Download Synoptic physics paraphrased

Gravitational fields
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Force on masses
Kepler’s laws:
o The orbit of each planet is an ellipse with the sun at one force. This law can only be
obeyed if there is an inverse square law for the attraction between.
o The line drawn from the sun to a planet sweeps out equal areas in equal intervals of
time.
o The squares of the period of revolution are proportional to the cubes of the average
distances from the sun. r3= K T2
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F  mg 
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mv 2
As gravitational force must be equal to centripetal force.
r
GMm
GM
F 2
V2 
r
r
but also V 
2 r
4 2 r 2 GM
so v 2 

T
T2
r
Gravitational potential
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A route taken that does not affect the change in potential energy is called a conservative field
Potential energy of a mass at a point is numerically equal to the work done in bringing that
mass from infinity to that point. E p 

GMm GMm
(
)
r1
r2
The Potential at a point is numerically equal to the work done in bringing a unit mass from
infinity to that point. Vg 
Ep
m
 Vg  
GM
r
GM
)
r2
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g field strength = negative potential gradient (differentiate and equate to
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F force=gradient of potential energy
From cons of energy loss of KE= gain of EP, therefore
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Escape speed
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Role of gravity in determining whether Universe is open, Critical or closed. The edge of the
Universe should decrease in acceleration because there is less gravitational force acting on
those bodies. How much mass/density there is in the Universe will determine whether
expansion will be indefinite, critical, or definite.
Ve 
2GM
r
(& thermal energy transformations)
Electric fields
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Force on charged bodies
Touching a plate with a sphere causes the sphere to have the same polarity and the pendulum
will be repelled by it. Knowing the weight of the sphere and the angle of deflection, the
electric force could be calculated.
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The strength of the electric field is dependent on the p.d. on the plates and the separation of
the plates. E 
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F
q
E is electric field strength, the force acting per unit charge.
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Energy gained by charge = Vq, the electrical energy gained by the charge is the charge Q
multiplied by the difference in potential V.
Fd=Vq
The work done.
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E is the force acting per unit charge, therefore E 
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Coulomb’s law: The force between two point charges is directly proportional to the square of
the distance between.
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F
F V
 , only need voltmeter & ruler.
q d
kQ1Q2
Charges may repel or attract.
r2
The 4π in k is introduces because we are dealing with a radial rather than uniform field.
Electric potential
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A p.d. of one volt exist between two points in a circuit if 1J is required to transfer one
coulomb of charge between the two points. If we define one part as being zero potential, we
can give a value to the other part. For practical reasons the Earth is taken as the zero of
potential.
Potential is negative as with gravity if the charges attract from infinity (where zero).
But if the charges actually repel, potential becomes positive in bringing from infinity as work
is being done against the direction of the field.
Same polarity- repulsive- result in decrease in potential energy with increasing separation.
Opposite polarity- an increase in potential energy with increasing separation.
When we move a charge against an electrical field its potential will rise because work is being
done on the charge. Vq
Electroscope measures electrical potential. Outside of case is earthed and charge will flow on
or off the gold leaf. The greater the potential of the golf leaf the greater the deflection. The
gold leaf electroscope is calibrated by connecting wire loop to known voltages and marking
the deflection. V= k/r
k=600 Vm
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1 1
E p  w  kQ1Q2 (  )
r1 r2

Absolute potential- VE 
VE 
W
Q2
 negative as
1 1

r2 r1
kQ
at a distance r from a charge Q, raising from infinity
r
2
Strong nuclear force
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Neutrons are uncharged. Therefore when they pass a nucleus which is positively charged
there is no electromagnetic deflection. There is virtually no gravitational attraction since the
masses involved are very small. This explains why neutrons interact very weakly with atoms
and are difficult to absorb.
When neutrons travel very close to a nucleus it suddenly experiences a large force. It cannot
be an inverse square law like gravity, and it is not electromagnetic. There is a new force that
is only detected outside the immediate surroundings of the nucleus.
A proton is repelled by the electromagnetic force determined by the atomic number of the
nucleus and the charge of the proton +1.6x10-19, and this is stronger than the gravitational
attractive force.
However if a proton has sufficient Kinetic energy to get closer than one nuclear diameter.
1x10-14 a different strong attractive force comes into play, overcoming the repulsive force.
Leptons do not experience the strong force. The strong force exists between quarks.
The second neutron in helium contributes further to the strong force.
The binding effect due to the strong force falls very rapidly with distance. As the number of
protons increases the electrostatic force repulsive forces within the nucleus increase, and this
affects all the nucleons. However the strong with not increase with distance because of its
short range. This is why forces tending to disintegrate the nucleus become more dominate and
the nucleus becomes more unstable.
Nuclei with more than 82 protons with more than 82 protons are unstable or radioactive and
emit radiation.
Or the liquid drop model where the nucleus splits in two and nuclear fission takes place.
(When nucleus is constructed they have a negative potential energy, this make the nucleus
stable. To dismantle to constituent parts energy needs to be supplied to lift the nucleons out of
the potential well.)
Magnetic fields around electric currents
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Force on poles of a magnet
Right hand grip rule
Magnetic Field is at right angles to the current, concentric circles.
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Increasing the current increases the force acting on the on the wire in the magnetic field.
Increasing the strength of the field increases the force acting on the current.
Reversing the current means the direction of the force has been reversed.
Reversing the magnetic field (direction of N to S) reverses the direction of the force.
Fleming’s left hand rule
 Force, Field, Current. Thumb, First, SeCond.
Torque in motors. Current (thick, low resistivity), strength of magnet, number of turns.
F=BIL(n)
Flux density –B is directly proportional to mag. Field strength H
Top pan balance, or current balance.
Tesla- one tesla moves one Ampere of current one metre with one Newton of force. In a
magnetic field at right angles to the current. 1NA-1 m-1
B

A
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  BA (cosθ) flux whose unit is the weber Wb.
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Flux may be visualised as the total number of magnetic field lines.
Flux density may be visualised as the number of field lines than pass each square metre.
1T= Wb m-2
Electro- magnetic induction
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Moving a conductor through a magnetic field is said to induce a voltage across the ends of the
conductor. This voltage is called the induced e.m.f.
The greatest e.m.f. is induced when the coil moves vertically ‘cutting through’ the field lines.
When the direction of the movement of coil or magnet reverses, the direction of the induced
current is reversed.
Flux – flow. When the surface is parallel with the field no flux will pass through the surface.
Zero flux linked into it, and zero flux density.
As the magnet or coil moved faster the induced e.m.f. increased.
e.m.f. increases with magnetic flux increase and number of turns, and speed.
“Speed flux, number of turns” makes the induced current stronger.
Induced e.m.f. is proportional to:
 Magnetic flux density B
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
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Rate of cutting of lines of flux

t
 The number of turns on the coil N
The result may be summarised as Faraday’s first law of electromagnetic induction :
o
emf 
( N )
t
where ∆t is the time it took the flux to change
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Reversing the direction of movement reverses the direction of induced e.m.f., why?
Because of the direction the current will flow, the direction of the e.m.f opposes the change
that causes it, otherwise the conservation of energy would be violated.
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Transformers- When the current was switched on the primary coil created lines of magnetic
flux. These flux lines linked with the secondary coil and induced an e.m.f., causing the
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voltmeter to deflect to the right. When the current was switched off the flux linked the
secondary coil was zero. This caused a deflection in the opposite direction.
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VpIp=VsIs
Np/Ns =
Vp/Vs =
Is/Ip
The induced current in the secondary coil opposed the growth of flux, but when the current
turned off it tried to maintain the flux.
With a greater current flux was created at a faster rate, causing a larger induced e.m.f in the
secondary coil. This deflected the pointer further than before. There was a faster rate of
decrease in flux when the current was switched off, causing a larger deflection to the left.
When the flux did not change the induced e.m.f was zero. Ever-changing flux with AC.
AC charge flows in opposite directions at different times.
Energy losses:
o copper losses in the coils I2R
o Leakage loss, some magnetic flux will not reach the secondary coil. Air gaps cause
o Hysteresis losses magnetic flux created in the core, continual reversing of flux causes
some energy to be wasted in heating the iron core. Should choose a good iron alloy
with the best magnetic properties
o The change in magnetic flux will induce electric currents in any conductor it passes
through. These currents create heat as they flow in the conductor. Iron core is
laminated to reduce the size of the eddy currents.
Eddy currents circle at right angle to the flux. Gaps mean less energy wasted in heating.
Laminating insulates layers electrically from each other and so reduce heating.
Eddy currents can be useful: High frequency AC crates a rapidly changing magnetic flux
through the base of the aluminium pan.
(Transformers) Transmission, at high voltages large powers can be delivered in low currents,
this means power losses are low, high voltages are inappropriate for energy consummation.
Flemmings Right hand rule; motion, field, Induced current. Thumb, first, second
Factors: B, v speed of movement, length or no. of turns (of conductor) E= BLv
Generating electricity. The flux is not moving. As there is a speed in rotation, AC frequency
must increase, as the time taken for a cycle decreases. The peak e.m.f. also increases as the
rate at which the conductor cuts through the flux is greater at greater speeds; the rate at which
flux is linked to the coil is greater. When coil horizontal- maximum induced emf because flux
linked is most rapidly changing. At vertical emf is zero because the change in flux linked is
zero. (motor split rings DC, carbon brushes ----- generator slip rings AC)
Torque = BANI cosθ
 Emax= BANω
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Back e.m.f. As the motor turns it creates a back emf which opposes the emf of the power
supply. The faster the motor turn in a magnetic field, the greater the back emf and so the
smaller the current. The current varies with back emf not resistance as this doesn’t really
change. V-E =IR
Initially large current as back emf is zero, then current falls as the motor gains speed and the
back emf increases. So need starting resistance so not to burn out coils.
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Nuclear energy
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According to special relativity when at rest, mass is equivalent to an amount of
energy, where
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E  mc 2
It means a change in energy is a change in mass and vice-versa E  mc 2 ,
not that mass can be converted into energy and vice-versa
Einstein’s theory states when there is a loss of energy from the system, there must be an
equivalent loss of mass from the system. The final mass of the chemical products would be
less than the initial mass of the reactants. A gain in KE = a gain in mass
On the nuclear level the decrease in mass can be a significant fraction of the original mass.
Binding energy and mass defect: It takes energy to separate nucleons from a nucleus, and it
also takes energy to bind the nucleons into a nucleus. However because of the short range
strong nuclear interaction, and the long range electromagnetic interaction, and the effective
changes in potential energy (their summation), a nucleus has less energy than its separate
constituent parts, by an amount called the binding energy. Work done in unbinding the
nucleus.
Einstein’s relationship means that the mass of a nucleus is less than the mass of its constituent
parts. The difference is called the mass defect. There is a far greater mass defect on the
nuclear level because the strong nuclear force is much greater than the electromagnetic force
involved on large scale.
Binding energy = mass defect x c2
Binding energy per nucleon is very useful to measure stability, since it is the average energy
need to remove one nucleon from the nucleus, against the net attractive force.
During fission and fusion, energy will be released instead. (could compare to GPE)
The steep gradient of fusion shows energy would be released than fission. However it is
difficult because of the strongly repulsive electromagnetic force. (The 2 main areas of
research are magnetic confinement of a plasma and inertial confinement of a deuterium
micro-pellets)
Radio-active decay Spontaneous decay: Occurs when a nucleus has enough energy to decay
unaided. This means the mass must be greater than the sum of the masses of the daughter
nucleus and the alpha particle.
Natural radio- active decay is spontaneous, but it is rare in fission. Instead thermal nucleons
are slowed down to thermal speeds
of particle Ek 
3
1
kT  m c 2 and the process is
2
2
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called induced fission.
Energy is released in the process because the binding energy of the daughter nucleus is
greater than the binding energy of the parent.
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The activity is portioned to the total number of nuclei.
N  N 0e
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 t
 A  A0e
A   N
 t
Critical size, drop oscillates, strong force overcome by electromagnetic force.
235 U to 236 U. Never undergoes spontaneous, then highly unstable.
Could have run-away effect of chain reaction. But the emitted neutrons do not interact easily,
or escape for a small mass. (more fission so more neutrons so more fissions)
Positive-feedback, output energy is made to reinforce input energy. (in –ve, decrease input E)
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Self-sustaining critical reaction if mass large enough. The min mass for a sphere is called
critical mass. For explosion must exceed the critical mass in very short time.
If the speed of the emitted neutrons is slowed to thermal speeds. Then they are thermal
neutrons. Slowed to match the same average KE as surrounding atoms 2200Ms-1
Supercritical-meltdown, subcritical-chain reaction dies out, critical – steady self sustaining.
Need a way to contain neutron flux the no. of neutrons passing a unit area in one second.
Internal energy
Mass- large no. of fuel rods, neutrons thermal-moderator, inelastic collisions.
Neutron flux- control rods, variably absorb neutrons.
Radioactive decay
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Exponential decay: capacitor discharge, absorption of E-M radiation, damping of oscillatory
motion, rate of heat loss form a body- N’s law of cooling.
The rate of decay is portioned to how much you have. A   N
Particle detectors
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Reveal the path of the particles. A magnetic field is used to deflect the –particles, thus
providing information on their charge and momentum. As with the mass spectrometer.
Cloud chambers- diffusion type. Tiny drop of alcohol condense around the ionisation trail
left by the particles. Supersaturated with alcohol vapour. The path is straight because alpha
particles have a relatively large mass and remove electrons without getting close to any
nucleus to be deflected. Length corresponds to KE of alpha particles. Thin-low ionisation
potential. Magnetic field can be said to curve high energy electrons.
Bubble chambers- more effective because far greater density of atoms in liquid hydrogen
compared to ethanol vapour. More collision per unit length, and not travel as far. Liquid
hydrogen is super heated. Any ion left in the path of the particle will ‘impure’ the liquid and
cause boiling or bubbles along the path.
r
mv
BQ
Particle Accelerators
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Mass spectrometer – Uses a magnetic field to deflect charged particles that have been
accelerated using an electric field. They measure charge to mass ratios.
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Electrons accelerated using an electron gun
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The force on a moving charge is given by F  BQv , this force supplies the centripetal force
needed for circular motion
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F
QV 
1 2
mv (1) as the EK gain=EP loss
2
mv 2
equate to v2 and substitute in (1)
r
Q
2V
 2 2
m Br
For a fixed magnetic field , as the accelerating voltage increases, the radius increases, as the
electrons have more momentum.
The magnitude of the electric force is F=QE, The magnitude of the magnetic force is F=BQv
So at certain speeds the forces cancel v=E/B in a velocity selector. Only ions of this speed...
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...pass into the deflection chamber where a magnetic field deflects ions of different masses
along circular paths whose radius depends on the ion mass r 
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Eliminating v gives
m
QrBd B
E
mv
Bd Q
(u unit is 1/12th of mass of Carbon-12)
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Problems of probing matter which requires vast energies:
o Need good insulation to achieve equivalent effect of extreme potential difference.
o As particles gain speed they increase mass above rest mass e=mc2
Electron tubeAbeam of electrons is produced in the evacuated tube. The beam glows
making it possible to make measurements of it. The coils on the side of the tube produce a
magnetic field, B. B can be found by measuring the current and coils between them. The
radius of the beam is found and the e/m ratio is found by equating force due to movement in
the magnetic field to the centripetal force.
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But the velocity of the electrons needs to be found. Electrons retain KE gained by being
accelerated in an electric field.
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mv 2
Bev 
r
eV 
1 2
Q
2V
mv   2 2
2
m B r
Types of particle accelerator
Linear:
o
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Insulation problems of van de graaff overcome, as acceleration is through a series of
relatively small potential differences. The produce well collimated ion beams and do
not require magnets as with cyclotron and synchrotron.
o However an increase in mass becomes problematic as there is more inertia the effect
of the accelerating force levels off.
Cyclotron:
a LINAC crled up to save space! There are two Dees. To keep ions between
the dees there is a large evacuated chamber between the poles of a large electromagnet. The
uniform magnetic field provides the centripetal force, although the radius of the circle
increases progressively due to the increasing speed.
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The high potential difference applied between the dees has a frequency such that the polarity
is reversed every time the ions cross from one dee to the other. This makes the ions accelerate
when they pass between them. In the dees they are not gaining KE, just changing direction, as
the electric field in the dees is zero. The frequency of the accelerating potential can
remain constant as the speed increase of the ions causes them to spiral outwards, taking a
longer path.
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In a semi circle s   r so the time to travel is t 
r
v
, but r 
mv
m
t 
Bd Q
BQ
However v and r don’t appear in this equation, which mean that time is independent
of them- same time for every revolution. T is twice the value (as full circle) and T=1/f:
f 
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BQ
This is constant.
2 m
However because of Einstein’s mass increases orbital time
will increase an the particles will get out of step with the
cyclotron frequency. Protons have more rest mass than
electrons so move more slowly for a given energy, thus
accelerated to higher energies before the same problem
occurs.
Synchrotron: Overcomes the problem of mass increase as electric and magnetic field are
synchronised to the mass increase.
Electric field accelerates .Magnetic field changes direction. Much larger structure.
As the ions accelerate the magnetic field is increased to keep them in the same orbit. However
they must take progressively less time to complete each orbit and so the frequency the
alternating potential difference must increase too.
Cannot be accelerated as a continuous beam – however head-on collisions of particles and
anti-particles (of opposite charge) double the available KE.
Collision rates are lower in fixed target experiments
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