Photoelectric effect: How to understand the revolutionary experiment just like Einstein did

Experiment: Setup for the photoelectric effect

To demonstrate the photoelectric effect, you basically need four things:

1. Monochromatic light source - with this you create single-color light (e.g. green light). A vapor lamp (with sodium or mercury vapor) is fine for this purpose.

2. Plate capacitor - consists of two metal plates (called electrodes) between which a voltage \(U\) can be applied. For the photoelectric effect, it should also be possible to change the polarity of the voltage. That means: It should be possible to charge one electrode positively (+) and the other negatively (-) and vice versa.

3. Ammeter - is a current meter that allows you to measure the electric current \(I_{\text p}\) between the two electrodes.

4. Voltmeter - is a voltage meter with which you can read off the set voltage \(U\) between the two electrodes.

Light particles for bombarding the electrode

The revolutionary aspect in the explanation of the photoelectric effect is the assumption that the light of the used light source does not propagate wave-like, but particle-like. We assume that the light consists of quite many light particles, which we call photons.

As soon as we illuminate one of the electrodes using the light source, many photons hit the electrode. This can trigger the photoelectric effect, because these light particles have a certain energy, which is then transferred to the electrode. We refer to the energy that a single photon has as photon energy \( W_{\text p} \). In monochromatic light, all photons have equal energy. This amount of energy that a photon has determines whether it can knock out an electron or not.

The light quantum hypothesis established by Einstein states that we can obtain the energy \( W_{\text p} \) of a single photon by multiplying the light frequency \(f\) with the Planck's constant \(h\):

The Planck's constant \( h = 6.626 \cdot 10^{-34} \, \mathrm{Js} \) is a physical constant which appears in the equations whenever nature shows quantum effects. In our case, light shows a quantum effect by occurring in 'small portions'.

Since the Planck's constant \(h\) does not change, you can read off from the equation 2 that the light frequency \( f \) alone determines how large the energy \( W_{\text p} \) of a photon is.

Sometimes instead of the light frequency \(f\), the light wavelength \( \lambda \) ("Lamda") is known. But this is not a problem, because the frequency is related to the wavelength by the speed of light \( c \) as follows: \( f = c / \lambda \). If instead of the frequency \( f \) the wavelength \( \lambda \) is given, then you can express the photon energy 2 with the help of the wavelength as follows:

The speed of light is also like the Planck's constant a physical constant and has the following value:

From the equation 3 you can see that photons with a smaller (shorter) wavelength carry a larger energy! Red light, for example, has a longer wavelength than blue light. The photons of the red light have a smaller energy than the photons of the blue light.

Accelerating and stopping voltage between the electrodes

In order to be able to answer these questions at all, we must first set up the capacitor's polarity appropriately.

If a voltage \(U\) is applied between the two electrodes, the electrodes can be polarized in two ways:

1. The illuminated electrode is negatively charged and the opposite electrode is positively charged. In this case, an electron released from the illuminated electrode would be attracted by the opposite positive electrode. With this polarity, the voltage is called accelerating voltage \(U_{\text B}\) (see illustration 4).

2. The illuminated electrode is positively charged and the opposite electrode is negatively charged. In this case, an electron released from the illuminated electrode would be repelled by the opposite negative electrode. With this polarity, the voltage is called stopping voltage \(U_{\text G}\) (see illustration 5).

When an electron moves from one charged electrode to the other, it gains or loses potential (electric) energy depending on whether accelerating voltage or stopping voltage is applied. The electron with elementary charge \(e\) gains / loses the following electrical energy \(W_{\text e}\) if it passes through the voltage \( U \):

As can be seen from Eq. 4, an electron loses/gains more energy when a larger voltage \(U\) is applied.

• Electron gains energy \( e \, U_{\text B} \) when accelerating voltage \( U_{\text B} \) is set between electrodes. The electron accelerates to the opposite electrode.

• Electron loses energy \( e \, U_{\text G} \) when the stopping voltage \( U_{\text G} \) is set between the electrodes. The electron is decelerated by the opposite electrode.

Ejected electrons generate photocurrent

To determine whether electrons are being ejected, we need an ammeter that measures the electric current between the two electrodes. If the electrode is now illuminated with a sufficiently high frequency of light, the ammeter will indicate a non-zero current. This is exactly the electric current that is used to prove that the light ejects the electrons.

If we switch off the light, then the photocurrent drops to zero. When we switch on the light again, the ammeter shows a value different from zero. The photocurrent \(I_{\text p}\) between the two electrodes can be controlled in two ways:

1. Vary the light intensity: For example, you can use an aperture to regulate the intensity of the incident light and thus vary the number of photons that hit the electrode. This also changes the number of electrons ejected. Doubling the light intensity, and thus doubling the number of photons, results in doubled photocurrent.

2. Change voltage: With the help of a higher accelerating voltage, you can accelerate slower electrons so that they also reach the opposite electrode and thus contribute to the higher photocurrent. With the help of a larger stopping voltage, on the other hand, you can slow down faster electrons, so that they do not reach the opposite electrode and thus the photocurrent decreases.

If the voltage source is switched off, you can still measure a photocurrent with a proper light frequency, because some electrons fly straight towards the opposite electrode after being released. However, keep in mind that not all of the released electrons - with the voltage switched off - land on the opposite electrode. The measured photocurrent is not maximized! The reason for this is that some electrons exit the illuminated electrode at an angle and miss the opposite electrode.

How can you make sure that ALL the electrons ejected end up on the electrode? Or in other words: How can you maximize the photocurrent? This is where the previously learned accelerating voltage comes into play! Increase the accelerating voltage \(U_{\text B}\). This makes the opposite plate even more positively charged. Of course, this results in a greater attractive force on all the electrons that are knocked out. Increase the voltage as long as the photocurrent \(I_{\text p}\) also increases. If you increase the voltage so that the photocurrent at the ammeter does not change any more, i.e., goes into saturation, then all electrons land on the opposite electrode. The electrical attraction of the opposite electrode on the electrons becomes so strong that even the obliquely emitted electrons are 'sucked in'. The photocurrent reaches a maximum possible value \(I_{\text{max}}\).

Overcome work function of the illuminated electrode

In the photoelectric effect experiment, it is found that not all light is able to eject electrons. The electrons can move freely within the metallic electrode, but they cannot move out of the electrode because they are bound to it. This binding of the electron to the electrode can be overcome if you add enough energy to the electron so that it can be knocked out of the bond. But if you don't add enough energy to the electron, then of course it will continue to be bound and will not come out of the electrode.

The work function \(W\) differs depending on which material the electrode is made of. An electrode made of nickelium has a larger work function than an electrode made of aluminum. Electrons are more strongly bound to the nickelium electrode than to the aluminum electrode. Consequently, it will be more difficult to release electrons from the nickelium material. Here are a few examples of different materials:

You probably noticed from Table 2 that the work function was not only given in Joules (J), but also in Electron volts (eV). This is a typical compact energy unit in which the work function in the photoelectric effect and other quantum mechanical effects is given.

• Divide the energy in Joules (J) by the value of the elementary charge \( 1.6 \cdot 10^{-19} \) to express it in Electronvolts (eV).

• Multiply the energy in Electronvolts (eV) by the value of the elementary charge \( 1.6 \cdot 10^{-19} \) to convert it to Joules (J).

How do you supply the electron with this required energy? So how do you overcome this work function and thus also the binding of the electron to the electrode? Now, photons come into play. You have learned that the photon carries an energy \( W_{\text p} \), which is determined by the light frequency \( f \) or wavelength \( \lambda \). Now there are two possibilities which can occur:

1. Photon energy is smaller than the work function: \(W_{\text p} < W \).
   In this case, the photon cannot eject an electron.

2. Photon energy is greater than the work function: \(W_{\text p} \geq W \).
   In this case, a photon ejects an electron.

Photoelectric effect equation as a consequence of conservation of energy

But what if you take a higher frequency of the light than the cutoff frequency \( f_0 \)? Or equivalently: What if the photon energy is much greater than the work function: \( W_{\text p} \gt W \)? A part of the photon energy (namely work function \( W \)) is needed to overcome the bond of the electron to release it from the electrode. We subtract this work function \(W\) from the photon energy. But there is still a remaining energy:

Where is this remaining energy \( \Delta W \)? After all, it cannot simply disappear according to the energy conservation law!

In the experiment, it is observed that if you use light with sufficiently large frequency, then the ammeter will show a non-zero electric current: \( I_{\text p} \neq 0 \). As it looks, electrons released still have a velocity in the direction of the opposite electrode. And there we have our remaining energy! The remaining energy \( \Delta W \) is given to the electron in the form of kinetic energy \( W_{\text{kin}} \). The remaining energy is the kinetic energy of the ejected electron: \( W_{\text{kin}} = \Delta W \).

You can substitute the photon energy 2 into eq. 10:

To find out the concrete velocity \( v \) of the released electrons, you use for \(W_{\text{kin}}\) the classical formula for kinetic energy:

Here \( m_{\text e} ~=~ 9.109 \cdot 10^{-31} \, \mathrm{kg} \) is the rest mass of the electron. Rearranged for the photon energy, you get the famous formula for the photoelectric effect, for which Albert Einstein got the Nobel Prize:

The velocity \(v\) in 13 is not directly measurable, so we express it with the measurable quantity, using the stopping voltage \( U_{\text G}\).

To determine the value of the stopping voltage, you must increase the stopping voltage very slowly in the experiment and observe how the photocurrent decreases. In this way, fewer and fewer electrons reach the opposite electrode. At some point, you will reach a value of the voltage at which the repulsive force of the opposite electrode is so large that not a single electron reaches the electrode. The photocurrent has now dropped to zero. You have slowed down the last fastest electrons. At this value of the stopping voltage the fastest electron has lost the energy \(e \, U_{\text G}\). This lost energy just corresponds to its kinetic energy:

You may replace in this way the kinetic energy in 13 with \(e \, U_{\text G}\):

Using Eq. 15, you can obtain the work function of the material for a given light frequency/wavelength and the stopping voltage read on the voltmeter. Or, for a given work function and stopping voltage, figure out the light frequency / wavelength.

Note, however, that you are not allowed to increase the stopping voltage further, so that the equation 14 is valid. Further increasing of the stopping voltage does not change the photocurrent, because it is already zero. You have to set exactly the reverse voltage value at which the photocurrent just becomes zero!

Energy-frequency graph for the photoelectric effect

You can illustrate the equation 11 in a graph:

Kinetic energy \( W_{\text{kin}} \) is plotted on the \(y\) axis and light frequency \( f \) is plotted on the \(x\) axis.

If you look closely at the equation 16, you will see that it is a straight line equation of the form \( y = mx+b \). In our case:

• \(y\)-axis is the kinetic energy: \( y = W_{\text{kin}}\).

• \(x\)-axis is the light frequency: \(x = f\).

• The slope \(m\) of the straight line is the Planck's constant \( m = h \).

• \(y\)-intercept \(b\) is the negative work function: \( b = -W \).

To get the intersection of the line with the \(x\) axis, you must set the kinetic energy to zero in Eq. 16: \( W_{\text{kin}} = 0 \). After rearranging the equation, you will find that the intersection point corresponds to the cutoff frequency \( f_0 \) (see illustration 6).

With this knowledge you are able to determine the Planck's constant \(h\) with the help of the photoelectric effect.

You hopefully know from mathematics how to determine the slope of a straight line:

In our case, the slope corresponds to the Planck's constant: \( m = h\). Here \(y_{2}\) and \(y_{1}\) are two arbitrary values \(W_{\text{kin,1}}\) and \(W_{\text{kin,2}}\) for kinetic energy of electrons. And \(x_{2}\), \(x_{1}\) are associated light frequencies \(f_2\), \(f_1\). So you can rewrite equation 17 like this:

Unfortunately, the kinetic energy of electrons is not directly experimentally accessible. Since you perform the experiment in a plate capacitor, you can use it to rewrite the kinetic energy with an experimentally accessible quantity. This quantity is the stopping voltage \(U_{\text G}\) between the two electrodes.

In this case, the kinetic energy corresponds to the electrical energy: \(W_{\text{kin}}=e \, U_{\text G}\). Here, \(U_{\text G}\) is the stopping voltage necessary to completely decelerate the electrons. Substituting electrical energy into eq. 19 yields:

Investigate the photoelectric effect with two different, known light frequencies \(f_1\) and \(f_2\), read off the corresponding stopping voltages \(U_{\text{G},1}\) and \(U_{\text{G},2}\). Use 19 to determine the Planck's constant \(h\) experimentally.

Photoelectric effect shows contradictions to wave theory

What is observed in the photoelectric effect experiment cannot be explained by the assumption that light is wave-like.

According to the classical idea, electrons should be able to be ejected with any light frequency \( f \). Something like a cutoff frequency \( f_0 \) does not exist in the wave theory of light. However, this contradicts what is observed in the experiment.

According to classical wave theory, you could simply illuminate the electrode with an arbitrary light frequency \( f \) for longer to supply more energy to the electrode. The photoelectric effect would then occur after a time delay. In the experiment, however, we observe no such behavior: No matter how long you illuminate the electrode, no electrons can escape if the light frequency is too small.

According to the classical wave theory, the kinetic energy of the electrons \( W_{\text{kin}} \) should decrease with increasing light frequency \( f \), because in the classical wave picture the following proportionality holds: \( W_{\text{kin}} \sim 1/f^2 \). The kinetic energy should even decrease quadratically. But what one finds in the experiment to the photoelectric effect is an increase and no decrease of the kinetic energy.

Moreover, according to the classical idea, the kinetic energy should increase quadratically with the amplitude of the incident light: \( W_{\text{kin}} \sim A^2 \). However, the experiment shows a different behavior: Increasing the amplitude \(A\) (increasing light intensity) has no effect on the kinetic energy of the electrons.