FINAL EXAM NOTES

CHARGE AND FIELD MODELS

Neutral objects can be pushed by a charged object

Neutral objects can produce

and electric field (dipole)

FORCE TWO CHARGES EXERT

ON EACH OTHER CAN BE THE

SAME EVEN IF THEY HAVE

DIFFERENT MAGNITUDES!

Electric Field of point charge

For electrostatics, field lines must

begin and end on charges, so we may

not just show field lines emanating

from a point.

In a plane of infinite charge

the electric field on a point

does not depend on the

distance it is away from the

plane.

Electric Field of line of charge example next page.

Electric Field Strengths

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The field of a sphere of charge is the same as if it were all concentrated at the centre and modelled

as a point charge.

Dipoles

In a uniform field

o A point charge moves with constant acceleration

o A dipole rotates until it is aligned with the field

o NOTE DIPOLES HAVE NO NET FORCE IN UNIFORM FIELD BUT DO ROTATE FROM

FIELD

In a non-uniform field

o Dipole moves toward region of

higher field strength.

GAUSS LAW AND CONDUCTORS IN EQUILLIBRIUM If numerous closed shapes enclose a net charge, the flux through all shapes is equal.

For Gauss Law to be effective need symmetry e.g. A surface over which the electric field is

constant. SYMMETRY OF FIELD MUST MATCH SYMMETRY OF OBJECT.

Note: flux is independent of radius and surface shape

In conductors the electric field inside is zero if in

electrostatic equilibrium (no moving charges)

In a conductor there is no net charge if there is no charge

inside the object, meaning all charge is on the surface

If there is a hole in an object then the flux inside this must

be zero as seen in figure to the right.

ELECTRIC POTENTIAL

Think of electric field like gravity (long range force), in uniform

field it can be said that

Energy of a SYSTEM not just one particle is given as (Note: +ve for

like charges, -ve for opposite)

Electric potential can be thought of as stretch in spring (or hands)

and the electric potential energy is the energy from this stretch

This potential depends solely on the source and its charges

Potential is the ability for something to happen (much like gravity)

if we get a charge showing up this is when we consider electric

potential energy (gravitational force).

Potential is Volts, Electric Potential Energy is Joules

Particles accelerate or deccelerate through potential

Electric potential inside a parallel plate capacitor (s is distance from negative plate)

Not this potential is present inside all points of the capacitor

NOTE AT NEGATIVE PLATE THERE CAN STILL BE CHARGE ONLY THE DIFFERENCE MATTERS

Electric potential of a point charge (where q is the point and q is the other charge) can be

modelled as

Electric Potential of a charged sphere (R is radius of sphere and r is distance away from

surface, V0 is potential sphere is charged too.)

Electric potential obeys superposition such that

Potential of the object is measured at z=0

ELECTRIC POTENTIAL EXAMPLES

POTENTIAL AND FIELD

Chapter 30

Electric potential and electric field are two different ways of thinking about how source

charges affect space around them.

Potential Energy is defined as work done by a force F on charge q as it moves from position

i to position f.

Can also think of potential energy as Vf=Vi (are under Es & s curve, see below)

If force and direction of motion are orthogonal there is no work done on the object

Seperation of charge creates potential difference.

Battery potential can be seen as

Finding the electric field from potential is extremely simple

Electrostatic force is CONSERVATIVE!

The electric potential inside a parallel

plate capacitor is from the

negative plate. This means the closer

the PROTON is to the negative plate

the larger its potential energy is.

Any two points inside a conductor will

have the same potential.

Electrostatic potential is known as the multiple of two

charges so the two situations have the same potential

(NOT ELECTRIC POTENTIAL!!!!!!!!).

With two spheres the one with the larger radius will therefore have the lowest electric

potential

Potential spreads evenly over two objects, e.g. These

two spheres have the same electric potential when the

switch is closed.

However the charge would be greater on Sphere 1 in

this diagram, there is a larger radius and as they have

the same potential we use the formula

to

realise it is a larger charge

In contrast the electric field will be greater on Sphere 2,

as it will have a greater surface charge density

Electric field is always perpendicular to the surface of a conductor in electrostatic equilibrium

CAPACITORS

When charged by at a battery the charge on capacitor slowly approaches battery charge

This means that the positive capacitor plate has same charge as positive battery terminal

and vice versa

IN SERIES CAPACITORS: (

)

PARALLEL CAPACITORS:

NOTE SUM OF ALL POTENTIAL DIFFERENCES AROUND A CLOSED PATH IS ZERO

Current and Resistance

Electron current is

Electric field pushes the current

Current is the rate at which charge flows

Current per square metre of cross section or current density is given by

Field strength in a wire is given by

Electric field points in direction of current (From higher potential to lower potential).

Light bulbs will have the same brightness whether in series or parallel, only current differs.

Magnetism

Magnetic fields always try to oppose each other.

Moving charges create magnetic field, using index to point to

point, thumb as velocity of particle and palm as field direction we

can say that.

(

)

Superposition holds

Can model a small segment of wire current as

Can also model a coil of wire with N

loops of ANY SHAPE as

Interesting to note that a magnet and

loop of current carrying wire create

same magnetic field

Magnets and current carrying wires just

two different types of magnetism

Magnetic Dipole moment is simply

Amperes Law is used to calculate the magnetic field created by wires and other objects as

It should be noted that it is independent of shae of the curve

Independent of where current passes through curve

Depends only on total amount of current through area enclose by integration path

CURVE MUST BE CLOSED

When considering a SOLENOID amperes law CHANGES to

Magnetic field can exert force on current carrying wire and hence charge.

ONLY MOVING CHARGES EXPERIENCE THIS FORCE

FORCE IS ALWAYS PERPENDICULAR TO VELOCITY AND FIELD

REVERSE ABOVE FORCES FOR NEGATIVE PARTICLE

Can have something called cyclotron motion (see right) in which

As said before magnetic fields can have forces on current carrying wires

Parallel wires also exert magnetic field on each other creating relationship

Current loop is much like a magnetic dipole so it can be seen that they will act like magnets

when near each other.

Torque is known to be force multiplied by distance from this we

know that

For loops to calculate torque we use the right hand rule where

pointing finger is B and the field created by the loop (current

carrying wire) is the thumb. In this example the torque is up and is

out of the page.

Electromagnetic Induction You can induce a current by changing size or orientation of circuit in stationary magnetic field OR

changing magnetic field through stationary current

Motion through magnetic field creates a potential difference

This motion creates an electric field to balance the forces (see above)

This in turn creates a motional emf (potential difference) of

From this It can be seen a current may be induced in a circuit, using right

hand rule and fact that current runs from positive to negative (like in figure

to the right) then we can find direction and magnitude using

In order for sliding wire to work and have constant velocity we must

realise that pulling force is present and is equal to the magnetic force

Also note that

Eddy current are when a loop of non

magnetic material is placed in between two

magnets. This induces a current in the loop

and means a force must be applied to pull the loop out of the

the magnetic field.

Magnetic flux is the amount of magnetic field passing through a

loop and is given by

In a non-uniform magnetic field the flux is stated as

Lens Law states:

The induced emf created by this changing flux is given by Faradays Law (-ve because opposes

the change of magnetic flux)

Changing magnetic field causes electric field in pinwheel shape.

This is called a non-coloumb electric field as it rotates.

This is created by magnetic field, where as coloumb electric field is

created by positive and negative charges.

We may calculate the electric field as

|

|

From this we can get the equation for a solenoid

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|

|

|

Changing electric field also causes a magnetic field, but OPPOSITE IN

DIRECTION TO LENS LAW (Maxwells)

Maxwell found that these electromagnetic waves have speed

Discovery used practically in generators

Transformers

Like capacitors, inductors create a uniform magnetic field by use of a solenoid

Instead of charge to voltage ratio, inductor uses flux to current

We can then find the potential difference across this using faradays law such that

|

|

Often used as a spark, closing a switch creates a large change in current.

Electromagnetic Fields and Waves

Overview of concepts

Gauss Law: For any object enclosing a charge Qin, net electric flux is

For a magnet, continuous lines therefore

Faradays Law: An electric field and thus an emf can be created by a changing magnetic field

Ampere Maxwell Law (note this is a correction to the previous Amperes Law): Current

creates a magnetic field and magnetic flux is also a magnetic field so, B through a surface is

(

)

The lorentz force law describes how magnetic and electric fields effect a moving particle

In Summary!

Constant current means increasing electric field but constant

magnetic field induced.

Magnetic field depends on rate of change of charge.

For an electromagnetic wave, pointer is magnetic, thumb is

electric, palm is velocity

Electromagnetic intensity decreases linearly with time, e.g at

20km the ampliturde is half what it was at 10km

We find that:

Should be noted that is in direction of

at any point on the wave

Electromagnetic waves have an intensity, which is the average energy transfer. Intensity is

Note that light exerts momentum

It follows that radiation pressure has an effect on an object, with pressure being

OPTICS Travelling Waves

Wave source that oscillates with simple harmonic motion generates a sinusoidal wave

Period of this wave is called, T

Useful to define something called the wave number

Displacement of a wave at time is equal to

PHASE DIFFERENCE BETWEEN TWO POINTS OF A WAVE DEPENDS ONLY ON RATIO OF

SEPERATION TO WAVELENGTH

PHASE DIFFERENCE BETWEEN TWO WAVEFRONTS IS ALWAYS 2

Index of refraction of material is a ratio

Wavelenght in material is

E.g. in material there will be a smaller wavelength

Doppler Effect

Different wave lengths because moving source

Wavelength of approaching source will be

NOTE THIS DIFFERS FOR THE PERSON MOVING TOWARDS SOURCE

This is because waves move through medium, we get the equations

For light waves if we calculate the wavelength from above we find that a receding

source has a red shift while an approaching source has a blue shift.

Superposition

Superposition applies to waves as well

Possible to have constructive and destructive interference

When two or more waves are present at a point in space, the displacement is equal to

the sum of the waves displacement

Interference depends on the phase difference of waves,

(

)

For sources with no phase shit max constructive and perfect destructive occur at

and (

) respectively

This is used in the real world for thin film coating on glasses (see example)

NOTE MOTION OF WAVES DOES NOT EFFECT POINTS OF INTERFERENCE

Same equations apply for three dimensions as one dimension with r replacing x

(

)

THESE TWO ONLY APPLY IF WAVES ARE IN PHASE OTHERWISE USE PREVIOUS TWO

Wave Optics

Light is modelled in several different ways

o Wave Model: Used under most

circumstances and true for most

o Ray Model: Light travels in a straight

line, useful for modelling lenses

o Photon Model: Deal with that later

Youngs double slit experiment

demonstrates lights wave like nature

Slits are typically 0.1mm wide and 0.5mm

apart

Light waves interfere constructively to

produce bright bands of light

Can see that

Follows that position of bright

fringes

Diffraction grating which is an opaque screen with n

slits follows the same principles

Often modelled as REFLECTION GRATING

Single slit diffraction:

Dominated by central maximum of

Ray Optics

Objects are either self-luminous (sun, torch) or reflective (tree, people) The ray model states:

o Light Rays travel in straight lines o Rays can cross o Travel forever unless contact with matter o Object is a source of light rays o Eye sees by diverging bundles of rays.

Rays originate from every point on an object

Object and image heights related by ratio

Light reflects at the same angle it collided with material at

Refraction occurs when material passes from one material to the other

Not all light is refracted, some is reflected

NOTE ANGLE IS MEASURED FROM LINE PERPENDICULAR TO SURFACE

A point is reached where all light is reflected, Total Internal Reflection

(

)

Vergence is the curvature of wavefront. Curvature of wavefront is

The radius of the vergence is directly related to the vergence itself such that

{

}

Mirrors are used to direct light, given by equation

Vergence for a mirror is

NOTE VERGENCE POSITIVE FOR CONVERGING NEGATIVE FOR DIVERGING

Lenses (spherical interfaces) are used to direct

light, given by equation

| |

Power can also be written as (noting that nin is the concave side and nout is the convex)

Magnification is quite simply defined for mirrors and lens as

Possible to define thin lens equation where there are two

interfaces next to each other (see figure to the right)

[

]

Eye can see to an angular size of

If a single converging lens is used this

angle can be even smaller such that

using small angle approximation

This leaves the magnification to be

With a microscope this can be improved on even more so that we have a magnification

The light gathering power of a microscope is also an important

aspect, measured in numerical aperture, NA, it is defined as

Telescopes are another useful optical aperture with their magnification given by

In magnification there is still diffraction (occurs passing through hole of any shape) as

the ray model of light is not complete

This means there is a central maximum, meaning there is a minimum spot size able to be

projected

In a real world situation it is difficult to produce a lens

with a focal length less than its diameter so in reality

Minimum angular resolution of a lens is called Rayleighs

criterion

X-ray diffraction occurs when rays of very short

wavelengths pass through the crystalline structures of

materials.

X-rays strongly reflect when their angle of incidence is

RELATIVITY Reference Frames are the main concept to understand, once understood relativity

should be easy, they can be said to

o Extend infinitely in all directions

o Experimenters are at rest in the frame

o Velocity and distance is measured accurately from this frame

Important to define an event as an activity that takes place at a definitive point at a

definitive time in space. They are what REALLY happen.

However measuring the event is the difficulty

In each reference frame to measure events you must use metre sticks and clocks in

that reference frame only

Note the time t is the time the event ACTUALLY happened.

SIMULTANEITY is when two events at different positions occur at the SAME time in a

reference frame.

Exploding firecracker on a moving cart experiment to explain simultaneity

S commonly used to refer to rest frame of the moving object.

Consider clock in moving frame. This clock measures the proper time such that measuring a

time outside this reference frame yields the result

Often said that the time interval between two ticks is the shortest in the reference frame

where the clock is at rest.

Considering twin paradox, both twins think they will have the younger sibling.

But this is not true as theory of relativity only holds in inertial reference frames and

the second sibling goes there and back so person on earth would be older

Length contraction also occurs given by relationship. Where is the proper length of

object

Can use the spacetime interval to compare events in all inertial reference frames as

it is the same in all intervals

Galilean transforms are not valid in reference

frames close to the speed of light

Must use Lorentz Transformations, which all use the

value

The equations are then given as

Should be noted that the x-values are only valid

where and when the event actually occurs not where they are after a certain time.

Should also be noted that the y and z values stay the same, e.g .

There is also variation between the traditional velocity

transforms as seen to the right

Where u is the frame of the moving object, v is the

velocity between frames and u is from a separate rest

frame.

Again there is variation between traditional

momentum and relativistic motion and we see that the

law of conservation of momentum does not hold at

large speeds. Momentum is actually given by

From this we can see that it is impossible for an object to go faster than the speed of

light as more and more energy is required to accelerate and object and an infinite

amount of energy is required to accelerate an object to the speed of light.

It follows that an object has a rest energy and a kinetic energy of total

In this equation it should be noted that E0

ALWAYS has an energy regardless of the

reference frame

The subscript p implies we are referring to

motion within a reference frame NOT motion

of two reference frames relative to each other

Note mass is not conserved, consider two colliding clay balls at same speed, if they collide

and mould there will be no velocity and due to conservations of energy

From this we learn that for conservation of energy to be true

( )

So mass and energy can be transferred but are NOT the same thing (think of fission).

Skipped most of End of Classical Physics Chapter but a few useful things

One electron volt is kinetic energy gained by an electron or proton if it accelerates

through a potential difference of one volt

Quantization

Electromagnetic waves can be artificially created

By shining light of a certain frequency at a substance you may

emit electrons. Consider following experiment.

Battery used to create potential difference and noted a few

discoveries

1. Current is directly proportional to the intensity of light

2. When light is applied current occurs spontaneously

3. Only emitted if a certain frequency, threshold frequency,

is exceeded.

4. Threshold frequency depends on metal that cathode is

made of.

5. Positive potential difference creates no change in current

as increase. Negative potential difference does change

current until at some point the change in voltage makes

the current stop

6. This value is the same for all light intensities.

This shows that a minimum energy is required to free an electron from a metal such that

We can consider the voltage required to stop an electron freed from the cathode as it

approaches the anode to be

However this classical interpretation does not account for a threshold frequency

Einstein saw this and created his model that accounts for a threshold frequency

EINSTEINS EXPLANATION

Atoms vibrate with a frequency of f

Energy of an atom vibrating with frequency, f, has to have energy

Einstein states that light arrives in light packets, called a Light Quantam, and each light

quantam has energy

Think of rain analogy, generally water flows like light, but with rain we get it in drops. The

difference is intensity e.g intense energy is lots of rain or lots of light

Einsteins three statements about light quanta

1. All photons with energy travel at the speed of light

2. Light quanta absorbed on all or nothing basis, e.g. substances can only emit or

absorb 1,2 or 3 quanta, not 1.5

3. Light quanta when absorbed by an electron gives ALL of its energy to ONE electron

This lead us to the discovery that

IF this where E0 is the energy required to remove an electron, then the

electron leaves the metal becoming a photoelectron.

This gives us the previously stated threshold frequency

If the frequency is less than this threshold frequency, even for high intensity light no

electrons will be ejected

{

}

NOTE: Threshold frequency is directly proportional to

the work function, metals with low work functions=low

threshold and vice versa.

This leads to us discovering that

These light quanta later given the name photons

No perfect analogy for a photon, some work well in

some situations but not in others, however the

energy created by photons arriving at a point is

defined as power:

It was postulated by De Broglie that matter can have a wave like nature

De Broglie then found that the wavelength of matter with momentum

This means that matter undergoes interference like double slit experiment

Considering a matter wave to be a travelling wave that is

confined to a box

This creates something called a standing wave that has

a constant path

Wavelength must be

To satisfy this and the de Broglie wavelength formula

This means that energy of a confined particle is quantised e.g. has certain energies

Can rewrite this as

Older models for atoms didnt explain stability or discrete spectra

Bohrs model does and implies that

1. Atom consists of negative electrons orbiting a positive nucleus (Rutherford)

2. Atoms can only exist in certain stationary states

3. Each of these states has a certain energy level En where E1< E2< E3< E4.

4. Ground state E1 is stable and will always exist, other states may not, called excited

states

5. Atom can jump from one state to the other by absorbing or emitting a photon

6. By absorbing photon it goes to a higher state

7. By emitting it goes to a lower state

8. Atom always seeks the lower state

Main additions to the Rutherford model are atoms can only exist at certain states

and that atoms can jump from one state to another by absorbing just the right

frequencies such that energy is conserved.

This model implies that

1. Matter is stable

2. Atoms only emit certain spectra due to the set frequency and energy levels

3. Spectra can be produced by collisions such as experiment at beginning of chapter

4. This causes only certain wavelengths to be seen in spectra (suns show certain

colours)

5. This means each element has a certain spectra

Particularly useful to observe a

hydrogen atom

After a long amount of calculations we

find that there are set values for atom

Note that the energies of the stationary states are negative as energy is required to

keep the proton and electron near each other, e.g. it would take 13.60eV of energy

to remove electron from ground state, hence E is called the ionisation energy

It follows that electron must also have a certain momentum of

We can find the wavelength emitted by a hydrogen atom when the state is altered

It turns out these formulas work for all hydrogen-like ions in which there is one

electron and there is an atomic number Z (He has Z=2, Li has Z=3)

Wave functions and Uncertainty

Considering the double slit experiment there is no

definite place where a photon will land.

It is all probability with

Combining this photon model with the wave model we

can decipher that

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This equation is the statement that the probability of detecting a photon at a

particular point is directly proportional to the light wave amplitude function

Now we need to make an assumption that there is a continuous wave function for

matter that is analogous for that of an electromagnetic amplitude function

| |

Only real difference here is that (x) is for particles and A(x) is for photons

From this we can realise that the position of an atomic size particle is not well

defined, however the function (x) defines it, it is like with x(t) in classical physics

Should be noted the particle has a 100% chance of hitting somewhere.

Can split up the place we are observing into many different slits to form an equation

for the number of photons that will hit the area

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| |

Previously we have described objects as either a particle

or a wave but wave packets can account for the various

properties of both

Localisation is particle like

Undergoes diffraction and interaction like wave

Not entirely accurate but helpful for visualising problems

Only approximately accurate because packets come in

many different shapes and no precise definition of the

time and frequency

It is interesting as you cannot define exactly when the wave packet arrives at a point

(would you consider the front or centre)

Because spread out over time you cannot say, therefore known formula is lower

limit and it is actually better to say

Matter has wave like characteristics so this equation must apply to matter also.

Time interval is duration of wave packet as it passes a point

Any sinusoidal wave must satisfy

This leads us to the conclusion that the position and momentum of a particle cannot

be known for certain with the statement

This statement also applies for the y-axis

As a basic this equation implies that if we measure a particles position x, with some

uncertainty then the

smaller this uncertainty

is the larger the

uncertainty for

momentum HAS to be

Schrodingers equation is a method of calculating the wave function as discussed

previously

Considering particle with mass m, and mechanical energy E that can have its

interactions with the surrounding environment characterised in one dimension by a

potential energy function U(x) we have an equation

[ ]

JUST PRAY TO GOD THIS DOESN'T COME UP IN THE EXAM

Permittivity of free space - 0=8.854*10^-12 F m-1

Permeability of free space - 0=4*10^-7 N A-2

Coulomb Constant - k=1/(40)=8.998*109 N M2 C-2

Electron charge - e=1.602*10-19 C

Electron mass - me=9.109*10-31 kg

Proton mass - mp=1.673*10-27 kg

Unified atomic mass - u=1.661*10-27 kg

Planck constant - h=6.626*10-34 J s

Reduced planck constant - hbar=1.055*10-34 J s

Avogadro Number - NA=6.022*1023 mol-1

EQUATIONS

ELECTRICS

Coulombs Law

Electric Field of a Pont Charge

Electric Field of Dipole on Axis

Electric Field of Dipole Perpendicular

Surface Charge Density

Linear Charge Density

Electric Field of Point Charge

Electric Field of Infinite Line

| |

Electric Field of Infinite Plane

Electric Field of Sphere

Parallel Plate Capacitor Electric Field

Note outside the plates E=0

Force on a moving charged particle

Acceleration on particle (assuming electric field is only force)

Circular Motion in a non-uniform electric field | |

Dipole Torque in Uniform Field

(torque greatest when p is perpendicular to E)

Electric Flux

Gausss Law

Charge on a conductor in equillibrium

Electrical Energy

Energy of a system (not particle)

Note: +ve for like charges, -ve for opposite

Potential energy of a Dipole

Theta is angle between dipole and field

Electric Potential Energy (V=Electric Potential)

Potential inside a parallel plate capacitor

Electric Potential of a point charge

Electric Potential of a Charged Sphere

Sum of Electric Potential

Potential Energy

Battery Potential Energy

Electric Potential from Field

Charge on Capacitor

Capacitor in Series (

)

Capacitor in Parallel

Electron current

Current Density

Field Strength in Wire

Resistivity & Conductivity

Resistance

Power of Battery

Energy of battery

MAGNETISM

Biot Savart Law (

)

Small segment of wire

Magnetic field at centre of coil

Magnetic field Dipole

Magnetic Dipole Moment created by current loop

Amperes Law

Magnetic field through solenoid

Magnetic field force on moving particle

Cyclotron Motion

Force on current carrying wires

Force on parallel wires

Torque

Motional emf

Induced current in a wire

Magnetic flux

Faradays Law (induced emf by changing flux)

Faradays Law alternative (for induced electric field)

|

|

Emf in generator

Inductance

Potential difference across an inductor

Ampere-Maxwell Law (

)

Lorentz force Law

Velocity of Electromagnetic Wave

Intensity of electromagnetic Wave

Radiation pressure from EM waves

IMPORTANT EQUATIONS

OPTICS

Velocity of wave

Wavelength of wave

Displacement of a wave at time zero in one dimension

Wave number

Displacement of wave at any time

Index of refraction

Wavelength in material

Doppler Effect for source approaching person

Doppler Effect for source receding from person

Doppler effect for person approaching source ( )

Doppler effect for person receding from source ( )

Maximum constructive interference

Perfect destructive interference

(

)

Youngs Double Slit Experiment

Diffraction Grating

Single Slit Diffraction

Distance and Height relationship

Snells Law of Refraction

Total Internal Reflection (

)

Vergences {

}

Focal Point

Mirror Equation

Mirror Vergence

Lens Vergence

| |

Power for Lens

In is concave, out is convex

Magnification for lens and mirrors

Thin Lens Equation

[

]

Power in air

Angular size of unaided eye

Angular size of object with lens

Angular Magnification

Microscope magnification

Light gathering power

Telescope Magnification

Central maximum

Minimum central maximum

Rayleighs Criterion (minimum resolution of a lens)

X-ray reflection

RELATIVITY

Time Dilation

Length Contraction

Spacetime Interval

Lorentz Transforms

Lorentz velocity Transforms

Relativistic motion

Relativistic energy

MODERN PHYSICS

Classical Photoelectric effect

Energy of a photon

Einstein Stopping Potential

Photon Power

De Broglie Wavelength

Energy of confined particle

Hydrogen Atom radii

Hydrogen Atom electron velocity

Hydrogen Atom Ionisation Energy

Hydrogen Electron Angular Momentum

Wavelength emitted in hydrogen atom

Hydrogen Like Ions (only one electron)

Photon experiment expected photons in an area

Photon experiment expected photons in an area w.r.t light intensity

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Wave Function for matter | |

Amount of matter in an area | |

Wave packet Lower Limit

Wave packet actuality

Heisenberg Principle

Build an empire on methamphetamines and

Schrodingers Equation

[ ]