To Find the Refractive Index of a Liquid Using a Convex Lens and Plane Mirror
1. Aim
To determine the refractive index of a given liquid by using a convex lens and a plane mirror.
2. Apparatus Used
- Convex lens
- Optical bench with uprights
- Plane mirror
- Spherometer
- Plano-concave cell (watch glass)
- Given liquid (water, glycerin, etc.)
- Object pin and image pin
- Half meter scale
- Thread
- Spirit level
- Clamp stand
3. Diagram
4. Theory
When a convex lens is placed on a plane mirror and an object pin is placed at a certain distance from the lens, two images are formed:
- First Image: Formed by reflection from the plane mirror after refraction through the lens.
- Second Image: Formed by two refractions through the lens and one reflection at the plane mirror.
When a liquid is introduced between the lens and the mirror, the position of the second image changes due to the change in the optical path. This change can be used to calculate the refractive index of the liquid.
The formula for the refractive index is derived using the lens formula and the principles of reflection and refraction. The focal length of the lens changes when used in conjunction with the liquid, and this change is directly related to the refractive index of the liquid.
For a convex lens in air, the focal length is denoted by \(f_a\).
When the same lens is partially immersed in a liquid, its focal length changes to \(f_l\).
The relationship between these focal lengths and the refractive index of the liquid (\(\mu_l\)) is given by:
\[\mu_l = \frac{\mu - 1}{\frac{f_a}{f_l} - 1} + 1\]
where \(\mu\) is the refractive index of the lens material.
For the special case where we measure the position of the second image with and without the liquid:
\[\mu_l = \frac{v_2 - v_1}{v_2 - v_0}\]
where:
- \(v_0\) is the distance of the object from the lens
- \(v_1\) is the distance of the second image when there is no liquid
- \(v_2\) is the distance of the second image when the liquid is introduced
5. Formula
The refractive index of the liquid is given by:
\[\mu_l = \frac{v_2 - v_1}{v_2 - v_0}\]
where:
- \(v_0\) = distance of the object from the lens
- \(v_1\) = distance of the second image from the lens without liquid
- \(v_2\) = distance of the second image from the lens with liquid
6. Procedure
- Set up the optical bench horizontally using the spirit level.
- Place the plane mirror on the optical bench at one end.
- Place the convex lens on an upright holder at a suitable distance from the mirror.
- Fix the object pin on another upright holder and place it at a distance greater than the focal length of the lens.
- Adjust the position of the object pin until its image coincides with the object itself when viewed through the lens and mirror system.
- Record the position of the object pin (\(v_0\)) and the lens on the optical bench.
- Now place an image pin and move it until it coincides with the second image formed by reflection from the mirror after double refraction through the lens. Record this position (\(v_1\)).
- Place the watch glass (plano-concave cell) on the mirror and fill it with the given liquid.
- The position of the second image will change. Move the image pin to locate this new position of the second image and record it (\(v_2\)).
- Repeat steps 5-9 for at least five different positions of the object pin.
7. Observation Table
| S.No. | Object Distance \(v_0\) (cm) |
Image Distance without Liquid \(v_1\) (cm) |
Image Distance with Liquid \(v_2\) (cm) |
Refractive Index \(\mu_l = \frac{v_2 - v_1}{v_2 - v_0}\) |
|---|---|---|---|---|
| 1 | ||||
| 2 | ||||
| 3 | ||||
| 4 | ||||
| 5 |
8. Calculations
For each observation, calculate the refractive index of the liquid using the formula:
\[\mu_l = \frac{v_2 - v_1}{v_2 - v_0}\]
Sample Calculation (for first observation):
Given:
\(v_0 = \_\_\_\_ \text{ cm}\)
\(v_1 = \_\_\_\_ \text{ cm}\)
\(v_2 = \_\_\_\_ \text{ cm}\)
Substituting in the formula:
\[\mu_l = \frac{v_2 - v_1}{v_2 - v_0} = \frac{\_\_\_\_ - \_\_\_\_}{\_\_\_\_ - \_\_\_\_} = \_\_\_\_\]
Calculate the mean value of the refractive index:
\[\mu_{\text{mean}} = \frac{\mu_1 + \mu_2 + \mu_3 + \mu_4 + \mu_5}{5}\]
9. Result
The refractive index of the given liquid is __________ (with an experimental error of ±__________).
This value can be compared with the standard value of the refractive index for the liquid used in the experiment:
- Water: 1.33
- Glycerin: 1.47
- Olive Oil: 1.46
Percentage error = \(\frac{|\text{Standard value} - \text{Experimental value}|}{\text{Standard value}} \times 100\%\)
10. Precautions
- The optical bench should be perfectly horizontal.
- The plane mirror should be perpendicular to the optical bench.
- The convex lens should be clean and free from dust or fingerprints.
- The object pin and image pin should be straight and vertical.
- Parallax error should be avoided while taking readings.
- The liquid should completely fill the space between the lens and mirror without air bubbles.
- Care should be taken to avoid spillage of the liquid.
- All measurements should be taken with care to minimize errors.
- The experiment should be performed in a well-lit room but avoid direct sunlight on the apparatus.
- The lens should be placed very close to the liquid surface when measurements with liquid are taken.
11. Sources of Error
- Parallax Error: This occurs when the position of the image is not accurately located due to the observer's position.
- Spherical Aberration: The convex lens may suffer from spherical aberration, which affects the image formation.
- Alignment Errors: If the optical axis of the lens is not perpendicular to the plane mirror.
- Temperature Variations: The refractive index of liquids varies with temperature.
- Measurement Errors: Inaccuracies in measuring the positions of the object and images.
- Impurities in Liquid: Any impurities in the liquid can affect its refractive index.
- Liquid Evaporation: During the experiment, some volatile liquids may evaporate, changing their concentration and refractive index.
- Air Bubbles: Presence of air bubbles in the liquid can affect the results.
- Lens Thickness: The formula used assumes a thin lens, but most practical lenses have finite thickness.
12. Viva Voice Questions
The refractive index of a medium is the ratio of the speed of light in vacuum to the speed of light in that medium. It indicates how much light is slowed down in the medium compared to vacuum.
When a liquid is introduced between the lens and the mirror, the optical path of light changes because light travels at different speeds in different media. This alters the effective focal length of the lens-liquid system, thereby changing the position of the image.
Generally, the refractive index of a liquid decreases with increasing temperature. This is because the density of most liquids decreases with temperature, and the refractive index is directly related to the density.
With a concave lens, the images would be virtual and diminished. The formula would need to be modified accordingly, and the experimental setup would be more complex to handle virtual images.
The refractive index of a medium varies with the wavelength of light used, a phenomenon known as dispersion. Generally, the refractive index decreases with increasing wavelength (from violet to red).
Snell's Law states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction equals the ratio of the refractive indices of the two media. In this experiment, Snell's Law governs the bending of light as it passes from air to the lens, lens to liquid, and liquid to mirror, and then back through these media.
Keeping the lens close to the liquid surface ensures that the lens is partially immersed in the liquid, which is essential for the experiment. It also minimizes the air gap that might affect the results and simplifies the calculations by adhering more closely to the theoretical model used.
If a liquid with a higher refractive index is used, the difference between \(v_1\) and \(v_2\) would be larger. This would result in a larger calculated value for the refractive index, reflecting the physical reality of the more significant bending of light in the higher refractive index medium.