## Parallel plate micrometer

**Description:
**

For precise levelling, the estimation of 1 mm is not sufficiently accurate.Aparallel plate glass micrometer in front of the object lens enables readings to be made direct to 0.1 mm, and estimated to 0.01 mm. The parallel plate micrometer works by refracting the image of a staff graduation to make it coincident with the cross-hair. There is, therefore, no estimation of the position of the cross-hair with respect to the graduation.

The principle of the attachment is seen from Figure. Had the parallel plate been vertical the line of sight would have passed through without deviation and the reading would have been 1.026 m, the final figure being estimated. However, by manipulating the micrometer the parallel plate is tilted until the line of sight is displaced to the nearest division marked on the staff, which is 1.02 m. The rotation of the micrometer drum is proportional to the displacement of the image of the staff. The amount of displacement s is measured on the micrometer and added to the exact reading to give 1.02647 m, only the last decimal place is estimated.

It can be seen from Figure 3.36 that the plate could have been moved in the opposite direction, displacing the line of sight up. Since the parallel plate micrometer run is normally equal to the gap between two successive divisions on the staff it will not be possible to gain coincidence on more than one division. The displacement is related to the rotation of the parallel plate as follows. In Figure 3.37 the plate pivots about A. The displacement is BC and the rotation is equal to the angle of incidence i. The thickness of the plate is t and the ray of light from the staff is refracted by an angle r. μ is the refractive index of the glass of the plate.

**Displacement= BC = AB sin(i − r)
**

**Parallel plate micrometer**

**Parallel plate displacement**

But :

**AB = t sec r so Displacement = t sin(i − r) sec r
**

**= t(sin i cos r − cos i sin r) sec r
**

**= t(sin i − cos i sin r sec r)**

But from Snell’s Law of refraction sin i = μsin r. So upon substitution and rearrangement the equation becomes

Displacement = t sin i1 − (1 − sin2 i) ^{1/2} (μ2 − sin2 i)^{−1/2}

If i is small then sin2 i is negligible compared with 1 or μ2 and sin i = i radians. So

Displacement = t (1 − μ −1)i

Since t and μ are fixed properties of the plate then displacement is directly proportional to rotation. Parallel plate micrometers are also manufactured for use with 5 mm graduations.