In this unit we concentrate on the interactions between ...



In this unit we concentrate on the interactions between photons and electrons. We start with some properties of each particle. Finally we look at how the two behave together. This period of history starts around the 1880’s and will take us to about 1927.

Properties of each

1 Electrons

1 J.J.Thomson

2 Robert Millikan

2 Photons

1 Max Planck

2 Albert Einstein

Photoelectric Effect

1 Experiment and Data

2 Results and Conclusions

Energy Level Diagrams

1 Absorption/Emission spectra

2 Bohr Model

De Broglie Wavelength

X-Ray Production

Compton Scattering

Electrons & Photons

We start this unit with a look at the two key particles that are affected by the ideas and concepts of Modern Physics. A particle model of light will be developed that describes each particle as a photon. In 1897 the electron was discovered. In this lesson we review the properties of both particles.

The Electron

In 1897 J.J. Thomson used a crude particle accelerator/ mass spectrometer to measure the properties of a negatively charged particle. The best that he could do was to arrive at a value of the charge to mass ratio of the particle. About a decade later, Robert Millikan developed the now famous “Millikan Oil Drop Experiment (pp 483-485). The result of this experiment determined the charge of an electron to be q = -1.6E-19C. Now coupled with the mass to charge ratio of Thomson’s work it was determined that electrons have a mass of +9.11E-31Kg with an immeasurably small size.

The Photon

In the last quarter of the 1800’s scientists had been working on a way to take what they knew about mechanics, thermodynamics, electricity & magnetism and optics to describe how hot objects cool by giving off light in various parts of the electromagnetic spectrum. The study of this effect is known as blackbody radiation.

If an object is hotter than its surroundings it will cool by giving off light. In order to study this effect scientist had to eliminate the other modes of cooling. Blocks of graphite were hollowed and a small hole was drilled into the carbon. Although the outside of the carbon block would cool by convection as well as radiation, the inside would cool by mostly radiation alone. The intensity of each wavelength of light emitted by the inside of the hot, black boxes was studied, hence the name- blackbody radiation. It was noticed that at each temperature there was a specific wavelength that was most intense. Wavelengths slightly higher or lower would be less intense so that radiation curves could be plotted for different temperatures of the carbon. See figure 27.2. Observe that as the temperature gets higher the curve becomes more peaked with a shorter width. Also notice that the maximum value of the peak moves to a shorter wavelength as the temperature rises. A scientist named Wien determined that you could determine the temperature of an object by merely measuring the brightest wavelength emitted and then using formula [27.1] to find the temperature of the object. Chemists, metallurgists and astronomers all used Wien’s formula but no classical theory would completely explain the other parts of the curve.

In 1900 Max Planck developed a mathematical model that fit the data without using any known theory. The idea was to find what function worked and let it tell him what the theory should be. Much to the surprise of Planck, the mathematical model that worked called for light to be jumping off the hot objects in

bits and pieces like particles instead of waves. Each bit of

energy, called “quanta then but photons now” had a specific

value of energy that is determined using the formula in the

box to the right. This is the equation for the energy of a

photon. Store the value of Planck’s constant, h=6.63E-34Jsec, in your calculator under [alpha], [H]. Planck publicly refused to believe his own conclusion and makes mention of using the equation “until a better idea is found”.

In 1905 Albert Einstein tells of a few more properties of the particle of light called “photon”. Photons travel at v=c in a vacuum and there is nothing you can do to make photons go faster or slower. Photons have no rest mass, which is unique to the photon so far as we know. Although photons seem to have no

mass they do carry momentum with them. In order

to calculate a photon momentum use the equation in

the box to the right. In 1905 the above equation worked for photons only. Particles that had mass still used p = mv. Do not attempt to use p =mv on a photon because you have no value for “m”. By combinations of E=hf

and c=f( one can arrive at finding photon energy

in terms of wavelength or momentum as well as

frequency. The product of Planck’s constant and the speed of light show up so often that the AP exam will have a value listed in the constants table. The above equations work only for the mass-less photon. Do not attempt to use these on particles that have mass like electrons and protons. Given any one of frequency, wavelength, momentum or energy of a photon you should be able to calculate the other three values.

Suppose that you are given information concerning the power of a monochromatic source. Could you find all of the above as well as the number of photons/sec being radiated? We finish this lesson with an example problem.

Example Problem #1

A 3-milliwatt pen laser radiates at 633 nm. Find values for the following:

a) Frequency of light emitted,

b) energy of a single photon in joules,

c) energy of a photon in electron volts,

d) momentum of a single photon,

e) number of photons emitted in a quarter second burst of laser light and

f) length of the beam.

a) c = f( or

f = c/( = 3E+8m/s / 633E-9m

f = 4.74E+14 Hz

b) E = hf = 6.63E-34Jsec(4.74E+14s-1)

E = 3.14E-19 J

c) E = hf = 4.14E-15eVs(4.74E+14s-1)

E = 1.96 eV

d) p = h/( = 6.63E-34Jsec/ 6.33E-9m

or p = E/c = 3.14E-19J / 3E8m/s

p = 1.05E-27 Nsec

e) Power = Nhf where N is photons/sec.

N= Power/(hf)= 3E-3J/sec/ 3.14E-19J

N = 9.55E+15 photons/sec

Total = 9.55E+15 photons/sec(0.25s

= 2.39E+15 photons

The length of the beam, (f) can be found using d=ct = 3E8m/s(0.25sec) = 75E+6m. Once you divide the length by the number of photons you find that there are about 20 photons/meter of beam.

Photoelectric Effect

While verifying that Maxwell’s electromagnetic waves are the same form as light waves Heinrich Hertz noticed that under the right conditions UV light could cause sparks to fly from metal surfaces. This phenomenon was labeled photoelectric effect. The irony of Hertz’s discovery is that while doing the very last step in the history of light as a wave he had also taken the first step towards for light as a particle nature!

Experimental Setup

Shown below is a slight variation on figure 27.4. The idea is that you have a light source that you can vary in both intensity and frequency. In the circuit you have a variable voltage source that is used to vary the amount of voltage needed to suppress electrons from jumping from the emitter to the collector.

▪ With the battery turned off and the light source turned on the light can cause electrons to be knocked off the surface of the emitter and onto the surface of the collector.

▪ The ammeter will read the photocurrent. If 20 million electrons/sec are ejected from the emitter to the collector then the ammeter will read a photocurrent of 3.2 pAmps.

▪ Increasing battery voltage creates a more negative collector reducing the photocurrent.

▪ Finding the voltage that reduces the photocurrent to zero yields the stopping voltage or VS. VS can be used to find the maximum kinetic energy of the ejected electrons according to e(VS) = KEMAX. If the stopping voltage is 3.4 volts for example then the maximum kinetic energy is 3.4 eV or 5.44E-19J. See equation [27.4].

The two independent variables are the light intensity

and light frequency or wavelength of the source.

These values are set and then the experiment is run in

order to measure dependent variables such as stopping

voltage and photocurrent. Some experiments with

time are also critical. The relationships among the four variables in the above table are briefly described. (1) The intensity of the light has no affect on the stopping voltage. This contradicts the wave model of light because a greater intensity is equivalent to waves of higher amplitude and therefore more kinetic energy of the ejected electrons. (2) The intensity of the light source is linear with the photocurrent. (3) The frequency of the light source is linear with the stopping voltage as long as the source is above a certain frequency. See figure 27.6. Below a certain frequency no electrons are ejected no matter how long the experiment runs and no matter how intense the light source is. The minimum frequency to cause ejected electrons is called the cut-off frequency. The cut-off frequency is different for different metals. (4) Frequency has no bearing on the photocurrent. (5) The photocurrent develops within nanoseconds of the light source being turned on or not at all. The “start-up” time for photocurrent should be frequency or intensity dependent according to wave models since a lower frequency requires more delivery time for waves in order to get enough energy to free electrons from a metal.

Albert Einstein’s Explanation

In 1905 Albert Einstein gave a very simple explanation of the photoelectric effect. He was eventually awarded the Nobel Prize in Physics for “His explanation of the photo-electric effect and other contributions in physics.” According to Einstein, light is acting like particles that eventually became known as photons. Each electron can absorb a single photon. When you increase the intensity of light more photons are created and liberate more electrons. This explains the liner relationship between the intensity of the light source and the photocurrent.

A single photon brings to the metal surface an energy of E( =hf using Planck’s equation for photon energy. Recall that only five years earlier Planck and the rest of the world refused to publicly accept that this equation was correct. Some of the photon energy is used to free the electron from the metal surface. The amount of energy needed to free the electron from the surface is called the “work

function” and is designated (. See Table 27.1. The value

of ( is different for different metals. Once the photon

liberates the electron from the surface, remaining energy shows up as kinetic energy of the ejected electron. Suppose that a 6 eV photon strikes the surface of a metal where the work function is 1.5 eV. What is the maximum kinetic energy of the ejected electron? The answer is 4.5 eV. Suppose that a 1.3eV photon strikes the same surface. What happens? Since the photon energy is below the value of the work function no electron is liberated. This is the way in which the cut-off frequency is explained. Replace the equation for the kinetic energy of the electron with e(VS).

Use algebra to isolate the stopping voltage. You get the

equation shown in the text box to the right. If Einstein is

correct then every metal should give a graph of stopping

voltage as a function of frequency that has the same slope. The slope should be Planck’s constant divided by the charge of the electron. This is exactly the slope of the graph for all metals so Einstein was correct. But wait, there’s more! Different metals should have a different y-intercept. According to Einstein the negative y-intercept of any metal is its work function divided by the charge of an electron. He was correct again. The cut-off frequency is the x-intercept of the graph. If Einstein is correct the cut-off frequency for any metal is the work function divided by Planck’s constant. Again this is true! Albert Einstein had the courage to use Planck’s equation in his explanation. A simple but bold explanation was presented to the world. Review example 27.4. Note that if you are in eV’s then the charge of the electron is 1 in the above equations and you must use the correct value of “h”.

X-Rays and Wave Particle Duality

After 1905 the world admits that electrons can absorb photons. But can electrons emit or radiate photons in a manner opposite to the photoelectric effect? A bigger question should have arisen immediately after 1905. Light can sometimes act like a wave and sometimes act like a particle; does any other particle also have a wave-particle duality? These two questions are the focus of today’s lesson.

X-Ray Production

In 1895 Wilhelm Roentgen generated the first man-made x-rays. X-rays are the highest energy photons that are associated with electron emission. The diagram of an x-ray machine is very simple. Refer to figure 27.8a. A high voltage is used to accelerate electrons from rest. The final kinetic energy of the electrons depends on the accelerating voltage according to W=q((V or e(V. Upon gaining all of the kinetic energy the electron can slow down by emitting one or more energetic photons. Suppose for example that an electron has 1000 eV of kinetic energy. The electron could emit a single 1000eV photon or two 500eV photons or three 333eV photons or so forth. In order to have the electron give up its energy it must decelerate or stop. In figure 27.10 electrons slam into a Tungsten target giving up most of their kinetic energy in a short amount of time by x-ray production. There is no way to predict if individual electrons will emit a single photon or two or three or so forth. The way to get around the problem is to add words like “maximum frequency” or “minimum wavelength”.

The use of superlatives for extremes implies that all

of the electron kinetic energy is emitted in a single

photon in order to produce the highest energy and smallest wavelength. Be sure to use the correct form of Planck’s constant in the above equations. Review example 27.5 and the following example problem.

Example Problem

A dental x-ray machine has been set to 120kV at 12 mA. Assume that a minimum number of photons are emitted in a 4(sec burst from the x-ray machine.

A) What is the wavelength for the x-rays produced?

B) How many x-ray photons were generated?

A) From above equation, ( = hc/(e(V) = (6.63E-34Js(3E+8m/s)/(1.6E-19C(120,000J/C) = 10.4pm. Alternatively, ( = hc/(e(V) = (4.14E-15eVs(3E+8m/s)/(120,000eV)=10.4pm.

B) (12E-3C/sec)((1electron/1.6E-19C)((1photon/1electron)(4(sec = 300E+9 photons or about 300 billion x-ray photons that are quite revealing!

Wave-particle Duality for All

After the study of blackbody radiation, photoelectric effect and Compton scattering (tomorrow’s lesson) there is no doubt that light sometimes behaves as a wave and sometimes behaves as a particle. Up until the early 1900’s it was believed that if something acted like one then it could not behave in the other sense. This is because a particle is confined to a very small, finite space and can be described as having a single position at any single value of time. Particles are concentrated in a small space. Waves on the other hand distribute their energy and their position over many points in space at the same instant. How can something have a single position for a single instant of time like a particle and for some other circumstance be everywhere at once? But light does indeed act sometimes like the particle and sometimes like a wave.

I would like to interject the following bit of advice. Return to figure 21.22 on page 675. You recall having to know the entire spread of the electromagnetic spectrum? At the longer wavelengths like radio and microwaves the wavelength is much too large to easily observe the particle nature. Physicists who work at this end of the spectrum think almost entirely as “light acts like a wave”. They are very much concerned with diffraction and interference and so forth. Consider the opposite end of the spectrum. An x-ray is on the order of the size of an atom and gamma rays are on the order of the size of the nucleus. Physicists at this end of the spectrum think almost exclusively that light is made of particles called photons. And particle views work very well here with the photons having such a short wavelength. Indeed the energy is focused in a very small region of space much like a particle. And guess where we are most observant as humans? In the middle where both wave and particle characteristics are both easily seen.

In the early 1920’s Louis de Broglie (Fresnel was also from Broglie, France) proposed that if wave-particle duality was good for light then why not for matter particles as well? The idea that momentum can be used to

determine the wavelength of any particle, even

those with mass such as electrons was being proposed.

If you study the two example problems from the text, examples 27.8 & 9, you will see that the authors stress the following idea. Objects with large, everyday masses will have very small wavelengths and appear almost exclusively to behave like particles. But as masses approach smaller values such as those for electrons the wavelengths can become quite noticeable. This is true because Planck’s constant is so small, (E-34Jsec). Had the value been larger we would have realized the ideas of quantum mechanics much sooner.

In 1927, Davisson and Germer performed experiments where low energy electron beams were fired at nickel crystals. The scattered electrons from the beam were distributed in a pattern similar to that of x-rays with the corresponding wavelength. Davisson and Germer had experimentally verified de Broglie’s hypothesis. Now all objects have wave-particle duality. This should cause a brief bit of panic and anxiety on the reader’s life. You do know what happens when a wave passes through vertical slit? According to diffraction the wave is spread horizontally across space. So what happens to a student walking through a vertical door if he/she exhibits their wave nature at the precise moment of entering the doorframe? Do they get diffracted all over the hallway? And who cleans up the mess? Do not worry. Your wavelength is much smaller than the width of the door so that diffraction problems are not a concern. But if you were to pass through a door at 1 m/s that had a width comparable to your de Broglie wavelength then you may actually diffract. To observe this however you must get through a door smaller than the width of an atom and that will never happen at your school!

Compton Scattering

In the early 1920’s Arthur Compton demonstrated that x-rays scattered by electrons or other charged particles will have a longer wavelength due to transfer of momentum and energy from the incident photon to the charged particle. Refer to figure 27.16. All of the initial momentum and energy is in the incident photon. The following statements can be made about the system. The initial momentum in the x direction is px = h/ (o. The initial momentum in the y-direction is py = 0. The initial energy is E = hc/ (o. The following three equations can be established from the Laws of Conservation of Momentum and Energy:

Conservation of Momentum for x-direction: h/ (o = h/ ( (cos () + meve (cos ()

Conservation of Momentum for y-direction: 0 = (h/ ( (sin () + meve (sin ()

For the previous equation use absolute value of angles since negative signs are already in

place.

Conservation of Energy: hc/ (o = hc/ ( + ½ mev2

The previous steps leave you with three equations for finding up to three unknowns. In tpast the AP exams equations are decoupled the by giving scattering angles and so forth.

There is an alternate approach to setting up equations using a momentum triangle for the vectors of initial and final momentum. After drawing the triangle you can use Law of Sines, Law of Cosines and Conservation of Energy to determine unknowns. If this approach is used write the electron momentum as pe and electron kinetic energy as pe2/2me.

p(o

( (

p( pe

Recall that p = hc/( for initial and final photons but p=mv for electron.

Using either the components method for momentum or the Law of Sines and Law of Cosines method for momentum leads to the following result as stated in your textbook without proof. If a wavelength is not the initial then it

must be the final wavelength. The equation shows the

shift in wavelength of the scattered photon with

reference to the photon scattering angle. The constants

in front of the equation combine to have a unit of length. The value is 2.426 picometers and is known as the Compton wavelength. Beware of scattering high-energy photons off of protons. That would require use of a different mass in the above box and would lead to a different coefficient for h/mc. Review example 27.7. Warning about doing the homework, the above-boxed equation is nice to use but it is not given on most exams where the previous equations may be more efficient. Try to use the basics on some of the homework problems (blue ones) in order to prepare for your exam.

Energy Level Diagrams

Atomic Structure

When the early work on the electron was competed by Thomason and Millikan a model of a neutral atom had to be constructed. The model had to explain a neutral charge state for the atom while acknowledging the existence of electrons. Thomson’s model, a.k.a. the plum pudding model, is shown in figure 28.1.

A few years later, two students of Ernest Rutherford scattered alpha particles off gold foil (figure 28.2) and determined that the atom had a very small, very dense, positively charged nucleus. A new model had to be constructed that accounted for the positive nucleus and the electrons. The Rutherford model proposed that the electrons orbited the positive nucleus of a neutral atom much like planets orbit the sun. Rutherford’s planetary model had one big flaw. According to the laws of mechanics and Maxwell’s equations accelerating electrons should emit light and lose energy. Any electrons orbiting the nucleus should continuously emit light spiraling into the nucleus. A new model had to be constructed that accommodated the positive nucleus, negative electrons and have stable orbits that did not spiral into the nucleus. The new model that replaced the planetary model is known as the Bohr Model after Niels Bohr. In this lesson we focus upon the structure of the atom and the orbiting electrons. In the next unit we will turn our attention to the nucleus of the atom.

Atomic Spectra

Since the mid 1800’s chemists and astronomers had known that each element of the periodic table would emit a unique color pattern when heated at low pressure. By passing light from a heated gas through a prism or diffraction grating the colors can be dispersed or resolved. The color patterns of hydrogen, mercury and neon are shown on page 867, figure 28.3. The top three patterns are known as emission spectra since they are the patterns of a gas emitting light. Since stars tend to heat gas we can look at light from a star and determine their chemical composition from their emission spectra. When this was done to our own star a pattern of light was noticed that had never been seen. A new element was discovered, Helium.

Other astronomers were using absorption spectra instead of emission spectra. What they had discovered is that if you pass white light through a cold gas all of the wavelengths will get through the gas except for those same wavelengths that were emitted for emission spectra. Compare the pattern of emission spectra of hydrogen at the top of figure 28.3 with the white light passed through cold hydrogen on the 4th row of figure 28.3. The absence of lines in a continuous pattern is known as absorption spectra.

Hydrogen had the simplest pattern of lines and was therefore thought to have the easiest atomic structure of all atoms. Hydrogen actually has three distinct groups of emission lines. One group of lines is entirely in the UV part of the spectrum and is known as the Lyman Series. A second group of lines is in the visible part of the spectrum and is known as the Balmer Series. A third group of lines falls in the infrared and is known as the Paschen Series. Figure 28.4 shows the visible lines of the Balmer series and some of the UV lines of the Lyman Series.

Bohr Model of Hydrogen

In 1913 Niels Bohr developed a model for hydrogen that explained the existence of electrons, a positive nucleus and the correct wavelengths of all three groups of the hydrogen emission spectrum. Although the AP exam no longer requires you to know the assumptions of the model they are still worth reviewing. The assumptions are listed on page 968. Most notable of these assumptions to us are the first three. They can be summarized as follows.

▪ Electron orbits are only allowed at certain distances from the nucleus. While in these allowed orbits the electron will not emit light.

▪ Electrons only emit light if they make a transition to a smaller orbit. The photon energy of the emitted light will account for the

electron energy difference between initial and

final orbits according to the formula to the right.

Energies to the right of the equals sign are for the orbital energies of the electron.

▪ Electrons can absorb photon energies but only if it is the exact amount to raise the electron to an allowed orbit of greater radius.

Using these assumptions and starting with classical mechanics Bohr was able to calculate the radius and the energy of each allowed orbit. The allowed orbits are numbered from n=1 to n=( with a smaller number representing a smaller orbit.

Allowed Radii

The electron for hydrogen can orbit in any of the following radii but nowhere in between these values.

The value of ao is known as the Bohr radius. Bohr was able to calculate the smallest radius of the hydrogen atom from the constants as shown in equation [28.10]. This value had already been measured due to corrections of the ideal gas law. Bohr knew he was on the right track when the correct value appeared.

Allowed Electron Energies

When in any of the orbital radii listed from the previous column the electron will have energies as listed below.

Again, the energy required to ionize hydrogen had already been known. This was another sign that Bohr was moving in the correct direction for describing atomic structure. Using known constants the ionization energy of 13.6eV was found from equation [28.12].

Energy Level Diagrams

Drawing the orbits of the electron to scale becomes a problem due to the “n2” nature of the function. But something had to be done to describe the relationship of the electron orbits before and after a transition. Both figures 28.7 and 28.8 demonstrate the idea of energy level diagrams although the scales have been exaggerated or compressed in different places.

Picture the electron trapped in an orbit about the nucleus of hydrogen as being stuck at the bottom of a well or hole in the ground. The depth of the hole represents how much more energy the electron needs to get free of the nucleus. If the electron is in the smallest orbit, n=1, then it needs the most energy, -13.6eV. If the electron is in orbit n=2 then it is farther from the nucleus and needs less energy to get free, -3.4eV. In other words it is no longer at the bottom of the hole but closer to the top, where E=0eV and the electron is free.

We can map out the energy level diagram by drawing horizontal lines at various depths to represent the various energies. Tape two sheets of paper together on the 8.5 inch edges to make one longer sheet about 21 inches long. Use a scale of 1eV = 3cm and construct the energy level diagram of hydrogen for the first 6 orbits of hydrogen. Compare your energy level diagram drawn to scale with figure 28.8.

Graphing Calculator Exercise

Energy level diagrams can be mapped out on a graphing calculator by taking the following steps. A) Go to [STAT], [1] and enter the numbers 1,2,3,4,5 under list one, L1. B) Now move the cursor of your calculator until it is exactly on top of the heading of list 2, L2. C) Type –13.6/ L12 and [ENTER]. Note that L1 is [2nd], [1] on your calculator. You now have in the second list energy values for the first five orbits. I have also listed the orbital radii for the first five orbits in nanometers in list 3.

We are drawing horizontal lines so the x scale is really not important but leave room for labeling n values and E values. I recommend the window below.

Graph the following in the [y=]:

Y1= L2/ (x ( 0) /(x ( 1)

Even with the graphing calculator, the pixel height becomes significant just like pencil width on your exercise with paper and meter stick.

Transitions

With energy level diagrams constructed electron jumps are easier to describe and calculate. In figure 28.8 for example the electron jumps for the Balmer series are shown. If an electron falls down in an energy level diagram photons are emitted equivalent to the difference in energy levels of before and after. If an electron moves up the energy level diagram then photons of the proper energy must be absorbed. Some examples must be described. Refer to figure 28.8 for all of these descriptions.

1) The electron in figure 28.8 jumps from n=3 to n=2; a 1.89 eV photon is emitted.

2) The electron going directly from n=4 to n=2 will emit a single, 2.55eV photon.

3) An electron from n=4 could do two jumps to get from n=4 to n=2. The electron could emit a single, 0.66eV photon to go from n=4 to n=3 then emit a single 1.89eV photon to go from n=3 to n=2.

4) To move the electron from n=2 to n=5 requires the hydrogen atoms to be exposed to 2.86eV photons.

5) If electrons in the n=2 state are exposed to photons with an energy of 5.0eV the electrons will be freed from the nucleus with a resulting kinetic energy of 1.6eV for the electrons leaving the atom in an ionic state.

Bohr Model Failures

With Bohr’s model he was able to account for all of the lines of the Lyman series as transition to n=1 states, hence the UV nature. The Balmer series are all transitions to n=2 states and the Paschen series are all transitions to n=3 states. He also gave us a theoretical value for the both hydrogen radius and ionization energy. But the model did not work for all atoms, merely hydrogen like atoms or atoms with a single electron.

Bohr made some mistakes in his assumption. The biggest was that he treated the electron as a particle instead of a wave. Almost one decade later de Broglie pointed out this flaw in the Bohr Model. A few years after that Schrodinger came up with a three dimensional wave equation that when applied to orbiting electrons gave us the entire periodic table and the fuzzy, electron-cloud model of the atom as we know it today. There were other minor corrections including relativistic effects. So why is this model so famous if it works so infrequently? We recognize his great intuition because he led us into the idea of photon emission and absorption only during electron transitions.

-----------------------

E = h f where

h = 6.63E-34 Js

h = 4.14E-15 eVs

E = hc/ ( or E = pc

p = h / (

Variable Light

Source emitter

V

Collector

A

|Variables |VS |IPC |

|Intensity |1 |2 |

|Frequency |3 |4 |

hf = KEe + (

e((V) = hfMAX = hc/ (MIN

p = mv = h/(

(( = h/(mec) (1 - cos()

where h/(mc) = 2.426pm

for electrons

hf = Eo - Ef

rn = n2 ao where ao = 0.0529nm and n = 1, 2, 3, 4, …

En = ( 13.6 eV where n=1,2,3…

n2

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