Determination of Radius of Curvature of a Spherical Surface Using a Spherometer
1. Aim
To determine the radius of curvature of a given spherical surface using a spherometer.
2. Apparatus Used
- Spherometer
- Spherical surface (convex lens or watch glass)
- Plane glass plate
- Screw gauge (for measuring distance between legs)
- Vernier calipers (optional)
- Tissue paper or soft cloth for cleaning
3. Diagram
4. Theory
A spherometer is a precision instrument used to measure the radius of curvature of a spherical surface. It consists of a central screw with a fine pitch, surrounded by three equidistant legs that form an equilateral triangle. The basic principle involves measuring the "sagitta" (height) of the spherical segment formed when the spherometer is placed on the spherical surface.
When the spherometer is placed on a spherical surface:
- The three legs of the spherometer rest on the spherical surface, forming a circle of contact.
- The central screw is adjusted until its tip just touches the surface.
- The difference in the reading of the central screw when placed on a plane surface and when placed on the spherical surface gives the sagitta (h).
Using geometric principles, we can establish a relationship between:
- The radius of curvature (R) of the spherical surface
- The sagitta (h)
- The radius (a) of the circle formed by the three legs
When a spherometer is placed on a spherical surface, from the geometry of a circle:
$$R^2 = (R-h)^2 + a^2$$
Expanding the expression:
$$R^2 = R^2 - 2Rh + h^2 + a^2$$
Simplifying:
$$2Rh = h^2 + a^2$$
Since \(h\) is very small compared to \(R\), \(h^2\) can be neglected:
$$2Rh \approx a^2$$
Therefore:
$$R \approx \frac{a^2}{2h}$$
For a spherometer with three legs forming an equilateral triangle with side length \(L\), the radius \(a\) of the circle passing through the three points is:
$$a = \frac{L}{\sqrt{3}}$$
5. Formula
The radius of curvature (R) of the spherical surface is given by:
$$R = \frac{a^2 + h^2}{2h}$$
Since \(h\) is very small compared to \(a\), we can simplify it to:
$$R \approx \frac{a^2}{2h}$$
Where:
- \(R\) = Radius of curvature of the spherical surface
- \(a\) = Radius of the circle formed by the three legs of the spherometer
- \(h\) = Sagitta (height) measured by the spherometer
For a spherometer with three legs forming an equilateral triangle with side length \(L\):
$$a = \frac{L}{\sqrt{3}}$$
Substituting this value of \(a\) in the formula for \(R\):
$$R = \frac{L^2}{6h}$$
6. Procedure
- Measurement of distance between legs (L):
- Measure the distance between each pair of legs using a screw gauge or vernier calipers.
- Take multiple readings and find the average value of L.
- Determination of least count:
- Note the pitch of the screw (p) - distance moved by the screw in one complete rotation.
- Count the number of divisions (n) on the circular scale.
- Calculate the least count using the formula: Least Count = p/n.
- Zero error determination:
- Place the spherometer on a plane glass plate.
- Adjust the central screw until it just touches the glass surface (a characteristic sound will be heard when the tip gently touches and slides on the surface).
- Note the reading on the main scale and circular scale.
- This gives the zero error. If the reading is not zero, note it as positive or negative zero error.
- Measurement of sagitta (h):
- Place the spherometer on the given spherical surface with the three legs resting on it.
- Adjust the central screw until its tip just touches the spherical surface.
- Note the reading on the main scale and circular scale.
- Calculate the sagitta by taking the difference between this reading and the reading on the plane surface, after applying zero correction.
- Repeating measurements:
- Repeat steps 3 and 4 at least five times to minimize random errors.
- Calculate the average value of h.
- Calculation:
- Calculate the radius of curvature using the formula R = L²/6h.
7. Observation Table
Table 1: Measurement of the distance between legs (L)
| S.No. | Legs Pair | Distance (cm) |
|---|---|---|
| 1 | Between leg 1 and leg 2 | ... |
| 2 | Between leg 2 and leg 3 | ... |
| 3 | Between leg 3 and leg 1 | ... |
Mean value of L = _______ cm
Table 2: Determination of least count
| Pitch of the screw (p) | _______ mm |
| Number of divisions on circular scale (n) | _______ divisions |
| Least count = p/n | _______ mm/division |
Table 3: Determination of zero error
| S.No. | Main Scale Reading (mm) | Circular Scale Reading × Least Count (mm) | Total Reading (mm) |
|---|---|---|---|
| 1 | ... | ... | ... |
| 2 | ... | ... | ... |
| 3 | ... | ... | ... |
Mean zero error = _______ mm
Table 4: Measurement of sagitta (h)
| S.No. | Reading on Spherical Surface (mm) | Reading on Plane Surface (mm) | Sagitta (h) = Difference in Readings (mm) | Corrected Sagitta after Zero Error Correction (mm) |
|---|---|---|---|---|
| 1 | ... | ... | ... | ... |
| 2 | ... | ... | ... | ... |
| 3 | ... | ... | ... | ... |
| 4 | ... | ... | ... | ... |
| 5 | ... | ... | ... | ... |
Mean value of h = _______ mm = _______ cm
8. Calculations
Given:
- Mean distance between legs, L = _______ cm
- Mean value of sagitta, h = _______ cm
Using the formula:
$$R = \frac{L^2}{6h}$$
Substituting the values:
$$R = \frac{(\_\_\_\_\_)^2}{6 \times (\_\_\_\_\_)}$$
$$R = \_\_\_\_\_ \text{ cm}$$
9. Result
The radius of curvature of the given spherical surface is _______ cm.
Percentage error = _______ %
10. Precautions
- The spherometer should be placed gently on the spherical surface to avoid any damage.
- The spherical surface and the plane glass plate should be cleaned properly before use.
- The central screw should be adjusted carefully until it just touches the surface - excessive pressure can damage both the instrument and the surface.
- The spherometer should be placed at the center of the spherical surface to ensure accurate measurements.
- The reading should be taken with the line of sight perpendicular to the scale to avoid parallax error.
- While measuring the distance between legs, care should be taken to measure from the center of each leg.
- Zero error should be determined and applied to all readings.
- The central screw should be rotated in only one direction to avoid backlash error.
- Multiple readings should be taken and averaged to minimize random errors.
11. Sources of Error
- Instrumental Errors:
- Backlash error in the screw mechanism
- Uneven spacing between the legs
- Non-uniform pitch of the central screw
- Blunt tip of the central screw
- Observational Errors:
- Parallax error while taking readings
- Error in judging when the central screw just touches the surface
- Error in measuring the distance between legs
- Environmental Errors:
- Temperature variations affecting the dimensions of the spherometer
- Dust particles on the surface affecting the contact
- Theoretical Approximations:
- Neglecting the term \(h^2\) in the formula \(R = \frac{a^2 + h^2}{2h}\)
- Assumption of perfect sphericity of the surface
12. Viva Voice Questions
Q1. What is a spherometer and what is its principle?
A spherometer is an instrument used to measure the radius of curvature of spherical surfaces. Its principle is based on measuring the sagitta (height) of a spherical segment formed when the spherometer's three legs rest on the spherical surface.
Q2. How would you differentiate between convex and concave surfaces using a spherometer?
For a convex surface, the reading on the spherical surface will be less than the reading on the plane surface, resulting in a negative value of the sagitta. For a concave surface, the reading on the spherical surface will be greater than the reading on the plane surface, resulting in a positive value of the sagitta.
Q3. Why do we neglect the term \(h^2\) in the formula \(R = \frac{a^2 + h^2}{2h}\)?
The term \(h^2\) is neglected because the sagitta (h) is very small compared to the radius of curvature (R) and the radius of the circle formed by the three legs (a). Thus, \(h^2\) becomes negligibly small compared to \(a^2\), allowing us to simplify the formula to \(R \approx \frac{a^2}{2h}\).
Q4. What is backlash error and how can it be minimized?
Backlash error occurs due to the play between the screw and the nut in the spherometer. When the direction of rotation of the screw is reversed, there is a small movement of the screw without corresponding movement of the nut. To minimize this error, readings should always be taken while rotating the screw in the same direction (preferably clockwise).
Q5. How does the least count of a spherometer affect the accuracy of measurement?
The least count of a spherometer determines the smallest measurement it can accurately record. A smaller least count allows for more precise measurements, reducing the uncertainty in the determined radius of curvature. The least count depends on the pitch of the screw and the number of divisions on the circular scale.
Q6. Why is it important to place the spherometer at the center of the spherical surface?
Placing the spherometer at the center ensures that the measurement is taken at the vertex of the spherical surface, where the curvature is most uniform. If placed off-center, the measured sagitta would correspond to a different section of the sphere, leading to inaccurate results.
Q7. Can a spherometer be used to measure the refractive index of a material?
Not directly. However, if used to measure the radius of curvature of a lens surface, the spherometer's measurements can be used in conjunction with other optical measurements (like focal length) to calculate the refractive index using the lens maker's formula.
Q8. What are the limitations of a spherometer?
Limitations include: inability to measure very large radii of curvature accurately, requirement for a smooth and clean surface, difficulty in determining the exact point of contact with the surface, and assumptions in the theoretical formula that may introduce systematic errors.