# The Photoelectric Effect – Work Function, Threshold Frequency, Planck Constant, Photoelectron, Photon, Stopping Potential, Saturation Current

The Photoelectric Effect: It is a phenomenon in which electrons are emitted when a metal is illuminated with light of frequency greater than a certain frequency called ‘Threshold Frequency’

## Experiments

• Observation by Heinrich Hertz [1857 – 1894]
• Hallwach’s and Lenard’s Observations [1886 – 1902]

## Terminology

• Work Function – The amount of energy required to knock-off the least tightly bound electron of a metal.
• Threshold Frequency – The minimum frequency of the incident light below which photoelectric effect does not happen, even though the intensity of light is raised very high
• Planck Constant   $h ~=~ 6.63 \times 10^{-34} ~J . s$
• Photoelectron – The electron emitted by photons
Photon – ‘The Particle” of light. It has energy $h \nu$
• Stopping Potential – The negative potential which is required to stop the most-energetic electron.
• Saturation Current – The maximum photoelectric current at a particular frequency and intensity of light.

## Essential Formulae / Points

The following points are central to ‘The Photoelectric Effect’

1. Birth of a photon and the concept of ‘Duality’ of light
2. One-photon-one-electron interaction.
3. $Time~Lag ~\tau \approx 10^{-9} s$
4. $i_s ~=~ f ( \nu , V, I, \phi_0 )$
5. $\Sigma E ~=~ 0$
6. $h \nu ~=~ \phi_0 ~+~ (KE)_{max}$
7. ${KE}_{max} ~=~ f ( \nu ) ~\neq~ f ( I )$
8. $e V_0 ~=~ h (\nu - \nu_0)$
9. $\nu \ge \nu_0$
10. $i_s \propto I$
11. $\phi_0 ~=~ f(Material)$
12. $V_0 ~=~ (\frac {h} {e}) \nu - (\frac{h}{e})\nu_0$
• $slope ~=~ (\frac {h} {e}) ~=~ constant ~for ~all ~materials$
• $at~ \nu=\nu_0, ~V_0 = 0$

## Explanation and Graph

Failure of Wave Theory of Light: To explain photoelectric effect, the wave theory has proved inadequate. Wave theory could not explain why there should be a specific ‘threshold frequency’ below which no photoelectric effect takes place. Wave theory says that if the intensity of low frequency light is sufficiently raised, photoelectrons should emit. Secondly, there should be a time lag of the order of hours in emitting electrons.

### Birth of Photon and ‘Duality of Light’

Point (1): In order to explain, Einstein has predicted the presence of a ‘particle’ of light, which we now call – photon.

Point (2): This photon interacts with electron one-to-one.

Point (3): $Time~Lag ~\tau \approx 10^{-9} s$.
Since now one photon ‘hits’ one electron, it knocks that electron off the influence of the atom, and hence the time lag is very small (of the order of $\approx 10^{-9} s$

Point (4): $i_s ~=~ f ( \nu , V, I, \phi_0 )$
Hallwach’s and Lenard’s Observation was that the saturation current depends on frequency, voltage, intensity and metal’s work-function.

Point (5): $\Sigma E ~=~ 0$
Einstein started to explain the effect based on the law of conservation of energy in photon-electron interaction.

Point (6): $h \nu ~=~ \phi_0 ~+~ (KE)_{max}$
Photons energy is partly used to overcome binding energy (work function) and the remaining energy is imparted to the released electron as kinetic energy.

Point (7): ${KE}_{max} ~=~ f ( \nu ) ~\neq~ f ( I )$
Kinetics energy of the photoelectron depends on frequency and not the intensity of incident light.

Point (8): $e V_0 ~=~ h (\nu - \nu_0)$
Einsteins photoelectric equation can also be written as above. This is a very simple and eloquent equation which says that frequency has to greater than a ‘threshold frequency’ for this effect to occur.

Point (9): $\nu \ge \nu_0$
Existence of threshold frequency.

Point (10): $i_s \propto I$
For any frequency (greater than $\nu_0$ , saturation current linearly increases with intensity of light.

Point (11): $\phi_0 ~=~ f(Material)$
Work function is purely a material property. It does not depend on anything else but the material.

Point (12): $V_0 ~=~ (\frac {h} {e}) \nu - (\frac{h}{e})\nu_0$
Einstein’s equation can also written in this form. If frequency is independent variable and stopping potential is dependent variable, then it represents a line (linear equation, y = m.x + c), for a particular material.

Point (12.a) $slope ~=~ (\frac {h} {e}) ~=~ constant$
The slope (m) is constant and independent of the material. This means the graphs for different material would form a set of parallel lines, with different intercept-on-y-axis.

Point (12.b) $at~ \nu=\nu_0, ~V_0 = 0$
At frequency equal to threshold frequency, the stopping potential is zero.

## Applications of Photoelectric Effect

1. Photocell: Used to measure intensity of light, like in light-meters in camera
2. Automatic door opener
3. Used in counting devices like counting packets, people entering, etc
4. Burglar alarm
5. Fire alarm
6. Reproduction of sound in motion pictures
7. Detecting minor flaws or holes in metal sheets

The above 12 points briefly summarizes the photoelectric effect, metal’s work function, threshold frequency, photoelectron, birth of photon, stopping voltage, saturation current and its dependence on intensity.