Einstein's Photoelectric Equation and Millikan's Experiment

Laws of Photoelectric Emission

1. The minimum frequency, which can cause photoelectric emission, is called the threshold frequency.
2. The rate of emission is directly proportional to the intensity of the incident light, provided the frequency is greater than the threshold frequency.
3. The velocity and hence the energy of the emitted photoelectrons is independent of the intensity of the light and depends only in the frequency of the incident of light and the nature of the metal.
4. There is an instantaneous emission of photo-electrons within the limits of experimental accuracy.
5. The maximum velocity and hence the stopping potential are independent of the intensity of the incident light but are directly proportional to the frequency of the incident radiation for a given metal.

Einstein’s Photoelectric Equation

According to Einstein, light of frequency f consists of a photons each of energy hf. When a photon of light of frequency f is incident on a metal, the energy is completely transferred to a free electron in the metal. A part of the energy acquired by the electron is used to pull out the electron from the surface of the metal and the rest of its utilized in imparting K.E. to the emitted electron. Let W be the energy used up in librating the electrons from the surface of metal (work function) and \(\frac 12 mv^2 \) is K.E. of the photoelectron, then,

\begin{align*} \text {Energy of the photon} &= \text {Work function} + \text {K.E. of the electron} \\ \text {or} \: hf &= \phi + \frac 12 mv^2 \dots (i)\end{align*} This relation is known as the Einstein’s photoelectric equation. If f₀is the threshold frequency which just ejects an electron from the metal surface without any velocity, then. \begin{align*}\phi &= hf_0 \\ \therefore hf &= hf_0 + \frac 12 mv_{max}^2 \dots (ii) \\ \text {where}\: V_{max}\: \text {is the maximum velocity acquired by the electron} \\ \text {or,} \: \frac 12 mv_{max}^2 = h(f-f_0) \\ \end{align*}

Important Terms

1. Threshold frequency (f₀)
   The minimum frequency of incident radiation which is just sufficient to eject an electron (i.e. with zero velocity) from the surface of the metal is known as threshold frequency f₀ for that metal.
2. Threshold wavelength (λ₀)
   The wavelength corresponding to threshold frequency is known as threshold wavelength λ₀, The incident radiation with threshold wavelength is just capable of ejecting photoelectrons. If the wavelength is more than λ₀, no photoelectrons are emitted.
3. Work function
   The minimum energy of photon required to just liberate an electron from the metal surface with zero velocity is known as work function W of that metal.
   According to Einstein’s photoelectric equation,
   $$\text {i.e.} \: hf_0 = \phi + \frac 12 m v_{max} ^2 $$
   If the frequency of the incident radiation if f₀ , then the emitted electrons will have zero velocity.
   $$\text {i.e.} \: hf_0 = \phi + \frac 12 m (0)^2 $$
   $$\text {or,} \: \phi = hf_0 $$

Millikan’s Experiment and Measurement of Planck’s constant

In this experiment, alkali metal is used as emitters. Cylindrical blocks made by sodium, potassium or lithium are placed around a wheel W at the centre of the glass flask. To avoid tarnishing and the formation of oxide films on the metal surface, the metals were housed in a vacuum. Their surface was kept clean by a cutting knife K which could be moved and turned by the means of a magnet outside. When a beam of monochromatic light falls on the alkali metal, photoelectrons are emitted. The photoelectrons emitted reach the electrode C. the electrode C is usually kept negative with respect to the cylindrical blocks. Therefore, photo electrons will be repelled and only fast moving electrons will be able to reach the electrode C. Since the negative potential of electrode C is increased until the fastest moving photo electrons are repelled back and thud the current fall to zero. Therefore, the minimum potential and \(\frac 12 mV_{max}^2 \) is the K.E. of the fastest photoelectrons, then

\begin{align*} \frac 12 mV_{max}^2 &= eV_0 \dots (i) \\ \end{align*} where m, e, and v_maxbe the mass, charge and maximum velocity of the ejected electrons from Einstein’s photoelectric equation, we have \begin{align*} hf &= \phi + \frac 12 mV_{max}^2 \\ \text{or,} \: \frac 12 mV_{max}^2 &= hf – hf_0 \dots (ii) \\ \text {from equation} (i) \: \text {and} \: (ii), \text {we have,} \\ \text {or,} \: eV_0 &= hf –hf_0 \\ \text {or,} V_0 &= \frac he f - \frac he f_0 \dots (iii) \\ \end{align*}

Experimental graph between V₀ and f is in agreement with Einstein’s photoelectric equation. The slope of the straight line is

\begin{align*} m &= \tan \theta = \frac he \\ \therefore h &= e\tan \theta = 6.625 \times 10^{-34} Js \dots (iv) \end{align*}

Hence determining the slope of the curve V₀ and f. Planck’s constant h can be calculated by using equation (iv).

Uses of Photoelectric Effect

The photoelectric effect is used in photoelectric cells. The device used for converting light energy into electric energy is called the photoelectric cell. They are of three types:

1. Photo-emissive cells
2. Photo-voltaic cells
3. Photo-conductive cells

Photo-emissive Cells

It consists of two electrodes A and C called anode and cathode is a semi-cylindrical plate of metal coated with a photosensitive material of low work function such as cesium or silver oxide. The anode A is in a form of wire so that it does not obstruct the light falling on the cathode. A positive potential of about 100 volts is applied to the anode, the negative being connected through the galvanometer G.

When the light of frequency greater than the threshold frequency of cathode material is incident on the cathode C, photoelectrons are emitted from it. A small current flows through the cell and can be measured by the galvanometer.

Photo-voltaic Cells

It consists of a copper plate, on which a thin semiconductor layer of cuprous oxide is deposited. Over this, a very thin layer of silver or gold is deposited by the method of evaporation in a vacuum. When the light falls on the cell, it penetrates the film and ejects electrons from the interface between the gold film and the cuprous oxide layer. These electrons flow through the external low resistance R consulting a current, whose strength depends in the intensity of light.

The advantages of these cells are:

1. There is no need of an external battery.
2. It is sturdy
3. It is cheap

Photo-conductive Cell

The cell works on the principle that the electrical resistance of the semiconductor, such as cadmium sulphide, decreases with the increase in the intensity of light falling on it. A film of selenium, deposited on one side of the iron plate, is also used. When the light of varying intensity is made to fall on the film, a current flows through the circuit, containing a galvanometer and a battery.

Applications of Photoelectric Cells

1. They are used to switch on and off the automatic switches of the street light.
2. They are used to operate controls in electronic devices such as televisions, computers etc.
3. They are used in the reproduction of sound in cinematography.
4. Photoelectric cells are used in burglar alarm, fire alarm and television.
5. They are used in industry to locate minor flaws or holes in metal sheets.
6. Photoelectric cells are used for counting numbers.

Reference

Manu Kumar Khatry, Manoj Kumar Thapa, Bhesha Raj Adhikari, Arjun Kumar Gautam, Parashu Ram Poudel. Principle of Physics. Kathmandu: Ayam publication PVT LTD, 2010.

S.K. Gautam, J.M. Pradhan. A text Book of Physics. Kathmandu: Surya Publication, 2003.e