change in KE → no change in speed;
centripetal force cannot change the speed, only direction.
Examples: gravitational force on the moon,
magnetic force on the moving charge.
Conservation of energy law: Energy cannot be created or destroyed;
it can only be changed from one form to another.
For the system that has only mechanical energy
(ME = potential energies + kinetic energy)
and there is no frictional force acting on it,
so no mechanical energy is converted into thermal energy,
mechanical energy is conserved
ME1 = ME2 = ME3 = ME4
mgh1 + ½ mv12 = mgh2 + ½ mv22 = • • • • • •
f friction cannot be neglected we have to take into account work
done by friction force which doesn’t belong to the object alone
but is shared with environment as thermal energy.
Friction converts part of kinetic energy of the object into thermal
energy.
Frictional force has dissipated energy:
ME1 – Ffr d = ME2
(Wfr = – Ffr d)
Power is the rate at which work is performed or the rate at which energy is
transmitted/converted.
𝑃=
∆𝐸
𝑡
=
𝑊
𝑡
scalar (1 W(Watt) = 1 J/ 1s )
Another way to calculate power
𝑃 = 𝐹𝑣
𝑊 𝐹𝑑
𝑑
𝑃=
=
=𝐹
𝑡
𝑡
𝑡
Efficiency is the ratio of how much work, energy or power we get out of a
system compared to how much is put in.
Wout
Eout
Pout
eff =
=
=
Win
Ein
Pin
Centripetal acceleration causes change in direction of velocity, but doesn’t
change speed. It is always directed toward the center of the circle.
v2
ac =
r
Centripetal force
mv 2
Fc = mac =
r
Period T: time required for one complete
revolution/circle
speed around circle of radius r:
2πR
v=
T
Macroscopic level: temperature gives indication of the degree of hotness or coldness
of a body, measured by thermometer
Thermal equilibrium occurs when all parts of the system are at the same
temperature. There is no exchange of thermal energy/please do not mention heat.
(This is how a thermometer works)
Thermal energy of a system = internal energy—the sum total of the potential energy
and kinetic energy of the particles making up the system.
Potential energy of the molecules arises from the forces (bonds/ because of
intermolecular forces) between them.
Kinetic energy of the molecules arises from the translational, rotational, ad vibrational
motion of the particles.
Microscopic level: (absolute) temperature directly proportional to the average kinetic
energy of the molecules of a substance: (KE)avg = 3 2 kT.
k is Boltzmann constant
Heat is the thermal energy that flows/is transferred from one body or system of higher
temperature to another of lower temperature.
Relative atomic mass is the mass of an atom in units of 1/12 of the mass
of a carbon-12 atom.
The mole is the amount of substance that contains the same number
of atoms/molecules as 0.012 kg of carbon-12.
Molar Mass is the mass of one mole of a substance (kg/mol).
1 mole of a gas at STP occupies 22.4 l (dm3) and contains
6.02 x 1023 molecules/mol.
Heat/Thermal Capacity is the amount of thermal energy needed to raise the
temperature of a substance/object by one degree Kelvin.
C =
Q
∆T
→
Q = C ΔT
unit: (C) = J K-1
Specific heat capacity is the quantity of thermal energy required to raise the
temperature of one kilogram of a substance by one degree Kelvin.
c =
𝑄
𝑚∆𝑇
→
Q = mc ΔT
unit: (c) = J kg-1 K-1
amount of thermal energy needed to increase the temperature of
m kg of a substance with specific heat capacity c by ΔT amount
of thermal energy released when the temperature of m kg of a
substance with specific heat capacity c decreases by ΔT
homogeneous substance: C = mc
Latent heat is the thermal energy that a substance/body absorbs or releases during a
phase change at constant temperature.
L = Qat const. temp.
unit: J
Specific latent heat is the thermal energy required for a unit mass of a substance to
undergo a phase change.
L=
Q
m
→
Q = mL
unit: (L) = J/kg
If electrical energy is converted into increase of
internal energy of the system, then:
Qadded = electrical energy = Pt = IVt = Qabosorbed
P – power, I – current, V – voltage, t - time
4 Phases (States) of Matter
solid, liquid, gas and plasma; ordinary matter – only three phases
Characteristic
Solid
Liquid
Gas
Volume and shape
definite volume and definite shape
definite volume but its shape can change – it takes the
shape of their containers.
neither definite volume
nor definite shape
Compressibility
Almost Incompressible
Very Slightly Compressible
Highly Compressible
Bonds =
intermolecular forces
characterized by high density and the
molecules are held in fixed position by
strong bonds. Molecules vibrate around a
mean (equilibrium) position.
density is lower and molecules are further apart without fixed
positions
Molecules experience little resistance to motion and move
freely about.
There are still strong forces between the molecules but they
are free to move around each other.
the forces between molecules are very
weak – molecules are essentially
independent of one another but they do
occasionally collide
Comparative Density
High
High
Low
Kinetic Energy
Vibrational
Vibrational, rotational,
some translational
Mostly translational, higher rotational and
vibrational
Potential Energy
High
Higher
Highest (ideal gas – zero)
Mean molecular
Separation
r0 ( size of the particle)
> r0
10 - 100 r0
Phase transition is the transformation of a thermodynamic
system from one phase to another.
Changes:
S→L
melting or fusion
L → S freezing or solidification
S → G sublimation
G → S deposition or desublimation
L → G vaporization
G → L condensation
includes boiling and evaporation
While melting, vibrational kinetic energy increases and
particles gain enough thermal energy to break from fixed
positions. Potential energy of system increases.
Melting point of a solid is the temperature at which it
changes state from solid to liquid. Once at the melting
point, any additional heat supplied does not increase the
temperature. Instead is used to overcome the forces
between the solid molecules increasing potential energy.
◌ At the melting point the solid and liquid phase exist in equilibrium.
While freezing, particles lose potential energy until thermal energy of the system is
unable to support distance between particles and is overcome by the attraction force
between them. Kinetic energy changes form from vibrational, rotational and part
translational to merely vibrational.
Potential energy decreases
(It is negative!!! = attraction: intermolecular forces become stronger).
While boiling, substance gains enough potential energy to break free from inter-particle
forces. Similar to evaporation, the only difference being that energy is supplied from
external source so there is no decrease in temperature.
While condensing, the energy changes are opposite to that of boiling.
The distinguishing characteristic of a phase transition is an abrupt change
in one or more physical properties, in particular the heat capacity, and the
strength of intermolecular forces.
During a phase change, the thermal energy added or released is
used to change (increase/decrease) the potential energy of the
particles to either overcome or succumb to the inter-molecular force
that pulls particles together. In the process, the average kinetic
energy will not change, so temperature will not change.
Evaporation is a change of phase from the liquid state to the gaseous state that
occurs at a temperature below the boiling point.
Evaporation causes cooling.
A liquid at a particular temperature has a range of particle energies, so at any instant, a
small fraction of the particles will have KE considerably greater than the average value. If
these particles are near the surface of the liquid, they will have enough KE to overcome
the attractive forces of the neighbouring particles and escape from the liquid as a gas. The
escape of the higher-energy particles will lower the average kinetic energy and thus lower
the temperature.
The rate of evaporation is the number of molecules escaping the liquid per second.
Evaporation can be increased by
• increasing temperature/more particles have a higher KE
• Increasing surface area/more particles closer to the surface
• Increasing air flow above the surface (gives the particles somewhere
to go to)/ decreasing the pressure of the air above liquid
Kinetic Model of an Ideal Gas
PV = NkT=nRT (for Jerry)
P – pressure,
V – volume,
N – number of particles,
k – Boltzmann constant,
T - temperature
Gas pressure is the force gas molecules exert due to
their collisions (with a wall – imaginary or real), per unit area.
𝐹
P=
𝐴
Assumptions of the kinetic model of an ideal gas.
• Gases consist of tiny hard spheres/particles called atoms or molecules.
• The total number of molecules in any sample of a gas is extremely large.
• The molecules are in constant random motion.
• The range of the intermolecular forces is small compared to the average separation
of the molecules
• The size of the particles is relatively small compared with the distance between them
• No forces act between particles except when they collide, and hence particles
move in straight lines.
• Between collisions the molecules obey Newton’s Laws of motion.
• Collisions of short duration occur between molecules and the walls of the container
and the collisions are perfectly elastic
(no loss of kinetic energy).
Temperature is a measure of the average
random kinetic energy of the molecules
of an ideal gas.
KEavg =
3
2
kT
Macroscopic behavior of an ideal gas in terms of a molecular model.
• Increase in temperature is equivalent of an increase in average kinetic energy
(greater average speed). This leads to more collisions and collisions with greater
impulse. Thus resulting in higher pressure.
• Decrease in volume results in a smaller space for gas particles to move, and thus a
greater frequency of collisions. This results in an increase in pressure. Also, depending
on the speed at which the volume decreases, particles colliding with the moving
container wall may bounce back at greater speeds. This would lead to an increase in
average kinetic energy and thus an increase in temperature.
• An increase in volume would have an opposite effect.
Application of the "Kinetic Molecular Theory" to the Gas Laws
Microscopic justification of the laws
Pressure Law (Gay-Lussac’s Law)
Effect of a pressure increase at a constant volume
Macroscopically:
at constant volume the pressure of a gas is proportional to its temperature:
PV = NkT → P = (const) T
Microscopically:
∎ As T increases, KE of molecules increase
∎ That implies greater change in momentum when they hit the wall of the container
∎ Thus microscopic force from each molecule on the wall will be greater
∎ As the molecules are moving faster on average they will hit the wall more often
∎ The total force will increase, therefore the pressure will increase
The Charles’s law
Effect of a volume increase at a constant pressure
Macroscopically:
at constant pressure, volume of a gas is proportional to its temperature:
PV = NkT → V = (const) T
Microscopically:
∎ An increase in temperature means an increase in the average kinetic energy
of the gas molecules, thus an increase in speed
∎ There will be more collisions per unit time, furthermore, the momentum
of each collision increases (molecules strike the wall harder)
∎ Therefore, there would be an increase in pressure
∎ If we allow the volume to change to maintain constant pressure,
the volume will increase with increasing temperature
Boyle-Marriott’s Law
Effect of a pressure decrease at a constant temperature
Macroscopically:
at constant temperature the pressure of a gas is inversely proportional to its volume:
PV = NkT → P = (const)/V
Microscopically:
∎ Constant T means that the average KE of the gas molecules remains constant
∎ This means that the average speed of the molecules, v, remains unchanged
∎ If the average speed remains unchanged, but the volume increases, this means
that there will be fewer collisions with the container walls over a given time
∎ Therefore, the pressure will decrease
OSCILLATIONS and WAVES
What do all of them have in common? Oscillatory motion, but not just any.
IT IS SIMPLE HARMONIC MOTION
Graphical treatment and math
To analyse these oscillations further, we can plot graphs for
these motions.
 You can plot a displacement – time graph by attaching a pen
to a pendulum and moving paper beneath it at a constant
velocity, or by shining the light on and oscillating spring.

the shadow should
look like this graph
The shape of this displacement – time graph is cosine curve.
The amplitude is x0 is initial displacement
displacement x = x0 cos ωt
where angular frequency is: ω =
2πf = 2π/T
Simple Harmonic Motion is periodic motion in which the
acceleration/ restoring force is proportional to and in opposite
direction of the displacement.
a = − ω2x.
spring: a = F/m = – kx/ m = – (k/m)x = = − ω2x
ω2 = k/m
General equation for the position of a particle undergoing simple
harmonic motion: x = x0 cos ωt.
x0 – amplitude, f – frequency f = 1/T
T – period for full oscillation
angular frequency  = 2 f (how many full circles 2 per second) = 2/T
Dependence on time
x = x0 cos ωt.
v = dx/dt = –  x0 sin ωt = – v0 sin ωt
a = dv/dt = – 2x0 cos ωt = – a0 cos t
a = – 2x
x0, v0, a0 positive maximum values
Data booklet
Dependence on position
v = = –  x0 sin ωt = ±  x0
1 − cos2 ωt = ±  x0
x2
1− 2
x0
2
v = ±  x2
−
x
0
2)
EK = ½ m v2 = ½ m2 ( x2
−
x
0
EK for x = 0 → EK(max) = ½ m2 x2
0
Data booklet
EK(max) = EP(max) = ETotal
potential energy at any moment = total energy – KE
EP = ½ m2 x2
Energy (total) of SHM is proportional to amplitude2 ( x2
0)
Period does not depend on amplitude!!!!!!
Damping: Due to the presence of resistance/friction forces on oscillations in the
opposite direction to the direction of motion of the oscillating particle
Amplitude of oscillations decreases
Friction force is a dissipative force.
"to damp" is to decrease the amplitude of an oscillation.
Decreasing the amplitude doesn’t change period.
Light/under damping: The decay in amplitude is
relatively slow and the oscillator will make quite a few
oscillations before finally coming to rest.
Critical/heavy damping: occurs if the resistive force is
so big that the system returns to its equilibrium position
without passing through it. The mass comes to rest at
its equilibrium position without oscillating. The friction
forces acting are such that they prevent oscillations.
Over-damping: the system returns to
equilibrium without oscillations, but much
slower than in the case of critical damping.
Natural frequency is the frequency an object will vibrate with after
an external disturbance.
Forced oscillations: when an external periodic force with frequency fD is applied on a free
system with a natural frequency f0 , the system may respond by switching to oscillations with a
frequency equal to the driving frequency fD.
Variation of the amplitude of vibrations of an object
subjected to the forced frequency close to its natural
frequency of vibration.
Qualitative description of the factors that affect the
frequency response and sharpness of the curve.
▪ For a small degree of damping, the peak of the curve occurs at the
natural frequency of the system.
▪ The lower the degree of damping, the higher and narrower the curve.
▪ At very heavy damping, the amplitude is essentially constant.
Resonance is increase in amplitude of oscillation of a system exposed to a periodic
driving force with a frequency equal to the natural frequency of the system.
WAVES
▪ Useful: microwave oven, radio. .
▪ Harmful: bridges, aero plane wings, internal organs in the case of heavy machinery .
When a wave (energy) propagates through a medium, oscillations of the particles of
the medium are simple harmonic.
Progressive waves transfer energy through a distortion that travels away from the
source of distortion. There is no net transfer of medium.
▪ Transverse waves are waves in which the particles of the medium oscillate
perpendicular to the direction in which the wave is traveling.
◌ EM waves: light, radio waves, microwaves:
need no medium, electric and magnetic field oscillate perpendicular
to each other and to the direction of wave propagation.
◌ Earthquake secondary waves, waves on a stringed musical instrument,
waves on the rope.
◌ Transverse wave can not propagate in a gas ( and actually pure transverse
can not propagate in liquid either)
▪ Longitudinal waves are waves in which the particles of the medium vibrate parallel to the
direction in which the wave is traveling.
◌ Sound waves in any medium, shock waves in an earthquake,
compression waves along a spring
A wavefront is set of points having the same phase/displacement.
A ray is an arrow drawn on a diagram to show the direction of propagation of waves.
It is always at right angles to the wavefront.
Energy of a wave of amplitude A is proportional
: E ∞ A2
to the amplitude2
Although the speed of a wave depends only on the
medium, there is a relationship between wavelngth 𝜆 ,
frequency f (period T) and the speed

v= = f
T
Waves that need medium to travel through are called mechanical waves.
Electromagnetic wave is made up of changing electric and magnetic fields perpendicular
to each other and to the propagation of the wave.
They travel through vacuum with the SAME SPEED!
speed of light c ≈ 3 x 108 m/s
Index of refraction n of a medium is the ratio of the speed of light in a
vacuum, c, and the speed of light, v, in that medium:
n=
c
v
As the speed of light in air is almost equal to c, nair ≈ 1
Refraction: When a wave passes from one medium to another, its speed changes
resulting in a change in direction of the refracted wave
Snell’s law states that for a given pair of
media, the ratio of the sines of the angles of
incidence and refraction is equal to the ratio
of velocities in the two media
sinθ1 n2
v1
λ1 𝑓
λ1
=
=
=
=
v2
λ2 𝑓
λ2
sinθ2 n1
v = 𝜆 f ; frequency doesn’t change in
refraction, so in the medium with
smaller wave speed, wavelength will
be smaller.
The refracted ray is
refracted more in the
medium with greater n,
slower speed of light
Chromatic dispersion is phenomenon in which the index of refraction depends on
wavelength/frequency, so the speed of light through a material varies slightly with the
frequency of the light and each λ is refracted at a slightly different angle.
The longer λ, the smaller index of refraction.
nred < nblue , red light is refracted less than blue light
Dispersion is the phenomenon which gives separation of
colours in prism/rainbow and undesirable chromatic
aberration in lenses.
Diffraction is the spreading of a wave into a region behind an
obstruction (into a region of geometrical shadow).
Diffraction effects are more obvious when wavelength of the wave is
similar in size to aperture/obstacle or bigger.
remember: big λ (compared to aperture or obstacle), big diffraction effects
Interference is the addition (superposition) of two or more waves overlapping that results
in a new wave pattern.
Principle of superposition: When two or more waves overlap, the resultant
displacement at any point and at any instant is the vector sum of the displacements of the
individual waves at that point: y = y1 + y2
PD = path difference is the difference in distances traveled by waves
from two sources to a point P: PD = d2 – d1
Two coherent waves traveling along two
different paths to the same point will:
interfere constructively if the difference in distance traveled
is equal to a whole number of wavelengths:
PD = n λ
n = 0, ± 1, ± 2, ± 3, …
interfere destructively if the difference in distance traveled
is equal to a half number of wavelengths:
PD = (n + ½ ) λ
n = 0, ± 1, ± 2, ± 3, …
▪ charge is quantized and conserved
▪ Coulomb’s law: Force between TWO POINT
charges q1 and q2 at distance r :
F=k
q1q2
r2
▪ Electric field is the force per unit positive charge . 𝐸 =
𝐹
𝑞
(E) = NC-1
Direction of the field is the direction of the force on
a positive test charge placed at that point.
▪ Electric field lines show direction of the force on positive charge
▪ always point away from + charges, towards – charges…
▪ positive charge placed in the field would move along field line,
negative charge would move opposite
Essentially, potential energy is capacity for doing work which
arises from position or configuration.
In the electrical case, a charge placed in electric field will have electric potential
energy. If left on its own it will start accelerating due to electric force acting on it.
We say it has electric potential energy which will be converted into kinetic energy.
On the other hand if external force pushes the charge against electric field, the
work done by that force will be stored as potential energy in the charge.
Potential energy difference between two points (∆𝑼) is equal to the work done on a
charge in order to move it from one point to the other.
Potential difference between two points (∆𝑽) is equal to the work done per unit positive
charge in order to move it from one point to the other.
V 
W U

q
q
1 𝑉𝑜𝑙𝑡 =
1 𝐽𝑜𝑢𝑙𝑒
1 𝐶𝑜𝑢𝑙𝑜𝑚𝑏
Capacitor: uniform electric field (the one that has constant magnitude and
direction) is generated between two oppositely charged parallel plates. Edge
effect is minimized when the length is long compared to their separation.
▪ ① Work done by external force on charge against an electric field between two
points is stored in the charge as the change in electrical potential energy, U.
W = ∆U
Fext d = q ∆V
qEd = q ∆V ⇒ 𝑬 =
∆𝑽
𝒅
⇒ (E) = NC-1 = Vm-1
▪ ② Work done by electric force
If a charge, q, is moved on its own through a potential difference, ∆V,
the work done on it by electric force is equal to the decrease in its electric
potential energy which is converted into kinetic energy:
W = ∆U = Fd = q Ed = q ∆V = ½ mv2
▪ Positive charge accelerates from higher to lower potential (from positive to negative).
▪ Negative charge accelerates from lower to higher potential. (from negative to positive)
▪ 1 electron-Volt (eV) is the amount of energy/work it takes to move an electron through a
potential difference of 1volt.
1 eV = W = ∆U = q ∆V = (1.6x10-19 C) (1V)
1 eV = 1.6x10-19 J
▪ Current is the rate at which charge flows past a given cross-section
𝐼 =
𝑄
𝑡
1A =
1C
1s
▪ Electrical resistance , R is the ratio of the potential difference across the resistor/conductor
to the current that flows through it.
𝑅=
𝑉
𝐼
1 (ohm) = 1V/1A
▪ The resistance of a conducting wire depends on four main factors:
• length, L
• cross-sectional area, A
• material/resistivity, ρ
• temperature, T:
• if temperature is kept constant: 𝑅 = 𝜌
𝐿
𝐴
▪ OHM’S LAW: Current through resistor is proportional to potential difference across the resistor
if the temperature of a resistor is constant (the resistance of a conductor is constant).
or:
𝐼 =
𝑉
𝑅
I – current through resistor,
V – potential difference across R
Ohmic and Non-Ohmic conductors
How does the current varies with potential difference for some typical devices?
diode
current
filament lamp
current
current
metal at const. temp.
potential
difference
𝐼
1
=
𝑖𝑠 𝑐𝑜𝑛𝑠𝑡. → 𝑅 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡.
𝑉
𝑅
potential
difference
potential
difference
devices are non-ohmic if
resistance changes
Devices for which current through them is directly proportional to the potential difference across device
are said to be ‘ohmic devices’ or ‘ohmic conductors’ or simply resistors. There are very few devices that
are trully ohmic. However, many useful devices obey the law at least over a reasonable range.
▪ When a current is flowing through a load such as a resistor, it dissipates energy in it. In collision with
lattice ions electrons’ kinetic energy is transferred to the ions, and as a result the amplitude of
vibrations of the ions increases and therefore the temperature of the device increases.
KE is transferred to thermal energy.
▪ Electric power is the rate at which energy is supplied to or used by a device.
▪ Electric power is the rate at which electric energy is converted into another form such as
mechanical energy, thermal energy, or light.
Power dissipated in a resistor/circuit:
W = qV → P = qV/t
P=IV
and I = q/t, so P = I V
▪ Electromotive force (V), (E or EMF or ε) is the work done per unit charge in moving charge
completely around the circuit.
electric potential energy
electric potential 
∆V = ∆U/Q
charge
▪ Electromotive force, (E or EMF or ε) is the power supplied to the circuit per unit current
power
P = I V ⟹ V = P/ I
electric potential =
current
▪ Energy supplied by the source = ε Q
(coming from U = Q ∆V)
Total energy supplied by the source = energy used in the resistors: ε Q = V1 Q + V2 Q + …..
divide it by time ⟹ ε I = V1 I + V2 I + …..
ex: the cell supplies 8.0 kJ of energy when 4 kC of charge moves completely across the circuit with
constant current. Find ε:
ε = energy/charge = 2 V
▪ Power supplied by the source will be dissipated in the circuit:
Power of the source = sum of the powers across the resistors: Pout = Σ Pi
⟹ ε I = V1 I + V2 I + …..
Resistors in Series: connected in such a way that all components have
the same current through them.
Burning out of one of the lamp filaments in series or simply opening
the switch causes a break.
Equivalent resistance is greater that the greatest resistance in series.
𝑅𝑒𝑞 = 𝑅1 + 𝑅2 + ⋯
Resistors in Parallel: are connected to the same two points of an
electric circuit, so all resistors in parallel have the same potential
difference across them.
The current flowing into the point of splitting is equal to the sum of
the currents flowing out at that point: I = I1 + I2 + …
A break in any one path does not interrupt the flow of charge in the
other paths. Each device operates independently of the other devices.
The greater resistance, the smaller current. Equivalent resistance is
smaller than the smallest resistance in series.
1
1
1
=
+
+⋯
𝑅𝑒𝑞 𝑅1 𝑅2
Internal resistance, r: some of the power/energy delivered by a cell is used/dissipated
in driving the current though the cell itself
Terminal voltage (the actual voltage delivered to the circuit): V = ε – Ir
𝜀
𝜀
𝑉
I=
=
=
𝑅𝑒𝑞 𝑅+𝑟
𝑅
To measure the current, we use an AMMETER.
• Ammeter is in SERIES with resistor R in order that whatever current passes through the resistor also passes
through the ammeter. Ram << R, so it doesn’t change the current being measured.
• would ideally have no resistance with no potential difference across it so no energy would be dissipated in it.
To measure the potential difference across resistor, we use a VOLTMETER.
• Voltmeter is in PARALLEL with the resistor we are measuring. Rvm >> R so that it takes very little current
from the device whose potential difference is being measured.
• an ideal voltmeter would have infinite resistance with no current passing through it and no energy
would be dissipated in it.
A photoresistor or light dependent resistor (LDR) is a resistor whose resistance decreases with
increasing incident light intensity; in other words, it exhibits photoconductivity
• Potentiometer – variable resistor
COULOMB’S/ NEWTON’S LAW
Electric (Coulomb’s law)/gravitational force (Newton’s law of
gravitation) between two POINT charges/masses is proportional
to the product of two charges/masses and
inversely proportional to the distance between them squared.
Both laws can be applied to the objects that are spherically
symmetrical (do not look like potatoes).
In that case we consider that the charge/mass is concentrated
at the center.
Electric /gravitational force is a vector. If two or more point
charges/masses act on some charge/mass, the net force on that
charge/mass is a vector sum of all individual forces acting on
that charge/mass.
The electric/gravitational field at point P is defined as the force per unit charge/mass placed at that point.
F
=E
q
F
= g
m
𝑀
g = G 𝑅2
M – mass of the planet
R – radius of the planet
𝑀
gEarth = G 2 = 9.80 m s-2
𝑅
Direction of electric field due to charge Q at
some point P is equal to the direction of the
force on a positive charge placed at that point.
Electric force acting on a charge q placed there is:
𝐹=𝑞𝐸
Direction of gravitational field due to mass M
at some point P is toward mass M.
Gravitational force acting on a mass m placed there is:
𝐹 =𝑚𝑔
commonly called “weight of mass m”
▪ Uniform electric field, E
▪ Uniform gravitational field, g
If a positive charge q is released at point A,
force F = qE will accelerate it in the direction of
the field toward point B.
If a mass q is released at point A, force F = mg will
accelerate it in the direction of the field toward point B.
Work done on charge by force F along
displacement d is converted into kinetic energy.
W = Fd = qEd = ½ mv2
(remember: const E, const. F so W = Fd)
Work done on mass by force F along displacement d is
converted into kinetic energy.
W = Fd = mgd = ½ mv2
(remember: const E, const. F so W = Fd)
A magnetic field is a vector field that permeates space and which can exert a magnetic
force on moving electric charges and on magnetic dipoles.
We define the magnitude of the magnetic field by measuring the magnetic force on a
moving charge q:
B=
Fmag
qv
N∙s
1 T(Tesla) = C∙m
Direction at any location is the direction in which the north pole of the
compass needle points at that location.
Outside magnet: N → S
Inside magnet: S → N
(always closed loops)
① An electric charge experiences a magnetic force when moving in a magnetic field.
Magnetic force acting on a charge q
in a magnetic Field B: F = qvB sinq
Magnetic force on a wire carrying current I
in a magnetic field B: F = I LB sinq
q = charge [C]
v = velocity [m/s]
B = magnetic field [T]
𝜃 = angle between v and B
I = current [A]
L = length [m]
B = magnetic field [T]
𝜃 = angle between I and B
R-H-R 1: The direction of the magnetic force on a charge/current is given by the right-hand rule 1:
Outstretch fingers in the direction of v (or current I).
Curl fingers as if rotating vector v (I ) into vector B.
Magnetic force on a positive charge (or I) is in
the direction of the thumb.
Magnetic force on a negative charge
points in opposite direction.
Charge q in elec. field E and mag. field B
The electric force: Felec = Eq
● is always parallel to the direction of the
electric field.
● acts on a charged particle independent of the
particle’s velocity (even at rest).
● does the work when moving charge.
The work, W = Fel d cos θ1, is converted into
kinetic energy which is, in the case of conductors,
transferred to thermal energy through collisions
with the lattice ions causing increased amplitude
of vibrations seen as temperature rise
The magnetic force: Fmag = qvB sin
● is always perpendicular to the direction
of the magnetic field
● acts on a charged particle only when the particle
is in motion and only if v and B do not point in
the same or opposite direction
(sin 00 = sin 1800 = 0).
● Force is perpendicular to the direction
of the motion, so the work done by
magnetic force is zero.
W = Fmag d cos θ1 = 0 (cos 900 = 0).
W = ΔKE = 0
Hence change in kinetic energy of the charge is 0,
and that means that mag. force cannot change
the speed of the charge. Magnetic force can only
change direction of the velocity – therefore it acts
as centripetal force.
θ1 is angle between F and direction of motion
Examples of the Lorentz Force
1) the trajectory of a charged particle in a uniform magnetic field and
2) the force on a current-carrying conductor.
1) The trajectory of a charge q in a uniform magnetic field B
● Force is perpendicular to B,v
● Magnetic force does no work! (W = F d cos θ1 = 0 )
● Speed is constant (W = Δ KE = 0 )
● Circular motion
Charged particle in a magnetic field when v  B:
𝜃 𝑖𝑠 900 → sin 900 = 1
In the case the charge q is subject to the uniform field B, centripetal force Fc
is magnetic force forcing the charge to move in a circle:
Positive charge q in magnetic field B
𝐹𝑐 = 𝐹𝑚𝑎𝑔
𝑣2
𝑚
= 𝑞𝑣𝐵
𝑅
𝑅=
𝑚𝑣
𝑞𝐵
● massive or fast charges – large circles
● large charges and/or large B – small circles
B = magnetic field [T]
R =is the radius of the path
Fma is magnetic force on the charge directed toward
the centre of the circular path
m = mass [kg]
v = velocity [m/s]
q = charge [C]
②A moving charge produces a magnetic field.
R-H-R 2: The direction of the magnetic field produced by electric current is given
by the right-hand rule 2:
If a wire is grasped in the right hand with the thumb in the direction of current flow,
the fingers will curl in the direction of the magnetic field.
Magnetic field B around a wire with current I
B=
𝜇0 𝐼
2𝜋 𝑟
𝜇0 = the permeability of free space 4p×10-7 T·m/A
I = current [A]
r = distance from the center of the conductor
n = N/L number of turns of wire per unit length
Magnetic Field B Inside of a Solenoid
B = 𝜇0 n I
The magnetic field is concentrated into a nearly
uniform field in the centre of a long solenoid.
The field outside is weak and diverging
Currents in same direction attract!
Currents in opposite direction repel!
One Ampere is defined as that current flowing in each of two infinitely-long parallel
wires of negligible cross-sectional area separated by a distance of one metre in a
vacuum that results in a force of exactly 2 x 10-7 N per metre of length of each wire.
What is the direction of the force on the top wire,
due to the two below?
1) Left 2) Right 3) Up 4) Down 5) Zero
What is the direction of the force on
the midlle wire, due to the two
others?
I
1) Left
I
2) Right 3) Up
4) Down 5) zero5) Zero
I
What is the direction of the force on
the left, due to the two others?
I
1) Left
I
I
2) Right 3) Up
4) Down 5) Zero
Other way: 1. find magnetic field due the other two and
then use RHR1
What is the direction of the force on
the midlle wire, due to the two
others?
I
1) Left
2I
2) Right 3) Up
4) Down 5) Zero
3I
What is the direction of the force on
the midlle wire, due to the two
others?
I
1) Left
I
I
2) Right 3) Up
4) Down 5) Zero
Electric and Magnetic Field
Source:
Electric
Charges
Magnetic
Moving Charges
Act on:
Charges
Moving Charges
Magnitude:
F=qE
F = q v B sin θ
Direction:
Parallel to E
Perpendicular to v,B
Direction:
Opposites
Charges Attract
Currents Repel
Rutherford/Planetary/Nuclear Model of atom: the atom
consists of a very tiny but very massive positive nucleus,
surrounded by electrons that orbit the nucleus as result of
electrostatic attraction between the electrons and the nucleus
evidence: Geiger and Marsden’s experiment:
• Alpha particles bombarded at a sheet of gold foil mostly passed through—atoms mostly
consist of empty space.
• Particles that were deflected bounced straight back from the foil—the huge deflection of the
alpha particles must have been caused by electrostatic repulsions between the positive
alpha particles and a dense, positive nucleus.
limitations/problems:
• According to Maxwell, any accelerating charge will generate an EM wave
• This EM wave would be a release of energy and give off light at all sorts of
wavelengths.
• electron releasing energy should slow down and eventually spiral into the nucleus .
Bohr’s Model:
• Electrons orbit at specific energy levels (“discrete states”), called “stationary states.”
– we say energy of is quantized
• Electrons in these stationary states do not emit EM waves as they orbit.
• Photon must be first absorbed in order to be emitted. Absorbed photon has the energy equal
to the difference between excited and ground state.
• Photon is emitted when an electron jumps from excited state to a lower energy state.
Energy of that photon is equal to the energy difference between two states. Ephoton = ΔE,
and frequency is given by Einstein’s relationship between energy and frequency of a photon:
E = hf
h = Planck's constant = 6.627x10-34 Js
Modern approach - Schrodinger wave function:
• energy level model - give possible energies of electrons and probability to find
electrons somewhere is given by wave function.
Continuous Spectrum
•
•
•
•
(without prism white light)
a spectrum having all wavelengths over a comparatively wide range
All possible frequencies of EM waves are present.
Generally, solids, liquids, or high pressured (dense) gases emit a continuous spectrum when heated.
Generally, strong interaction between molecules (spreading energy band)
Discrete/ Line spectrum - Evidence of electron energy levels
Pattern of distinct lines of color, corresponding to particular wavelengths.
Generally, weak or no interaction between molecules (no spreading of energy band)
• Emission Spectrum
• Set of frequencies of the electromagnetic waves emitted by atoms of a particular element.
• A hot, low-density / low pressure gas (gas in the atomic state) produces
an emission-line spectrum – energy only at specific λ.
• Absorption Spectrum
• Pattern of dark lines against a continuous spectrum background that results from
the absorption of selected frequencies by an atom or molecule.
• An absorption spectrum occurs when light passes through a cold, dilute gas and atoms in the
gas absorb at characteristic frequencies; since the re-emitted light is unlikely to be emitted
in the same direction as the absorbed photon, this gives rise to dark lines (absence of light)
in the spectrum.
Nucleon: a proton or neutron.
Nuclide: A particular combination of protons and neutrons that form a nucleus.
It is used to distinguish isotopes among nuclei.
A
Z
X
•
•
•
•
X is chemical symbol of the element
Z is the atomic number = number of electrons or protons
A is the nucleon (mass) number = number of neutrons + protons
A – Z = number of neutrons
Isotopes: Nuclides contains the same number of protons but different number of neutrons.
Isotopes are evidence for the existence of neutrons
Repulsive electromagnetic forces between the protons would cause the nucleus to disintegrate
if it were the only force.
Strong nuclear force is an attractive force, which exists between all nucleons to hold them
together. It is effective only over a very short range.
Weak nuclear force exists only in the nucleus and is responsible for the disintegration of
a neutron into a proton and an electron in beta decay.
Nuclear Stability: depends on the neutron-proton ratio
Nuclei are held together by a strong nuclear force, which counteracts the repulsive force among protons
contained within it. As long as the attractive nuclear forces between all nucleons win over the repulsive
Coulomb forces between the protons the nucleus is stable. It happens as long as the number of protons is
not too high. Atomic nuclei are stable subject to the condition that they contain an adequate number of
neutrons, in order to "dilute" the concentration of positive charges
♦ Small nuclei- tend to have equal number of neutrons and protons
♦ Large nuclei- tend to have more neutrons to counterbalance repulsive Coulomb force.
238
The most massive isotope found in nature 92 U is uranium isotope
For more massive nuclei strong nuclear force can’t overcome electric repulsion.
Radioactive Decay
Spontaneous decay of unstable nuclei.
• process in which unstable nucleus loses energy by emitting “radiation” in form of particles or EM waves,
resulting in transformation of parent nuclide into daughter nuclide.
• three common radiations - alpha, beta, gamma
• they differ in charge, ionization and penetration power.
Alpha decay:
• nucleus ejects an α particle, the atomic number is decreased by two and the atomic mass is decreased by four
• charge is + 2e
• the most ionizing and therefore the least penetrating (a few cm of air)
• Governed by strong nuclear force = α decay occurs primarily among heavy elements because the nucleus has too
many protons which cause excessive repulsion. In an attempt to reduce the repulsion, a helium nucleus is
emitted. Mass of parent > mass of daughter + mass of alpha
• difference = kinetic energy
Beta decay
• nucleus spontaneously emits beta particle and an antineutrino
• the most common decay occurs when the neutron to proton ratio is too great in the nucleus and
causes instability
• In β− decay, the weak interaction converts a neutron into a proton while emitting an electron and
an anti-neutrino:
1
1
−
1
0
+ ν 10n → 1p + −10e β− + ν
0n → 1p + −1e β
• charge is – e
• medium ionizing and therefore medium penetrating (a few mm of metal)
Gamma Decay
• EM waves (high-energy photons) are emitted from a nucleus in an excited state dropping to a
lower energy state (more stable)
• charge is 0
• no ionizing and therefore highly penetrating (a few cm of lead)
Biological effects of ionizing radiation
• Prompt effects: effects, including radiation sickness and radiation burns, seen immediately
after large doses of radiation delivered over short periods of time.
• Delayed effects: effects such as cataract formation and cancer induction that may appear months
or years after a radiation exposure
Radioactive Decay is a random process on the atomic level, in that it is impossible to predict when a
particular atom will decay, but given a large number of similar atoms, the decay rate, on average, is
predictable. The rate of decay decreases exponentially with time.
Half – life T 1 /2 is the interval of time required for one-half of the atomic nuclei of a radioactive
sample to decay.
Activity (becquerel - Bq) of a radioactive sample is the average number of disintegrations per
second.
Nuclear Reactions – A reaction that occurs whenever the number of protons or neutrons changes.
Nuclear reactions include natural and artificial transmutation, fission, and fusion.
Transmutation – Change of one element into another.
In natural transmutations the nucleus decays spontaneously. There is only one nucleus that undergoes
the transformation.
Artificial transmutation is induced by the bombardment of the nucleus by high-energy particles
(Uranium atoms bombarded with neutrons to start fission reaction.)
Unified Atomic Mass Unit (u) is 1/12 of the mass of one atom of carbon-12 atom (6p+6n+6e)
• 1 u = 1.66056655 x 10-27 kg
• 1 u = 931.5 MeV c-2
due to relationship E = mc2
• 1 u of mass converts into 931.5 MeV
Mass Defect (∆m) is the difference between the total mass of all nucleons in the nucleus
and the mass of the nucleus itself
∆m = Zmp + Nmn - Mnucleus
(can be calculated in kg or u) . Equivalent to binding energy.
Binding Energy is the work/energy required to completely separate the nucleons of a nucleus
/ energy released when nucleons form a nucleus.
nuclear binding energy is actually energy that corresponds to mass defect
1. BE in MeV:
• find mass defect in u and multiply it by 931.5 MeV
2. ME in J:
• BE = ∆m c2 (c = 3x108 m/s, ∆m = mass defect in kg)
In order to balance nuclear reaction the total mass/energy and total charge number of the
reactants has to equal the total mass/energy and total charge number of the products.
Energy released/required in a nuclear reaction/artificial transmutation
Nuclear reactions A + B → C + D can either release energy or requires energy input.
• release energy: Energy will be released in nuclear reaction if Δm = LHS – RHS > 0
The total amount of energy released will be in the form of kinetic energy of products.
If there was initial kinetic energy, that will be added up to released energy.
energy released in nuclear reaction is found the same way as binding energy:
first find mass difference and then equivalent energy
Δm in u, E = (Δm) x 931.5 (MeV)
or
Δm in kg, E = (Δm) c2 (J)
• energy input: if Δm = LHS – RHS < 0, reaction cannot be spontaneous.
For example, some nuclei will decay only if energy is supplied to it - collision with fast
moving α particle:
α particle must have enough KE to make up for imbalance in masses, and to provide
for kinetic energy of products.
Energy released in a decay - conservation of total energy (energy + mass). as always
226
88
Ra 
222
86
Rn 
4
2

M > m1 + m2 , but
total energy on the left = total energy on the right
Mc2 = m1 c2 + m2 c2 + KE1 + KE2
• spontaneous decay: M > m1 + m2 → binding energy of the decaying nucleus < binding energies
of the product nuclei.
This is why radioactive decay happens with heavy elements lying to the right of maximum
in the binding energy curve.
Energy released is in the form of kinetic energy of the products.
Binding energy per nucleon: the work required to remove one nucleon from the nucleus;
roughly the binding energy divided by the total number of nucleons in nucleus.
The binding energy of a nucleus is a measure of how stable nucleus is.
Greater mass defect – higher binding energy – greater stability.
Most nuclei have a binding energy per nucleon of approximately 8 MeV.
Nuclear fission: process in which a large nucleus (A>200)
splits up into two smaller nuclei, generally accompanied
by the release of one or more neutrons and energy (as
gamma rays and as kinetic energy of the fragments).
Large amounts of energy produced, can be selfsustaining due to chain reactions. The total BE would
increase which means that the daughters are more
stable than parent. Spontaneous fission is very rare.
Nuclear fusion: joining of two small nuclei into a bigger
one, releasing great amounts of energy in the process.
High temperatures are required to provide sufficient
kinetic energy to approach each other, overcoming
electrostatic repulsion.
When two small nuclei the product of fusion would have more BE per nucleon.
The increases in binding energy per nucleon are much larger for fusion than for fission
reactions, because the graph increases more steeply for small nuclei, so fusion gives out
more energy per nucleon than fission.
Wind generator. air density ρ, wind speed v, area of
the turbine A
assumption: wind is stopped by the wind turbine,
which is not the case, so not all of KE of the wind is
turned into electricity.
To calculate how much energy there is in the wind, we
consider a cylinder of air with a radius the same as the radius
of the turbine as shown.
𝛾=
∆𝑉
𝑉0 ∆𝜃
(K-1 or oC-1)
γ-coefficient of
volume expansion
∆𝜃 increase in temp.
If the velocity of air is v then in ∆𝑡 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 it will move a
distance v ∆𝑡. The volume of air passing by the turbine per
second is v ∆𝑡 π r2 where r is the length of one of the turbine
blades.
The mass of this cylinder of air, m = ρ v ∆𝑡 π r2 where ρ
is the density of air.
The KE of this air = ½ mv2 = ½ ρ v ∆𝑡 π r2 v2
= ½ ρ π r2 v3 ∆𝑡
Since this is the KE of air moving past the turbine per ∆𝑡
second, the power in the wind is KE/ ∆𝑡.
P = ½ ρ π r2 v3
The principle of the
oscillating water column:
consists of a column that is
half full of water, such that
when a wave approaches it
pushes water up the column.
This compresses the air that
occupies the top half,
Wave power
oscillating water column (OWC)
ocean-wave energy converter
The energy in a wave alternates between PE
as the water is lifted tip, and KE as it falls.
pushing it through a turbine which drives an electric
generator. The turbine is specially designed so that it
also turns when the water drops back down the column,
pulling air into the chamber. The main components of an
oscillating water column generator.
The form of a wave can be approximated to a rectangle (length λ, height A and width L travelling at velocity v).
The PE of this mass of water is given by PE = mgh, where h = the average height of the wave = A/2
PE = mg A/2
density of water = ρ
m = ρ x volume = ρ λ A L
PE = ρ λ A L g A/2 = ρ λLgA2/2
Power = energy per unit time, so if the waves arrive every T seconds then
Power = ρ λLgA2/2T , but λ/T = wave velocity, v, so power = ρ vLgA2/2
The power per unit length of wavefront is
ρvgA 2
P=
2
Hydroelectric power
gravitational PE of water
→ KE of water → KE of turbines →
electrical energy
The energy stored in a lake is gravitational PE = mgh. h is the height difference between the
outlet from the lake and the turbine. Since not all of the water in the lake is the same height,
the average height is used (this is assuming the lake is rectangular in cross section).
The rate of change of the potential energy converted
into kinetic energy is
P=
mgh
(ρV)gh
V
=
= ρ gh = ρ Q g h
t
t
t
Q is known as the volume flow rate (m3/s )
ρ – density of the water
V – volume of the lake
Intensity I of the Sun’s radiation incident on a planet at
distance r from the Sun is the power radiated received at
distance r per unit area
Astrophysics: apparent brightness b
The power from the star received (incident) per m2 of the
L
Earth’s surface. If the energy radiated by a star is emitted
b = 4π𝑑2
uniformly in all directions, then apparent brightness is
where L is luminosity (power radiated) of the star and d its distance from the Earth.
Albedo: Some of the radiation received by a planet is
reflected straight back into space. The fraction that is
reflected back is called the albedo, 
α=
total (reflected) scaterred power
total incident power
Earth’s albedo varies daily and is dependent on season (cloud formations) and latitude. Oceans
have a low value but snow has a high value. The global annual mean albedo is 0.3 (30%) on Earth.
If the temperature of a planet is constant, then the power being absorbed by the planet must equal
the rate at which energy is being radiated into space. The planet is in thermal equilibrium.
Surface heat capacity is the energy required to raise
the temperature of unit area of a planet’s surface by
one degree, and is measured in J m-2 K-1
CS =
energy
area of surface x temperature change of surface
If the incoming radiation power and outgoing radiation power are not equal, then the change of the
planet’s temperature in a given period of time is:
ΔT =
(incoming radiation intensity − outgoing radiation intensity)× time
Cs
A black body is a theoretical object that absorbs all incident
electromagnetic radiation. Therefore it reflects no radiation and
appears perfectly black. It is also a perfect emitter of radiation.
It would emit at every wavelength of light, and the “black body
radiation” distribution as a function of wavelength, known as
Planck’s law, depends upon its temperature.
Although stars and planets are not perfect emitters, their radiation
spectrum is approximately the same as black-body radiation.
WIEN’S LAW
wavelength at which the intensity of the radiation is
a maximum, λmax, is inversely to the temperature of
the black body
2.9×10-3
max (m) 
T(K)
STEFAN - BOLTZMANN LAW
The total power ((total energy per unit time) radiated by a black
body is proportional to 4th power of surface temperature
(astrophysics: luminosity)
P = σAT4
 = Stephan - Boltzmann constant
A – surface area of the emitter
T – absolute temperature of the emitter (in Kelvin)
The Earth and its atmosphere are not a perfect black body.
Emissivity, e, is defined as the ratio of power radiated by an
object to the power radiated by a black body at the same temperature.
e=
power radiated by an object
power radiated by black body at the same temperature
There are only two ways to transfer energy from one body to another — either by doing work or
by transferring thermal energy.
Thermal energy may be completely converted to work in a single process, but that continuous
conversion of this energy into work requires a cyclical process (use of machines that are
continuously repeating their actions in a fixed cycle) and the transfer of some energy from the
system (to the surroundings and therefore no longer available to perform useful work).
Degraded energy is energy that has become less useful (unavailable), i.e. cannot perform
mechanical work due to being transformed
into other forms of energy, e.g. thermal energy (in accordance with the second law of
thermodynamics)
Sankey diagrams are used to represent different ways of producing useful energy.
Fuel is a substance that can release energy by changing its chemical or nuclear structure.
All possible sources of energy:
▪ The Sun’s radiated energy
▪ Gravitational energy of the Sun and the Moon
▪ Nuclear energy stored within atoms
▪ The Earth’s internal heat energy
○ The Sun is the prime energy source for the world’s energy.
Energy density is the amount of energy that can be extracted per kilogram of fuel. Unit: J kg -1
Chain reaction: ▪ Energy is required to split a U – 236 nucleus. This can be supplied by adding a
neutron to the U –235 nuclei, which destabilizes the nucleus U – 236
(formed after a neutron is caught by U – 235) and causes it to split in two.
▪ Extra neutrons are produced, which can go on to react with other U – 235 nuclei
in a self-sustaining chain reaction.
However neutrons must be first slowed down to less than 1 eV.
Too fast neutrons are not likely to make reaction.
Critical mass: the minimum mass required for a chain reaction. (atomic bomb: mass > critical mass)
Fuel enrichment: ▪ Uranium comes naturally as 99.3% U-238. However only U – 235 is used
in the reaction process.
▪ The process of increasing the percentage of U-235 in the material to make
nuclear fission more likely is called enrichment.
▪ 3% U-235 must be reached in order to power a nuclear reactor.
Controlled nuclear fission (power production) and uncontrolled nuclear fission (nuclear weapons)
Main energy transformations in a nuclear power station:
nuclear energy → thermal energy → mechanical energy → electrical energy
Three important components in the design of all nuclear reactors are
moderator, control rods and heat exchanger.
▪ Moderator is a medium that slows down fast neutrons to make them suitable for reaction
(water, graphite, heavy water).
▪ Control rods are movable rods that readily absorb neutrons. They can be introduced or
removed from reaction chamber in order to control the rate of fission of
uranium and plutonium. Made of chemical elements capable of absorbing
many neutrons without fissioning themselves (cadmium, hafnium, boron, etc)
▪ Heat exchanger is used to seal off the place where nuclear reactions take place from the rest
of the environment. In some nuclear power plants, the steam from the
reactor goes through a heat exchanger to convert another loop of water
to steam, which drives the turbine.
The advantage to this design is that the radioactive water/steam never
contacts the turbine.
Neutron capture by a nucleus of uranium-238 results in the production of a nucleus of
plutonium-239
In addition to uranium – 235, plutonium – 239 is also capable of sustaining fission reactions. This
nuclide is formed as a by - product of a conventional nuclear reactor. A uranium – 238 nucleus
can capture fast moving neutrons to form uranium – 239. This undergoes β – decay to neptunium
– 239 which undergoes further undergoes further β – decay to plutonium – 239
238
239U
1
U
+
𝑛
→
0
92
92
239
239
U → 93Np + −10𝛽 + 𝜈
92
239
93
Np →
239
0
94Pu + −1𝛽 + 𝜈
Plutonium-239 is used as a fuel in other types of reactors.
Problems associated with producing nuclear power using nuclear fusion: the reaction
requires creating temperatures high enough to ionize atomic hydrogen into a plasma state.
Currently the principal design challenges are associated with maintain and confining the
plasma at sufficiently high temperature and density for fusion to take place.
Solar Power
▪ Solar panel (active solar heater) is used for central heating or for making hot water for
household use, placed on roofs of houses, consisting of metal absorber, water pipes,
and glass. It converts solar energy into thermal energy of water.
▪ A photovoltaic cell converts solar radiation into electrical energy. Produces very small voltage
The greenhouse effect is the warming of a planet due its atmosphere allowing in ultraviolet
radiation from the Sun, but trapping the infrared radiation emitted by the warm Earth.
Temperature of the Earth’s surface will be constant if the rate at which it radiates energy
equals the rate at which it absorbs energy.
Short wavelength radiation is received from the sun and causes the surface of the Earth to
warm up. The Earth will emit infra-red radiation (longer wavelengths than the radiation
coming from the sun because the Earth is cooler than the sun). Some of this infra-red
radiation is absorbed by gases in the atmosphere and re-radiated in all directions.