Spherometer: Working Principle, Solved Examples, and Applications

Spherometer is a versatile and precise instrument used to measure the curvature or radius of curvature of spherical surfaces. It is particularly valuable in determining the focal length of lenses and the shape of concave or convex mirrors. The design of a spherometer typically consists of a solid metal or plastic base with three equally spaced legs or arms extending from it. 

• At the end of each leg, there is a small spherical tip, forming a tripod-like structure. These tips are carefully machined to ensure uniformity and accuracy.
• To measure the curvature of a spherical surface, the spherometer is placed on the surface with the three legs making contact. 
• The principle behind the spherometer's operation is based on the concept of the sagitta, which refers to the vertical distance between the apex of a spherical surface and a tangent plane passing through a point on its circumference. 

Read More: Dimensional Analysis

Key Terms: Spherometer, Radius, Curvature, Sagitta, Screw, Pitch, Least Count, Depth Gauge.

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What is Spherometer?

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A spherometer is a specialized instrument used to measure the curvature or radius of curvature of spherical surfaces. It consists of a solid base with three equally spaced legs or arms that extend from it, forming a tripod-like structure. At the end of each leg, there is a small spherical tip.

• The spherometer operates on the principle of measuring the sagitta, which is the vertical distance between the apex of a spherical surface and a tangent plane passing through a point on its circumference. 
• By placing the spherometer on the surface to be measured and adjusting a central screw, the instrument is raised or lowered until all three legs make simultaneous contact with the surface. 
• The central screw is then locked in position, and the distance or height between the base and the spherical surface is measured using a depth gauge or a calibrated micrometer.
• By obtaining multiple measurements of the sagitta at different points on the surface, the spherometer enables the calculation of the curvature and radius of curvature. 
• This information is valuable in various fields such as optics, where it aids in determining the focal length of lenses and the shape of mirrors. 
• It is also used in material science, geology, and mechanical engineering to measure the thickness of thin films, the shape of small spheres or cylinders, and the flatness of surfaces.

Read More: Measurement of length

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Spherometer Formula

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The formula to find the radius of curvature using a spherometer is given as

R = (S² / 6h) + h/2

Where:

R is the radius of curvature

S is the screw reading (the vertical distance traveled by the central screw)

h is the height or distance between the base of the spherometer and the spherical surface being measured.

• In this formula, the first term (S² / 6h) represents the contribution of the screw reading to the radius of curvature, and the second term (h/2) accounts for the height or distance between the base and the spherical surface.
• The least count of a spherometer refers to the smallest increment that can be measured using the instrument. It is typically determined by the pitch of the screw and the scale divisions on the depth gauge or micrometer used to measure the height.
• The pitch (P) of the screw is the axial distance traveled by the screw for one complete rotation. 
• It is usually expressed in millimeters per revolution. 
• The formula to calculate the least count (LC) of a spherometer is given by: LC = P / n, Where: LC is the least count, P is the pitch of the screw, and n is the number of divisions on the scale of the depth gauge or micrometer.
• For example, if the pitch of the screw is 1 mm per revolution and the depth gauge has 100 divisions, the least count would be 0.01 mm (1 mm / 100 divisions).

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Spherometer Parts

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Parts of the spherometer are mentioned below–

Spherometer Parts

1. Base: The base of a spherometer is a solid, flat platform that provides stability and support to the instrument. It is usually made of metal or plastic and has a circular shape.
2. Legs: The spherometer consists of three equally spaced legs or arms that extend vertically from the base. These legs form a tripod-like structure and provide stability to the instrument during measurements.
3. Spherical Tips: At the end of each leg, there is a small spherical tip. These tips are carefully machined and have a smooth, curved surfaces. They are in contact with the surface being measured and help ensure accurate readings.
4. Central Screw: The central screw is located at the center of the base and is perpendicular to it. It extends vertically upward from the base and has a threaded shaft. The central screw is used to adjust the height of the spherometer and make contact with the spherical surface being measured.
5. Depth Gauge or Micrometer: A depth gauge or a calibrated micrometer is used to measure the distance between the base and the spherical surface. It is attached to the base and has a scale or dial that provides precise readings.

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Working Principle of Spherometer

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The working principle of a spherometer is based on the measurement of the sagitta, which is the vertical distance between the apex of a spherical surface and a tangent plane passing through a point on its circumference. Here is a brief explanation of the working principle:

• When using a spherometer, the instrument is placed on the surface to be measured, and the central screw is adjusted to raise or lower the spherometer until all three legs make simultaneous contact with the surface. 
• The screw is then locked in position.
• By making contact at three equidistant points on the surface, the spherometer creates a triangle with the surface acting as the base. 
• This triangle allows for the calculation of the sagitta and, subsequently, the curvature or radius of curvature of the spherical surface.
• The height or distance between the base of the spherometer and the spherical surface is then measured using a depth gauge or a calibrated micrometer. 
• By obtaining multiple measurements of the sagitta at different points on the surface, the spherometer enables the calculation of the curvature and radius of curvature using appropriate mathematical formulas.
• The accuracy of the measurements obtained using a spherometer depends on various factors, such as the precision of the spherical tips, the stability of the instrument, and the resolution of the depth gauge or micrometer. 
• These factors collectively contribute to the instrument's ability to provide reliable and precise measurements of the curvature of spherical surfaces.

Read More: Accuracy and Precision Difference

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How to Use a Spherometer?

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There are a few steps described below about the correct method for using the spherometer– 

Step 1: Prepare the Spherometer: To prepare the spherometer at the proper position– 

• Ensure that the spherometer is clean and free from any debris or obstructions. 
• Verify that the spherical tips are in good condition and properly aligned. 
• If necessary, adjust the central screw to ensure it moves smoothly.

Step 2: Place the Spherometer: Next step is placing the apparatus for measurement–

• Position the spherometer on the surface you want to measure, such as a lens or a mirror. 
• Make sure the surface is clean and free from any dirt or smudges.

Step 3: Make Initial Contact: This happens by adjusting the spherometer in the mentioned way– 

• Gently lower the spherometer by turning the central screw until the spherical tips of all three legs make initial contact with the surface. 
• Avoid applying excessive force to prevent damage to the surface.

Step 4: Level the Spherometer: For the accuracy of reading the leveling is needed– 

• Adjust the central screw to ensure that all three legs are in contact with the surface simultaneously. 
• This can be done by carefully turning the screw clockwise or counterclockwise to raise or lower the spherometer until it is properly leveled.

Step 5: Lock the Central Screw: Now for the next step– 

• Once the spherometer is properly leveled and all three legs are making simultaneous contact with the surface, lock the central screw in position to maintain the contact. 
• This prevents any inadvertent movement or changes in the height during the measurement process.

Step 6: Measure the Height: To obtain the reading– 

• Use a depth gauge or a calibrated micrometer attached to the spherometer to measure the height or distance between the base of the instrument and the surface being measured. 
• Read and record the measurement indicated by the depth gauge or micrometer.

Step 7: Repeat Measurements: To be sure about the obtained measurements– 

• The process is repeated by making measurements at different points on the surface. 
• Adjust the position of the spherometer as necessary, ensuring simultaneous contact of all three legs, and take additional height measurements using the depth gauge or micrometer.

Step 8: Calculate Radius of Curvature– Now for the final measurement step–

• Once the multiple height measurements at different points on the surface are obtained, use the appropriate mathematical formula to calculate the radius of curvature. 
• The formula commonly used is R = (S² / 6h) + h/2, where R is the radius of curvature, S is the screw reading, and h is the average height measurement.

Step 9: Analyze and Interpret Results– At the very final step– 

• Analyze the calculated radius of curvature and interpret the results based on the specific application or purpose. 
• This may involve comparing the measured radius of curvature to a desired specification or using it to determine the focal length of a lens or the shape of a mirror.

Read More: Micrometer

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Solved Examples

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Example 1: Calculate the radius of curvature of a convex lens using the spherometer formula. From the below given data:

Solution: To calculate the radius of curvature, the spherometer formula: R = (S² / 6h) + h/2 is used.

Assuming the screw reading (S) for all measurements is 1.5 mm, the radius of curvature for each measurement point using the formula can be calculated. 

By substituting the height (h) and screw reading (S) values into the spherometer formula, the corresponding values for the radius of curvature (R) for each measurement point are obtained.

Now, to determine the average radius of curvature, the mean value of the calculated radius of curvature is taken. Sum up all the calculated radius of curvature values and divide by the total number of measurements.

Average Radius of Curvature = (39.00 + 38.50 + 38.83 + 39.16 + 38.67) / 5 = 38.83 mm

The average radius of curvature for the convex lens, based on the spherometer measurements, is found to be 38.83 mm.

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Applications of a Spherometer

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A few applications of spherometer are–

1. Optics: Spherometers are extensively used in optics for the measurement of lens curvatures, focal lengths, and radii of curvature. 
2. Mirror Manufacturing: Spherometers are also used in the production and testing of mirrors. 
3. Surface Flatness Analysis: Spherometers are employed to measure the flatness of surfaces. 
4. Thin Film Thickness Measurement: Spherometers find applications in measuring the thickness of thin films. 
5. Geology and Earth Sciences: Spherometers are utilized in geological studies to measure the curvature and radius of spherical structures, such as rock formations, concretions, and fossilized objects. 
6. Quality Control and Metrology: Spherometers play a vital role in quality control and metrology by providing accurate and precise measurements of curvature and radius of curvature. 

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Things to Remember

• A spherometer is an instrument used to measure the curvature, radius of curvature, and sagitta of spherical surfaces.
• It consists of a base, three legs with spherical tips, a central screw, and a depth gauge or micrometer.
• The formula R = (S² / 6h) + h/2 is used to calculate the radius of curvature, where R is the radius, S is the screw reading, and h is the height or distance between the base and the surface being measured.
• The least count of a spherometer is determined by the pitch of the screw and the number of divisions on the depth gauge or micrometer.
• Spherometers find applications in optics for measuring lens curvatures, focal lengths, and radii of curvature.
• They are used in mirror manufacturing to ensure proper shaping and curvature for optimal reflection.
• Spherometers assist in analyzing surface flatness by measuring the sagitta at different points on a surface.
• They are utilized for thin film thickness measurement in industries such as optics, electronics, and material science.