An optical flat is used to measure the height difference between a reference slip gauge A and a slip gauge B. Upon viewing via the optical flat using a monochromatic light of wavelength 0.5 µm, 12 fringes were observed over a length of 15 mm of gauge B. If the gauges are placed 45 mm apart, the height difference of the gauges is ______________ µm. (Answer in integer)

The correct answer is 9.

1. Understanding the Principle:
When an optical flat is placed over two slip gauges of different heights (reference gauge A and test gauge B) on a flat reference surface, a wedge-shaped air film is formed. Monochromatic light directed normally onto the setup reflects from both the bottom surface of the optical flat and the top surfaces of the slip gauges, resulting in interference fringes (alternating light and dark bands).

The wedge angle, $\theta$, formed by the tilt of the optical flat can be expressed in two ways based on the geometry of the system:

$\theta =\frac{h}{L}$

where:
• $h$ is the height difference between gauge A and gauge B.
• $L$ is the distance between the two gauges ($L=45\text{mm}$).

The wedge angle can also be defined by the fringe pattern observed over gauge B:

$\theta =\frac{n\cdot \left(\frac{\lambda }{2}\right)}{W}$

where:
• $n$ is the number of fringes observed over the gauge ($n=12$).
• $\lambda$ is the wavelength of the monochromatic light ($\lambda =0.5\text{μm}$).
• $W$ is the length (or width) of gauge B over which the fringes are observed ($W=15\text{mm}$).

2. Equating the Formulations:
Equating both expressions for the wedge angle $\theta$ gives:

$\frac{h}{L}=\frac{n\cdot \lambda }{2\cdot W}$

Rearranging the equation to solve for the height difference $h$:

$h=\frac{n\cdot \lambda \cdot L}{2\cdot W}$

3. Calculation:
Substituting the given values into the formula:

$h=\frac{12\cdot 0.5\text{μm}\cdot 45\text{mm}}{2\cdot 15\text{mm}}$

Simplifying the numerator and denominator:

$h=\frac{6\cdot 45}{30}\text{μm}$

$h=\frac{270}{30}\text{μm}=9\text{μm}$

Thus, the height difference between the slip gauges is 9 µm.