Class 12/CBSE/Dual Nature of Radiation and Matter
Introduction
Planck’s constant and the work function of a metal are two fundamental quantities in the study of quantum physics. They can be determined experimentally by studying the photoelectric effect and measuring the stopping potential for different frequencies of incident light.
The photoelectric effect demonstrates that light transfers energy in discrete packets called photons. According to Einstein’s photoelectric equation, the energy of an incident photon is partly used to overcome the work function of the metal, while the remaining energy appears as the maximum kinetic energy of the emitted photoelectron.
By measuring the stopping potential at different frequencies and plotting a suitable graph, we can determine both Planck’s constant (h) and the work function (φ) of the emitting material.
This experiment is important for CBSE Class 12 Physics practicals and provides a direct experimental verification of Einstein’s photoelectric equation.
1. Aim of the Experiment
To determine:
- Planck’s constant using the photoelectric effect.
- The work function of the photosensitive material.
- The threshold frequency of the material, if required.
2. Apparatus Required
The experiment may be performed using a photoelectric-effect experimental setup consisting of:
- Photoelectric cell with a photosensitive cathode.
- Light sources of different known frequencies, or a monochromatic light source with suitable wavelength selection.
- Variable DC power supply.
- Microammeter or sensitive current-measuring device.
- Voltmeter or digital voltage-measuring instrument.
- Connecting wires and switching arrangement.
- Optical filters or a monochromator, where required.
In a typical school or college laboratory, the apparatus may be provided as an integrated photoelectric-effect setup.
3. Theory and Principle
When electromagnetic radiation of sufficiently high frequency falls on a photosensitive metal surface, electrons are emitted. These electrons are called photoelectrons.
The maximum kinetic energy of the emitted photoelectrons depends on the frequency of the incident radiation and the work function of the metal.
According to Einstein’s photoelectric equation:
hν = φ + Kₘₐₓ
Therefore,
Kₘₐₓ = hν − φ
Here:
- h = Planck’s constant.
- ν = Frequency of incident light.
- φ = Work function of the photosensitive material.
- Kₘₐₓ = Maximum kinetic energy of the photoelectrons.
The maximum kinetic energy can be determined using the stopping potential.
When a reverse potential is applied between the electrodes, the photoelectric current decreases. At a particular reverse potential, even the most energetic photoelectrons cannot reach the collector, and the photoelectric current becomes zero. This potential is called the stopping potential, represented by V₀.
The maximum kinetic energy is then:
Kₘₐₓ = eV₀
where e is the magnitude of the electronic charge.
Substituting this into Einstein’s equation:
eV₀ = hν − φ
Dividing by e:
V₀ = (h/e)ν − φ/e
This is the working equation for determining Planck’s constant and the work function.
4. Principle of Graphical Determination
The equation
V₀ = (h/e)ν − φ/e
has the same mathematical form as the equation of a straight line:
y = mx + c
Comparing the two equations:
- y corresponds to stopping potential, V₀.
- x corresponds to frequency, ν.
- Slope, m = h/e.
- Vertical intercept, c = −φ/e.
Therefore, a graph of stopping potential against frequency is a straight line under ideal experimental conditions.
From its slope, Planck’s constant can be calculated. From its vertical intercept, the work function can be determined.
Alternatively, the frequency-axis intercept gives the threshold frequency directly.
5. Experimental Procedure
Step 1: Set Up the Apparatus
Connect the photoelectric cell, variable voltage supply and current-measuring instrument according to the laboratory instructions.
Ensure that the photosensitive cathode is illuminated properly and that all electrical connections are secure.
Step 2: Select a Known Frequency
Illuminate the photosensitive surface using light of a known frequency.
If the wavelength is given instead, calculate the frequency using:
ν = c/λ
where c is the speed of light in vacuum and λ is the wavelength of the incident radiation.
Use consistent units throughout the calculations.
Step 3: Measure the Photoelectric Current
Allow the light to fall on the photosensitive surface and observe the photoelectric current.
Apply a reverse potential between the electrodes and gradually increase its magnitude.
The photoelectric current decreases as the reverse potential increases.
Step 4: Determine the Stopping Potential
Continue increasing the reverse potential until the photoelectric current becomes zero or reaches the experimentally defined cutoff level.
Record the magnitude of this reverse potential as the stopping potential, V₀.
Repeat the measurement if necessary to improve reliability.
Step 5: Repeat for Different Frequencies
Repeat the experiment for several different known frequencies of incident radiation.
For each frequency, determine the corresponding stopping potential.
Use the same photosensitive material and maintain suitable experimental conditions throughout the measurements.
Step 6: Tabulate the Observations
Record the wavelength or frequency and the corresponding stopping potential in a table.
Calculate the frequency for each wavelength if it is not already known.
Step 7: Plot the Graph
Plot frequency, ν, on the horizontal axis and stopping potential, V₀, on the vertical axis.
Draw the best-fit straight line through the experimental data points.
Determine the slope and the intercept of the graph.
Step 8: Calculate Planck’s Constant and Work Function
Use the slope of the graph to calculate Planck’s constant.
Use the vertical intercept or threshold frequency to calculate the work function.
Compare the experimental value of Planck’s constant with its accepted value.
6. Observation Table and Graph
The following is an illustrative observation table. These are sample values for demonstrating the calculation method, not actual measurements from a laboratory.
| S.No | Frequency |
Stopping Potential (Vo) |
| 1 | 5.0 | 0.28 |
| 2 | 5.5 | 0.48 |
| 3 | 6.0 | 0.69 |
| 4 | 6.5 | 0.90 |
| 5 | 7.0 | 1.10 |
The graph of stopping potential V₀ against frequency ν is a straight line described by:
V₀ = (h/e)ν − φ/e
The important features of the graph are:
- Slope: The slope is h/e.
- Vertical intercept: The intercept is −φ/e when the line is extrapolated to zero frequency.
- Frequency-axis intercept: The frequency-axis intercept is the threshold frequency ν₀, where the stopping potential becomes zero.
- Linear relationship: The stopping potential increases linearly with frequency above the threshold frequency.
The graph provides an experimental method for determining Planck’s constant and the work function.
7.Determination of Planck’s Constant
From Einstein’s photoelectric equation:
eV₀ = hν − φ
Therefore,
V₀ = (h/e)ν − φ/e
Comparing this with the straight-line equation y = mx + c, we obtain:
Slope = h/e
Hence,
h = e × slope
If the slope of the V₀ versus ν graph is m, then:
h = em
The electronic charge is approximately:
e = 1.602 × 10⁻¹⁹ C
The accepted value of Planck’s constant is approximately:
h = 6.626 × 10⁻³⁴ J s
Calculation Using Two Points on the Graph
If two points on the best-fit line are (ν₁, V₁) and (ν₂, V₂), then:
Slope = (V₂ − V₁)/(ν₂ − ν₁)
Therefore,
h = e(V₂ − V₁)/(ν₂ − ν₁)
The frequency difference must be expressed in hertz to obtain Planck’s constant in joule-seconds.
Illustrative Calculation
Suppose the slope of the stopping-potential-versus-frequency graph is:
m = 4.14 × 10⁻¹⁵ V s
Then:
h = em
h = (1.602 × 10⁻¹⁹)(4.14 × 10⁻¹⁵)
Therefore,
h ≈ 6.63 × 10⁻³⁴ J s
This value is close to the accepted value of Planck’s constant.
8.Determination of the Work Function
The work function is the minimum energy required to remove an electron from the surface of a material.
It is represented by φ.
From Einstein’s photoelectric equation:
eV₀ = hν − φ
Rearranging:
φ = hν − eV₀
Thus, the work function can be calculated if the frequency of incident radiation, the corresponding stopping potential and Planck’s constant are known.
Method 1: Using the Vertical Intercept
The graph equation is:
V₀ = (h/e)ν − φ/e
If the vertical intercept is c, then:
c = −φ/e
Therefore,
φ = −ec
The work function is positive even though the vertical intercept is negative.
Method 2: Using the Threshold Frequency
At the threshold frequency, the maximum kinetic energy of the emitted electrons is zero.
Therefore,
φ = hν₀
Hence,
φ = h × threshold frequency
If the threshold frequency is known, the work function can be determined directly.
Expressing the Work Function in Electronvolts
The work function is often expressed in electronvolts (eV).
Since one electronvolt is the energy gained by an electron when it moves through a potential difference of one volt:
1 eV = 1.602 × 10⁻¹⁹ J
To convert the work function from joules to electronvolts, divide its value in joules by 1.602 × 10⁻¹⁹.
9. Determination of Threshold Frequency and Threshold Wavelength
The threshold frequency is the minimum frequency at which photoelectric emission begins.
From the photoelectric equation:
φ = hν₀
Therefore,
ν₀ = φ/h
The threshold wavelength is the maximum wavelength that can produce photoelectric emission under the ordinary single-photon process.
Using c = ν₀λ₀:
λ₀ = c/ν₀
Therefore,
λ₀ = hc/φ
These relationships show that the threshold frequency and threshold wavelength depend on the work function of the material.
10. Sample Calculation of Work Function
Suppose the threshold frequency of a photosensitive material is measured to be:
ν₀ = 5.0 × 10¹⁴ Hz
Using the accepted value of Planck’s constant:
h = 6.626 × 10⁻³⁴ J s
The work function is:
φ = hν₀
φ = (6.626 × 10⁻³⁴)(5.0 × 10¹⁴)
φ = 3.313 × 10⁻¹⁹ J
Converting into electronvolts:
φ = (3.313 × 10⁻¹⁹)/(1.602 × 10⁻¹⁹)
Therefore,
φ ≈ 2.07 eV
This is an illustrative calculation. The actual work function must be determined from the experimental data for the material used.
11. Sources of Experimental Error
The accuracy of the experiment may be affected by several factors:
- Uncertainty in wavelength: Incorrect wavelength values produce errors in the calculated frequencies.
- Stopping-potential measurement: Instrument resolution and difficulty identifying the zero-current condition can affect the result.
- Surface contamination: Contamination or oxidation can change the effective work function of the photosensitive surface.
- Ambient light: Unwanted light may produce additional photoelectric current.
- Contact potential: Differences in electrode work functions can introduce a potential offset.
- Instrument calibration: Incorrect voltage readings affect the slope and intercept of the graph.
- Limited frequency range: A narrow range of frequencies can increase uncertainty in the fitted slope.
- Graph plotting errors: Incorrect scale selection or inaccurate plotting can affect the calculated values.
For better accuracy, use calibrated instruments, shield the setup from unwanted light, record multiple observations and determine the slope using a best-fit line rather than relying on a single pair of data points.
12. Precautions
- Make all electrical connections according to the experimental instructions.
- Ensure that the photosensitive surface is illuminated uniformly.
- Use known wavelengths or frequencies.
- Keep the same photosensitive material throughout the experiment.
- Increase the reverse potential gradually.
- Avoid parallax errors when reading analogue instruments.
- Record the magnitude of the stopping potential as a positive value when using the equation Kₘₐₓ = eV₀.
- Take repeated readings and use a best-fit graph.
- Ensure that the frequency is expressed in hertz during calculations.
- Convert the work function into electronvolts only after obtaining the energy in joules, if required.
13. Using the photoelectric effect and the graph of stopping potential versus frequency:
- Planck’s constant is determined from the slope using h = e × slope.
- The work function is determined from the vertical intercept using φ = −e × intercept, or from the threshold frequency using φ = hν₀.
- The threshold wavelength can be calculated using λ₀ = c/ν₀.
The experimental value of Planck’s constant should be compared with the accepted value of approximately 6.626 × 10⁻³⁴ J s. Any difference should be discussed in terms of measurement uncertainty and experimental limitations.
14. Viva-Voce Questions and Answers
Q1. What is Planck’s constant?
Planck’s constant is a fundamental constant that relates the energy of a photon to its frequency. Its accepted value is approximately 6.626 × 10⁻³⁴ J s.
Q2. What is the work function?
The work function is the minimum energy required to remove an electron from the surface of a material.
Q3. What is the stopping potential?
It is the minimum reverse potential needed to stop the most energetic photoelectrons from reaching the collector.
Q4. Why is the stopping potential measured?
It allows the maximum kinetic energy of photoelectrons to be determined using Kₘₐₓ = eV₀.
Q5. Which graph is used to determine Planck’s constant?
A graph of stopping potential against frequency of incident radiation.
Q6. What does the slope of the graph represent?
The slope represents h/e. Multiplying it by the magnitude of the electronic charge gives Planck’s constant.
Q7. How can the work function be determined from the graph?
It can be determined from the negative vertical intercept, or by multiplying Planck’s constant by the threshold frequency.
Q8. What is the threshold frequency?
It is the minimum frequency of incident radiation required for ordinary photoelectric emission from the material.
Q9. Does the stopping potential depend on the intensity of incident light?
For a fixed frequency above the threshold, the stopping potential is ideally independent of intensity.
Q10. Why is the graph a straight line?
Einstein’s equation gives V₀ = (h/e)ν − φ/e, which is a linear relationship between stopping potential and frequency.
Q11. What is the SI unit of the work function?
The SI unit is the joule. It is also commonly expressed in electronvolts.
Q12. Why might the experimental value of Planck’s constant differ from the accepted value?
Possible reasons include voltage measurement errors, uncertainty in wavelength, contact potentials, surface contamination and graph-fitting errors.
15. Important Formulae for Quick Revision
- Photon energy: E = hν
- Einstein’s photoelectric equation: hν = φ + Kₘₐₓ
- Maximum kinetic energy: Kₘₐₓ = eV₀
- Stopping-potential equation: V₀ = (h/e)ν − φ/e
- Planck’s constant from graph slope: h = e × slope
- Work function from a known stopping potential: φ = hν − eV₀
- Work function from threshold frequency: φ = hν₀
- Threshold frequency: ν₀ = φ/h
- Threshold wavelength: λ₀ = c/ν₀
Conclusion
The photoelectric effect provides an effective experimental method for determining Planck’s constant and the work function of a photosensitive material. By measuring stopping potentials for different frequencies and plotting stopping potential against frequency, the slope gives Planck’s constant while the intercept or threshold frequency gives the work function.
This experiment verifies the linear relationship predicted by Einstein’s photoelectric equation and demonstrates the quantised nature of light-matter interactions. It is an important experiment for understanding modern physics and preparing for CBSE Class 12 practical examinations, NEET and JEE.
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