Determination of Wavelength of Sodium Light using Michelson's Interferometer

Determination of Wavelength of Sodium Light using Michelson's Interferometer

1. Aim

To determine the wavelength of sodium light using Michelson's Interferometer .

2. Apparatus Used

  • Michelson Interferometer setup
  • Sodium lamp as a light source
  • Micrometer with least count 0.01 mm
  • Telescope for fringe observation
  • Optically flat mirrors (M₁ and M₂)
  • Half-silvered glass plate (Beam splitter)
  • Compensating plate
  • Mounting stand and accessories
  • Micrometer screw arrangements

3. Diagram

Michelson Interferometer Setup

Fig.1: Schematic diagram of Michelson's Interferometer setup

4. Theory

Michelson's interferometer is based on the principle of division of amplitude and interference of light waves. It consists of two perpendicular plane mirrors M₁ and M₂, one of which (M₂) can be moved precisely with a micrometer screw. A beam splitter (half-silvered glass plate G) is placed at an angle of 45° to both mirrors.

When monochromatic light from a source S falls on the beam splitter, it gets divided into two parts:

  • One part is reflected towards mirror M₁ and then reflected back to the beam splitter
  • The other part is transmitted through the beam splitter towards mirror M₂ and then reflected back

These two beams combine at the beam splitter and create an interference pattern observable through the telescope T. The compensating plate C ensures that both beams travel through equal thicknesses of glass.

When mirror M₂ is moved by a small distance, the optical path difference between the two beams changes, causing the fringe pattern to shift. By counting the number of fringes that cross a reference point in the field of view as the mirror is moved, we can determine the wavelength of light.

The optical path difference between the two beams is given by:

$\Delta = 2d$

Where $d$ is the distance mirror M₂ is moved.

For constructive interference to occur, the path difference must be an integral multiple of the wavelength:

$\Delta = n\lambda$

Where $n$ is an integer and $\lambda$ is the wavelength of light used.

Therefore, $2d = n\lambda$

When the mirror M₂ is moved a distance $d$ such that $n$ complete fringes cross the field of view:

$2d = n\lambda$

$\lambda = \frac{2d}{n}$

5. Formula

The wavelength of the sodium light is given by:

$$\lambda = \frac{2d}{n}$$

Where:

  • $\lambda$ = Wavelength of sodium light
  • $d$ = Distance moved by the mirror M₂
  • $n$ = Number of fringes crossing the reference point

6. Procedure

  1. Setup: Set up the Michelson's interferometer on a vibration-free table. Ensure that all optical components are clean and properly aligned.
  2. Initial Adjustment: Turn on the sodium lamp and allow it to stabilize for a few minutes. The sodium lamp emits yellow light with a predominant wavelength of approximately 589.3 nm.
  3. Alignment:
    • Adjust the mirrors M₁ and M₂ to make them perpendicular to each other.
    • Adjust the position of the beam splitter G to ensure equal path lengths to both mirrors.
    • Fine-tune the angles of M₁ and M₂ until clear, straight, and evenly spaced fringes are observed in the telescope.
  4. Locating Fringes: Look through the telescope and locate the circular or straight-line interference fringes. If fringes are not visible, slightly adjust the angle of one of the mirrors until the fringes appear.
  5. Fringe Counting:
    • Note the initial reading of the micrometer attached to mirror M₂.
    • Choose a reference point in the field of view (e.g., the center of the crosshair).
    • Slowly turn the micrometer screw to move mirror M₂, and count the number of fringes (n) that cross the reference point.
    • Stop at a convenient point and record the final micrometer reading.
  6. Repeat: Repeat step 5 for different distances of mirror movement to ensure consistency in results. Take at least 5 readings.
  7. Recording: For each reading, record:
    • Initial position of the micrometer
    • Final position of the micrometer
    • Number of fringes counted

7. Observation Table

S.No. Initial Micrometer Reading (mm) Final Micrometer Reading (mm) Distance Moved by Mirror, d (mm) Number of Fringes Crossed, n 2d/n (nm)
1
2
3
4
5

Note:

  • The distance moved by mirror (d) = Final reading - Initial reading
  • The value of d should be converted from mm to nm by multiplying by 10⁶ (1 mm = 10⁶ nm)

8. Calculations

For each observation:

1. Calculate the distance moved by mirror M₂:

$d = \text{Final micrometer reading} - \text{Initial micrometer reading}$ (in mm)

2. Convert d from mm to nm:

$d \text{ (in nm)} = d \text{ (in mm)} \times 10^6$

3. Calculate the wavelength for each observation:

$\lambda = \frac{2d}{n}$ (in nm)

4. Calculate the mean wavelength:

$\lambda_{mean} = \frac{\lambda_1 + \lambda_2 + \lambda_3 + \lambda_4 + \lambda_5}{5}$ (in nm)

Sample calculation (to be filled with actual data during the experiment):

For observation 1:

  • Initial micrometer reading = ______ mm
  • Final micrometer reading = ______ mm
  • Distance moved by mirror, d = ______ mm = ______ × 10⁶ nm
  • Number of fringes crossed, n = ______
  • Wavelength, λ = 2d/n = ______ nm

9. Result

The wavelength of sodium light using Michelson's Interferometer is found to be ______ nm.

The standard value of the wavelength of sodium light (D-line) is 589.3 nm.

Percentage error = $\frac{|\text{Measured value} - \text{Standard value}|}{\text{Standard value}} \times 100\% = ______\%$

10. Precautions

  1. The experiment should be performed on a vibration-free table to avoid any disturbance in the fringe pattern.
  2. The optical components (mirrors, beam splitter, compensating plate) should be clean and free from dust or fingerprints.
  3. The sodium lamp should be allowed to stabilize before taking readings.
  4. The mirrors should be perfectly perpendicular to each other for clear fringe observation.
  5. The micrometer should be turned very slowly and steadily to avoid missing any fringes.
  6. The eye should be positioned properly at the telescope to accurately count the fringes.
  7. The room should be darkened to improve the visibility of fringes.
  8. Air currents should be minimized as they can disturb the interference pattern.
  9. Temperature should be kept constant throughout the experiment as thermal expansion can affect the results.
  10. The micrometer reading should be taken without parallax error.

11. Viva Voice Questions

What is the principle behind Michelson's Interferometer?
Michelson's Interferometer works on the principle of division of amplitude and interference of light waves. It splits a beam of light into two paths, reflects them back, and recombines them to produce interference patterns that depend on the path difference between the two beams.
What is the function of the compensating plate in the interferometer?
The compensating plate ensures that both light beams pass through equal thicknesses of glass, compensating for the extra glass path that the reflected beam traverses in the beam splitter. This equalizes the optical path lengths and prevents dispersion effects that would otherwise affect the interference pattern.
Why is a monochromatic light source used in this experiment?
A monochromatic light source is used because it produces clear and distinct interference fringes. If white light were used, each wavelength would produce its own fringe pattern at slightly different positions, resulting in colored fringes that would be difficult to count accurately. Sodium light with its predominant yellow wavelength provides sharp, well-defined fringes.
What happens to the fringe pattern when mirror M₂ is moved?
When mirror M₂ is moved, the optical path difference between the two interfering beams changes. This causes the interference pattern to shift, with fringes moving across the field of view. Each fringe that crosses a reference point corresponds to a change in path difference equal to one wavelength of the light used.
What is the significance of counting fringes in this experiment?
Counting fringes allows us to measure the change in optical path difference in terms of wavelengths. Since each fringe corresponds to a path difference of one wavelength, by knowing the distance the mirror moved and the number of fringes counted, we can calculate the wavelength using the formula λ = 2d/n.
How does the Michelson Interferometer differ from the Young's double-slit experiment?
Young's double-slit experiment works on the principle of division of wavefront, where a single wavefront is divided into two parts by two slits, and the two resulting wavefronts interfere. In contrast, Michelson's Interferometer works on the principle of division of amplitude, where a beam splitter divides the amplitude of the incoming beam into two separate beams that travel different paths before recombining.
Can Michelson's Interferometer be used to measure the refractive index of a gas?
Yes, Michelson's Interferometer can be used to measure the refractive index of a gas. By placing a gas cell in one arm of the interferometer and evacuating it, the change in the optical path due to the introduction of gas will cause fringe shifts. From these shifts, the refractive index can be calculated.
What was the historical significance of the Michelson-Morley experiment using this interferometer?
The Michelson-Morley experiment (1887) was designed to detect the motion of Earth through the hypothetical "luminiferous ether" that was thought to pervade space. The experiment's null result (no detectable ether drift) contradicted the ether theory and became crucial evidence leading to Einstein's development of the special theory of relativity, which fundamentally changed our understanding of space, time, and the nature of light.
What factors can affect the accuracy of this experiment?
Factors affecting accuracy include: vibrations causing unwanted fringe movement, temperature fluctuations causing thermal expansion of components, air currents affecting the optical path, imperfect mirror alignment, errors in fringe counting, errors in micrometer readings, and non-monochromatic nature of the light source (sodium light has two closely spaced wavelengths).

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