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Subsection 53.4.1 Lenard and Millikan's Experiments

Figure 53.4.1 shows a schematic diagram of an apparatus similar to the one initially used by Lenard and later by Millikan to study various aspects of the photoelectric effect. A clean metal plate to be exposed to light, labeled E for emitter, is mounted in a vacuum tube along with a collector cup labeled C. The tube is also called a photocell. The two electrodes of the photocell are then connected to a variable power supply V and a galvanometer G as shown. The photocell is mounted so that light can shine on the emitter plate without illuminating the collector cup as well. The cleanliness of the metal surface was an important factor as oxidized metal surface have different characteristics. Even in best vacuum available in early 1900s the metal surfaces would get oxidized within tens of minutes which made it difficult to conduct experiments consistently and reliably.

When light strikes the emitter plate, it energizes the surface electrons sufficiently that some of the surface electrons escape the surface, leaving the surface positively charged. Some of these freed electrons, called photoelectrons, get pulled back to the plate while others with sufficient kinetic energy can overcome the attraction of the positive plate. When photoelectrons reach the collector they flow in the external circuit through the galvanometer, which registers the current, called the photocurrent, in their way to the emitter plate.

The connection to the external power supply can be made to either aid the flow of the photoelectrons inside the tube or impede the flow. If plate E is at a negative potential compared to cup C, the photoelectrons are accelerated towards the collector cup. That means if enough energy is absorbed so that any electron can be freed, they will make it to the collector, and therefore, there will be photocurrent. [Aside: If emitter is made sufficiently negative an electron can be ejected from the plate even without any light incident on the plate but that effect has nothing to do with the photoelectric phenomenon.]

On the other hand, if power supply is connected so that the collector plate is at a negative electric potential compared to the emitter plate, then the ejected electrons are pushed back to the emitter plate by the electric field. In this setting, the potential impedes the flow of electrons from the plate to the collector. This potential is therefore also called the impeding potential. When the impeding potential is increased, fewer and fewer electrons have high enough kinetic energy to make it to the collector and the photocurrent drops. At a particular potential \(V_0\text{,}\) called the \stopping potential, none of the freed electrons can make it to the collector and the current drops to zero. If the potential is increased further, the current remains zero. That means that the kinetic energy of the most energetic electron, \(K_{\textrm{max}}\text{,}\) is equal to the potential energy of the electron when \(V=V_0\text{.}\)

Result 1: Observation of Stopping Potential

Figure 53.4.2 shows several photocurrent vs \(V\) plots at different wavelengths incident on a metal plate similar to the graphs in Millikan's original paper. The figure shows the existence of stopping potential at these frequencies in the photoelectric effect and the role of intensity when photon has sufficient energy to cause an emission of an electron. Millikan's data shows that stopping potentials depend on the frequency of light. Millikan also observed that stopping potential depended on the material.

Result 2: Variation of Stopping Potential with Frequency

The stopping potential varies linearly with frequency as shown in Figure 53.4.3. That is, the maximum kinetic energy of freed electrons depend on the frequency of light and not on the intensity of light contrary to what would be expected from classical considerations. According to Einstein's photoelectric equation, Eq. (53.4.3) derived below, the slope of this graph is equal to \(h/e\text{,}\) where \(h\) is the Planck constant and \(e\) is the absolute value of the charge on an electron. Millikan used the slope with the value of \(e\) he had found from his oil-drop experiment to deduce the value of the Planck constant experimentally to be \(6.56\times 10^{-34}\:\textrm{J.s}\) , which was in good agreement with the theoretical value of \(6.63\times 10^{-34}\:\textrm{J.s}\text{.}\)

Summary of Various Results of Lenard

1. For a given metal, no electron is emitted unless the frequency of light is above a cutoff value, called the \textbf{threshold frequency} regardless of the intensity of the incident light. This is taken to define a property of each metal called the work function.
2. For a given metal, increase in frequency of the incident light above the threshold frequency increases the photocurrent.
3. For a given metal and for a particular frequency above the threshold frequency, the photocurrent increases linearly with the intensity of light.
4. For a given metal and for a particular frequency above the threshold frequency, the stopping potential \(V_0\) is independent of the intensity of light although more electrons are released with a bright source than with a dim source of the same frequency.
5. For a given metal the stopping potential \(V_0\) increases linearly with the frequency of the light above the threshold frequency.
6. The time-lag between the incidence of light and the emission of electron is too small to be measurable.
7. For linearly polarized light, the direction of emission is mostly towards the direction of the electric field.

Subsection 53.4.2 Einstein's Theory of Photoelectric Effect

In a groundbreaking paper in 1905 Einstein gave a very simple explanation of Lenard's results by assuming that light actually consisted of small packets of energy and each packet interacted with one electron at a time. These packets are now called photons. Einstein used Planck's idea of the energy packet in electromagnetic waves, \(E_{\textrm{photon}}=hf\text{,}\) where \(h\) is Planck constant and \(f\) is the frequency of ligh,

with the conservation of energy when a photon strikes a metal surface.

When a photon deposits an energy \(hf\) to an electron at the surface, a part of this energy (\(\phi\)) would go to overcome the binding of the electron to the metal, and the rest will end up as the kinetic energy (\(K\)) of the freed electron.

Rearranging we write kinetic energy of a photoelectron as

Experiments of Lenard and Millikan show that \(K_{\textrm{max}}\) is measurable by the method of impeding potential difference across the plates and given by \(eV_{0}\) where \(V_0\) is the stopping potential.

Subsection 53.4.3 How does Einstein's photoelectric equation explain experimental results?

1. Experimental result: No photoelectron unless frequency is greater than a threshold frequency. This is explained by Eq. (53.4.3) by the fact that kinetic energy is a positive quantity. That means,
   \begin{equation*} hf>W. \end{equation*}
   Therefore, no photoelectrons will be released unless \(f>f_0\) with \(f_0\) related to the work function by
   \begin{equation*} f_0 = W/h. \end{equation*}
   This also explains the dependence of the threshold frequency on the choice of metal for the emitter since different metals will have difference work functions.
2. Experimental result: For a given metal, increase in frequency of the incident light above the threshold frequency increases the photocurrent. According to Eq. (53.4.3), an increase in frequency will increase the kinetic energy, which would mean more electrons will arrive at the collector per unit time, and hence more current will be registered by the galvanometer.
3. Experimental result: For a given metal and for a particular frequency above the threshold frequency, the photocurrent increases linearly with the intensity of the light. More intensity will mean more photons will arrive at the plate per unit time. If we double the intensity, twice as many photons arrive at the plate per unit time, which will cause twice as many electrons to be emitted, thus, there will be twice as much photocurrent.
4. Experimental result: For a given metal and for a particular frequency above the threshold frequency, the stopping potential \(V_0\) is independent of the intensity of light. Equation (53.4.3) shows the max kinetic energy and hence the stopping potential is determined by interaction of individual photons and electrons and are not dependent on the total number of photons. Hence, the stopping potential \(V_0\) should be independent of the intensity of light as seen in experiments.
5. Experimental result: For a given metal the stopping potential \(V_0\) increases linearly with the frequency of the light above the threshold frequency. Einstein's equation Eq. (53.4.3) and the associated experimental equation Eq. (53.4.4) clearly show the that \(V_0\) is a linear function of \(f\text{,}\) the frequency of light.
6. Experimental result: The time-lag between the incidence of light and the emission of electron is too small to be measurable. Since in Einstein's explanation there is no buildup of energy required for the electron to be freed from the surface, no time lag is expected between the absorption of energy and the ejection of the electron.
7. Experimental result: For linearly polarized light, the direction of emission is mostly towards the direction of the electric field. This aspect of photoelectric effect is not addressed by Eq. (53.4.3). According to the wave aspect of light, the direction of the electric field will be the direction in which electron will be accelerated. Therefore, electrons should come out preferentially in that direction. This result shows that although photon picture helps answer most of the energetic aspects of the photoelectric effect, we still need the wave picture of light to understand some other aspects.