Angle of Minimum Deviation for a Given Prism
1. Aim
To determine the angle of minimum deviation for a given prism by plotting a graph between the angle of incidence and the angle of deviation.
2. Apparatus Used
- Spectrometer
- Glass prism
- Sodium vapor lamp / Mercury vapor lamp
- Reading lens
- Spirit level
- Graph paper
3. Diagram
Fig. 1: Experimental setup for determining the angle of minimum deviation
Fig. 2: Ray diagram showing refraction through prism and angle of deviation
4. Theory
When a ray of light passes through a prism, it undergoes refraction at both surfaces and emerges in a different direction. The angle between the incident ray and the emergent ray is called the angle of deviation (δ).
The deviation depends on:
- The angle of incidence (i)
- The refractive index of the prism material (μ)
- The angle of the prism (A)
For a given prism and a given wavelength of light, as the angle of incidence increases from zero, the angle of deviation first decreases, reaches a minimum value (δm), and then increases. This minimum value of deviation is called the angle of minimum deviation.
At the position of minimum deviation, the ray passes through the prism symmetrically, meaning:
When δ = δm, the angle of incidence (i) equals the angle of emergence (e)
Also, the refracted ray inside the prism becomes parallel to the base of the prism
The refractive index of the prism material can be calculated using the formula:
$\mu = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$
Where:
- μ is the refractive index of the material of the prism
- A is the angle of the prism
- δm is the angle of minimum deviation
5. Formula
From Snell's law and the geometry of the prism, we derive:
$\mu = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$
Where:
- μ = Refractive index of the prism material
- A = Angle of the prism
- δm = Angle of minimum deviation
For the relation between angle of incidence (i) and angle of deviation (δ):
$\sin(i) = \mu \sin\left(\frac{A + \delta - i}{2}\right)$
At minimum deviation:
$i = e$ (angle of incidence equals angle of emergence)
$r_1 = r_2 = \frac{A}{2}$ (both refracted angles are equal)
6. Procedure
- Preliminary Adjustments of the Spectrometer:
- Level the spectrometer using the leveling screws and spirit level.
- Adjust the eyepiece to get clear crosswires.
- Adjust the slit width to obtain a sharp image.
- Set the telescope for parallel rays by focusing it on a distant object.
- Align the telescope and collimator to make their axes perpendicular to the prism table axis.
- Determination of the Angle of the Prism (A):
- Place the prism on the prism table with its refracting edge towards the center.
- Rotate the telescope to obtain the reflection of the slit from one face of the prism and note the reading (θ1).
- Rotate the telescope to obtain the reflection from the second face and note the reading (θ2).
- The angle of the prism is given by: A = 180° - |θ1 - θ2|
- Determination of the Angle of Minimum Deviation (δm):
- Place the prism on the prism table with its refracting edge away from the collimator.
- Direct the light from the source through the slit and collimator onto one face of the prism.
- Rotate the telescope to observe the refracted image of the slit.
- Rotate the prism table slowly in either direction and simultaneously move the telescope to follow the refracted image.
- Note that as the prism is rotated, the refracted image first moves in one direction, becomes momentarily stationary, and then moves in the opposite direction.
- When the image is stationary, note the position of the telescope (θd).
- Remove the prism and align the telescope directly with the collimator to obtain the direct reading (θ0).
- The angle of minimum deviation is given by: δm = |θd - θ0|
- Plotting the Graph:
- For different positions of the prism, note the angle of incidence (i) and the corresponding angle of deviation (δ).
- Plot a graph between the angle of incidence (i) on the x-axis and the angle of deviation (δ) on the y-axis.
- The lowest point on the curve corresponds to the angle of minimum deviation (δm).
- From the graph, determine the angle of incidence corresponding to the minimum deviation.
7. Observation Table
Part A: Determination of the Angle of the Prism (A)
| Reflection from First Face (θ1) | Reflection from Second Face (θ2) | Angle of Prism A = 180° - |θ1 - θ2| |
|---|---|---|
| _______° | _______° | _______° |
Part B: Observation for Angle of Deviation (δ) at Different Angles of Incidence (i)
| S.No. | Angle of Incidence (i) | Position of Refracted Image (θr) | Position of Direct Image (θ0) | Angle of Deviation δ = |θr - θ0| |
|---|---|---|---|---|
| 1 | _______° | _______° | _______° | _______° |
| 2 | _______° | _______° | _______° | _______° |
| 3 | _______° | _______° | _______° | _______° |
| 4 | _______° | _______° | _______° | _______° |
| 5 | _______° | _______° | _______° | _______° |
| 6 | _______° | _______° | _______° | _______° |
| 7 | _______° | _______° | _______° | _______° |
8. Calculations
Step 1: Determine the angle of the prism (A):
A = 180° - |θ1 - θ2| = _______°
Step 2: Plot a graph between the angle of incidence (i) on the x-axis and the angle of deviation (δ) on the y-axis.
Step 3: From the graph, determine the minimum value of the angle of deviation (δm):
δm = _______°
Step 4: Calculate the refractive index of the prism material using the formula:
$\mu = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$
$\mu = \frac{\sin\left(\frac{_______° + _______°}{2}\right)}{\sin\left(\frac{_______°}{2}\right)}$
$\mu = \frac{\sin(_______°)}{\sin(_______°)}$
$\mu = _______$
9. Result
1. The angle of the prism (A) = _______°
2. The angle of minimum deviation (δm) = _______°
3. The refractive index of the prism material (μ) = _______
4. From the graph, it is observed that the angle of deviation first decreases with an increase in the angle of incidence, reaches a minimum value, and then increases. This verifies that there exists a minimum angle of deviation for a given prism.
10. Precautions
- The spectrometer should be properly leveled before starting the experiment.
- The slit should be narrow and vertical.
- The crosswires of the eyepiece should be clearly visible.
- The telescope and collimator should be properly adjusted for parallel rays.
- The prism should be placed with its refracting edge precisely at the center of the prism table.
- When finding the position of minimum deviation, the prism table should be rotated very slowly to accurately locate the position where the refracted image becomes momentarily stationary.
- Readings should be taken without parallax error.
- The monochromatic light source should be used for accurate results, as the refractive index depends on the wavelength of light.
- The prism faces should be clean and free from dust or fingerprints.
- The angle of incidence should be varied in small steps to get a smooth graph and accurately determine the minimum deviation.
11. Sources of Error
- Improper leveling of the spectrometer.
- Incorrect adjustment of the telescope and collimator for parallel rays.
- Parallax error in taking readings.
- Improper positioning of the prism on the prism table.
- Using a light source that is not perfectly monochromatic, which can lead to dispersion effects.
- Difficulty in locating the exact position of minimum deviation due to the slow rate of change of deviation near the minimum.
- Temperature variations affecting the refractive index of the prism material.
- Imperfections or non-uniformity in the prism material.
- Imprecise measurement of the angle of the prism.
- Human error in reading the vernier scale.
12. Viva Voice Questions
- What is the angle of minimum deviation?
The angle of minimum deviation is the smallest angle through which a ray of light deviates from its original path when passing through a prism. It occurs when the ray passes symmetrically through the prism. - What is the condition for minimum deviation in a prism?
At minimum deviation, the ray of light passes symmetrically through the prism, meaning the angle of incidence equals the angle of emergence, and the refracted ray inside the prism becomes parallel to the base of the prism. - How does the refractive index of a prism vary with the wavelength of light?
The refractive index of a prism decreases with increasing wavelength of light. This phenomenon is known as dispersion, which causes different colors (wavelengths) of light to bend at different angles when passing through a prism. - Why do we use a monochromatic light source in this experiment?
We use a monochromatic light source to avoid dispersion effects, as the refractive index of the prism material varies with wavelength. Using monochromatic light ensures we get a sharp refracted image and can accurately measure the angle of deviation. - What would happen if we use white light instead of monochromatic light?
If we use white light, which contains multiple wavelengths, each wavelength would have a different refractive index and hence a different angle of deviation. This would result in the formation of a spectrum rather than a single image, making it difficult to precisely measure the angle of deviation. - Why does the angle of deviation first decrease and then increase as the angle of incidence increases?
This behavior occurs due to the combined effect of refraction at both surfaces of the prism. As the angle of incidence increases, the angle of refraction at the first surface increases, while at the second surface, the angle of incidence (which is the angle of refraction from the first surface) decreases, causing complex variations in the overall deviation. Mathematically, it can be shown that there exists a minimum value of deviation. - How does the angle of minimum deviation change if we use a prism of higher refractive index?
For a prism with a higher refractive index, the angle of minimum deviation increases for the same prism angle. This is because a higher refractive index causes greater bending of light at each surface. - What is the relation between the angle of the prism, angle of minimum deviation, and refractive index?
The relation is given by the formula: $\mu = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$, where μ is the refractive index, A is the angle of the prism, and δm is the angle of minimum deviation. - Why is the spectrometer used in this experiment?
The spectrometer is used because it allows precise measurement of angles. It consists of a collimator to produce parallel rays, a prism table that can be rotated to adjust the angle of incidence, and a telescope that can be rotated to measure the angles accurately. - How would the angle of minimum deviation change if the experiment is performed in a medium other than air, like water?
If the experiment is performed in a medium other than air (e.g., water), the angle of minimum deviation would decrease. This is because the relative refractive index (ratio of refractive index of prism to that of the surrounding medium) would be lower, resulting in less bending of light.