To determine the radius of curvature of a given spherical surface by a spherometer.
A spherometer, a spherical surface such as a watch glass or a convex mirror and a plane glass plate of about 6 cm x 6 cm size.
A spherometer consists of a metallic triangular frame F supported on three legs of equal length A, B and C (Fig. E 3.1). The lower tips of the legs form three corners of an equilateral triangle ABC and lie on the periphery of a base circle of known radius, r. The spherometer also consists of a central leg OS (an accurately cut screw), which can be raised or lowered through a threaded hole V (nut) at the centre of the frame F. The lower tip of the central screw, when lowered to the plane (formed by the tips of legs A, B and C) touches the centre of triangle ABC. The central screw also carries a circular disc D at its top having a circular scale divided into 100 or 200 equal parts. A small vertical scale P marked in millimetres or half-millimetres, called main scale is also fixed parallel to the central screw, at one end of the frame F. This scale P is kept very close to the rim of disc D but it does not touch the disc D. This scale reads the vertical distance which the central leg moves through the hole V. This scale is also known as pitch scale.
Pitch: It is the vertical distance moved by the central screw in one complete rotation of the circular disc scale. Commonly used spherometers in school laboratories have graduations in millimetres on pitch scale and may have100 equal divisions on circular disc scale. In one rotation of the circular scale, the central screw advances or recedes by 1 mm. Thus, the pitch of the screw is 1 mm.
Least Count: Least count of a spherometer is the distance moved by the spherometer screw when it is turned through one division on the circular scale, i.e.,
Least count of the spherometer =Pitchof thespherometerscrew /Numberof divisions on the circular scale
The least count of commonly used spherometers is 0.01 mm. However, some spherometers have least count as small as 0.005 mm or 0.001 mm.
Formula for The Radius of Curvature of A Spherical Surface
Let the circle AOBXZY (Fig. E 3.2) represent the vertical section of sphere of radius R with E as its centre (The given spherical surface is a part of this sphere). Length OZ is the diameter (= 2R ) of this vertical section, which bisects the chord AB. Points A and B are the positions of the two spherometer legs on the given spherical surface. The position of the third spherometer leg is not shown in Fig. E 3.2. The point O is the point of contact of the tip of central screw with the spherical surface. Fig. E 3.3 shows the base circle and equilateral triangle ABC formed by the tips of the three spherometer legs. From this figure, it can be noted that the point M is not only the mid point of line AB but it is the centre of base circle and centre of the equilateral triangle ABC formed by the lower tips of the legs of the spherometer (Fig. E 3.1). In Fig. E 3.2 the distance OM is the height of central screw above the plane of the circular section ABC when its lower
tip just touches the spherical surface. This distance OM is also called sagitta. Let this be h. It is known that if two chords of a circle, such as AB and OZ, intersect at a point M then the areas of the rectangles described by the two parts of chords are equal. Then
AM.MB = OM.MZ
(AM)^{2} = OM (OZ - OM) as AM = MB
Let EZ (= OZ/2) = R, the radius of curvature of the given spherical surface and AM = r, the radius of base circle of the spherometer.
r^{2} = h (2R - h)
Thus, R = r^{2}/2h + h/2
Now, let l be the distance between any two legs of the spherometer or the side of the equilateral triangle ABC (Fig. E 3.3), then from geometry we have
Thus, r = 1/√ 3 , the radius of curvature (R) of the given spherical surface can be given by
R= ι ^{2}/6h + h/2
A. Pitch of the screw:
Least Count (L.C.) of the spherometer:
Determination of length l (from equilateral triangle ABC)
Table E 3.1 Measurement of sagitta h
S. No. | Spherometer Readings | (h_{1}-h_{2}) |
with sphherical surface | Pitch Scale reading x (cm)/Circular scale division coinciding with pitch scale y/Circular scale reading z =y x L.C. (cm)/Spherometer reading with spherical surface h1 = x + z (cm) | |
Horizontal plane surface | Pitch Scale reading x1 (cm)/Circular scale division coinciding with pitch scale y/Circular scale reading z' =y x L.C. (cm)/Spherometer reading with spherical surface h2=x' + z' (cm) |
Mean h = ... cm
Using the values of l and h, calculate the radius of curvature R from the formula:
R = ι ^{2}/6h + h/2;
the term h/2 may safely be dropped in case of surfaces of large radii of curvature (In this situation error in ι ^{2}/6h is of the order of h/2.)
The radius of curvature R of the given spherical surface is ... cm.
Get all latest content delivered to your email a few times a month.