Determination of Refractive Index of a Liquid Using Concave Mirror and Plane Mirror
1. Aim
To determine the refractive index of a given liquid using a concave mirror and a plane mirror.
2. Apparatus Used
- Concave mirror with holder
- Plane mirror
- Transparent glass cell or beaker
- Pin with stand (object pin)
- Clamp stand
- Scale/meter rod
- Traveling microscope (optional)
- Liquid whose refractive index is to be determined
- Water
- White screen or paper
3. Diagram
Fig 1: Experimental setup for determination of refractive index
4. Theory
This experiment utilizes the principles of reflection and refraction to determine the refractive index of a liquid. When a concave mirror is placed in a liquid, its focal length changes due to refraction at the air-liquid interface.
For a concave mirror in air, the relation between object distance (u) and image distance (v) is given by the mirror formula:
$$\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$$
where \(f\) is the focal length of the mirror in air.
When the concave mirror is immersed in a liquid of refractive index \(\mu\), the focal length changes to \(f'\), where:
$$f' = \frac{f}{\mu}$$
This is because when light passes from a denser medium to a rarer medium, the radius of curvature effectively changes, affecting the focal length.
In this experiment, we first determine the focal length of the concave mirror in air. Then, we fill the container with the liquid and determine the apparent focal length of the mirror when immersed in the liquid. The ratio of these two focal lengths gives us the refractive index of the liquid.
5. Formula
The refractive index of the liquid is calculated using the formula:
$$\mu = \frac{f}{f'}$$
where:
- \(\mu\) = Refractive index of the liquid
- \(f\) = Focal length of the concave mirror in air
- \(f'\) = Focal length of the concave mirror in the liquid
For determining the focal length, we use the mirror formula:
$$\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$$
If we adjust the object to a position where the image coincides with the object (i.e., \(u = v\)), then:
$$\frac{1}{f} = \frac{2}{u}$$
$$f = \frac{u}{2}$$
6. Procedure
- Part A: Determining the focal length of the concave mirror in air
- Place the concave mirror on the table with its reflecting surface facing upward.
- Place a pin (object) at a certain height above the mirror.
- Adjust the height of the pin until its image coincides with the object itself.
- Measure the distance between the pole of the mirror and the pin. This distance equals twice the focal length of the mirror in air.
- Record this distance as \(2f\).
- Calculate the focal length \(f = 2f/2\).
- Part B: Determining the focal length of the concave mirror in liquid
- Pour the liquid whose refractive index is to be determined into the transparent container.
- Place the concave mirror at the bottom of the container, with its reflecting surface facing upward.
- Place the plane mirror horizontally on top of the liquid surface.
- Position a pin (object) at a certain height above the plane mirror.
- Adjust the height of the pin until its image (formed by reflection from the plane mirror, then from the concave mirror, and again from the plane mirror) coincides with the object itself.
- Measure the distance between the plane mirror and the pin. This distance equals twice the focal length of the concave mirror in the liquid.
- Record this distance as \(2f'\).
- Calculate the focal length in liquid \(f' = 2f'/2\).
- Part C: Calculating the refractive index
- Calculate the refractive index of the liquid using the formula: \(\mu = \frac{f}{f'}\)
7. Observation Table
Table 1: Measurement of focal length of concave mirror in air
| S.No. | Position of object pin where image coincides with object (cm) | Distance between object and mirror (2f) (cm) | Focal length in air, f = 2f/2 (cm) |
|---|---|---|---|
| 1 | |||
| 2 | |||
| 3 | |||
| 4 | |||
| 5 | |||
| Mean value of focal length in air (f) | _____ cm | ||
Table 2: Measurement of focal length of concave mirror in liquid
| S.No. | Position of object pin where image coincides with object (cm) | Distance between object and plane mirror (2f') (cm) | Focal length in liquid, f' = 2f'/2 (cm) |
|---|---|---|---|
| 1 | |||
| 2 | |||
| 3 | |||
| 4 | |||
| 5 | |||
| Mean value of focal length in liquid (f') | _____ cm | ||
8. Calculations
From the experimental observations:
- Mean focal length of concave mirror in air (f) = _____ cm
- Mean focal length of concave mirror in liquid (f') = _____ cm
Using the formula for refractive index:
$$\mu = \frac{f}{f'}$$
$$\mu = \frac{\_\_\_\_ \text{ cm}}{\_\_\_\_ \text{ cm}}$$
$$\mu = \_\_\_\_$$
9. Result
The refractive index of the given liquid is _____, which is close to the expected value for [name of the liquid] which is _____.
The percentage error in the measurement is:
$$\text{Percentage Error} = \frac{|\text{Observed Value} - \text{Standard Value}|}{\text{Standard Value}} \times 100\%$$
$$\text{Percentage Error} = \frac{|\_\_\_\_ - \_\_\_\_|}{\_\_\_\_} \times 100\% = \_\_\_\_\%$$
10. Precautions
- The concave mirror should be clean and free from dust or scratches.
- The pin should be straight and sharp-pointed for accurate observations.
- While taking observations, avoid parallax error by keeping the eye at the same level as the object and its image.
- The liquid should be transparent and free from impurities.
- The plane mirror should be perfectly horizontal on the liquid surface.
- The liquid level should be sufficient to completely cover the concave mirror.
- Measurements should be taken multiple times and averaged to minimize random errors.
- The container should be transparent and have parallel walls to minimize distortion.
- Avoid disturbing the setup once it's aligned for measurements.
- Ensure that the room is adequately illuminated for clear visibility of the image.
11. Sources of Error
- Parallax Error: This occurs if the eye is not correctly positioned when reading measurements.
- Spherical Aberration: The concave mirror may introduce spherical aberration, affecting the clarity of the image.
- Impure Liquid: Impurities in the liquid can change its refractive index.
- Temperature Effects: The refractive index of a liquid varies with temperature, which might not be accounted for.
- Non-parallel Container Walls: If the container walls are not parallel, it can introduce additional refraction, affecting the results.
- Positioning Errors: Inaccurate positioning of the pin, mirror, or other components.
- Measurement Errors: Errors in measuring distances with the scale or meter rod.
- Non-horizontal Plane Mirror: If the plane mirror is not perfectly horizontal, it can introduce errors in the path of light.
- Evaporation: For volatile liquids, evaporation might change the concentration and hence the refractive index during the experiment.
- Illumination Issues: Poor lighting might make it difficult to accurately determine when the image coincides with the object.
12. Viva Voice Questions
A1: The focal length of a concave mirror changes in a liquid due to the change in the refractive index of the medium. When light passes from a denser medium (liquid) to a rarer medium (air), the effective radius of curvature changes, which in turn alters the focal length. The relationship is given by f' = f/μ, where f is the focal length in air and f' is the focal length in the liquid with refractive index μ.
A2: The plane mirror serves as a reference surface at the liquid-air interface. It reflects the incident light downward towards the concave mirror and then back up along the same path. This helps in creating a situation where the image formation can be analyzed without complications from refraction at multiple interfaces.
A3: Generally, the refractive index of a liquid decreases with increasing temperature. This is because as temperature increases, the liquid expands and becomes less dense, which typically leads to a lower refractive index. This is why precise measurements of refractive index should specify the temperature at which they were taken.
A4: When a concave mirror is placed in a medium with a higher refractive index than air, its focal length decreases. This is because the speed of light is slower in the medium with higher refractive index, which affects the path and convergence of light rays after reflection from the mirror.
A5: Multiple readings are taken to minimize random errors that might occur in individual measurements. By taking the average (mean) of several measurements, we can obtain a more reliable result as random errors tend to cancel out when multiple readings are averaged.
A6: If the concave mirror is not completely immersed in the liquid, part of the reflection would occur from a mirror in air and part from a mirror in liquid. This would lead to inconsistent focal points and incorrect determination of the focal length in liquid, ultimately resulting in an inaccurate value of the refractive index.
A7: This method can be used for transparent liquids that do not react with the mirror or container materials. Limitations include difficulty with very viscous liquids (which may not form a flat surface), highly volatile liquids (which evaporate quickly), or very dark or opaque liquids (which absorb too much light for clear reflection).
A8: In the self-coincidence method, the object is positioned such that its image coincides with the object itself. When this occurs, the object is at twice the focal length from the mirror (for a concave mirror in air). This principle simplifies the calculation as we can directly determine the focal length from the measured distance.