## Introduction: Measuring the Refractive Index of a Rectangular Glass Slab Without the Use of a Protractor.

One of the classical and least costly experiments performed by students for estimating the refractive index (RI) of glass using a rectangular glass slab without the use of a light box or laser light uses two pairs of pins. The glass slab is placed on a piece of paper which lies over some material like cork or corrugated cardboard into which pins can be readily inserted and removed. One pair of pins is inserted in the piece of paper so that a straight line drawn through the pin holes makes an angle (not a right angle) to one face of the glass slab. This line indicates the path of the ray entering the glass slab and is known as the incident ray. The other pair of pins is placed in line with the first pair of pins when viewed through the opposite face of the glass slab. The line joining these latter pin holes indicates the path of the ray leaving the glass slab and is known as the emergent or exiting ray.

Once the outline of the rectangular glass slab is drawn on the piece of paper and the pins removed, a number of steps are carried out which, in most descriptions of the experiment, require the measurement of angles with a protractor. Then the sines of these angles are computed using a calculator with trigonometric functions (or tables of trigonometric functions).

This *Instructables* points out that the steps involved in the measurement of angles with a protractor and then finding the sines of these angles are not necessary for computing the RI. Instead, one constructs various right-angled triangles, measures various lengths and then makes use of the definition of the sine of an angle in a right-angled triangle, namely, the sine of an angle is equal to the ratio of the length of the side opposite the angle whose sine is required divided by the length of the hypotenuse of the triangle.

Step 1 of this *Instructables* describes the conventional procedure used for working out the RI of glass using a rectangular glass slab. Steps 2 and 3 then present two of a number of different ways to perform the procedure without the use of a protractor.

The procedures described in this *Instructables* assume that the adjacent sides at the corners of the rectangular glass slab are perpendicular and can be used to draw lines at right angles to each other.

## Step 1: Description of the Usual Method for Determining the Refractive Index of a Rectangular Glass Slab Using a Protractor.

Once the outline of the rectangular glass slab is drawn on the piece of paper (thin lines in the above diagram) and the pins removed, the following steps are carried out (refer to the above diagram):

- two straight lines (AB and CD) are drawn with each line passing through one of the pairs of pin holes made by the pins and extended to the surface of the glass slab nearest the position of the two holes; these lines do not extend inside the glass slab; they represent the incident (AB) and emergent (CD) rays;
- straight lines (EF and GH) are drawn normal to the surfaces of the glass slab that pass through the point where the lines drawn in the previous step intersect the surfaces of the glass slab; these lines are drawn so that they lie outside and inside the glass slab;
- a straight line (BC) is drawn between each of the points of intersection made by the normal lines drawn in the previous step and the surfaces of the glass slab; this line represents the refracted ray;
- measure the angle of incidence (angle ABE) which is the angle between the incident ray and the normal that the incident ray makes with the glass surface;
- measure the angle of refraction (angle CBF or angle BCH) which is the angle between the refracted ray and either of the normals that the incident ray or the emergent ray makes with the glass surface;
- measure the angle of emergence (angle DCG) which is the angle between the emergent ray and the normal that the emergent ray makes with the glass surface; while this sub-step is not necessary for computing the RI, it is measured to show that the angle of incidence is equal to the angle of emergence and thus that the incident and emergent rays are parallel.

Following these measurements, the refractive index (RI) of the glass is obtained by taking the ratio of the sine of the angle of incidence to that of the sine of the angle of refraction.

This procedure is then repeated for a number of different incident angles to show that the ratio of the sine of the angle of incidence to that of the sine of the angle of refraction should remain constant when a number of different values are used for the incident angle.

## Step 2: Description of a Procedure for Finding the Sines of the Angles Made by the Incident and Refracted Rays Without Using a Protractor.

In this procedure it is not necessary to trace the complete outline of the glass slab; only the opposite surfaces through which the light rays travel need be indicated as shown in the above diagram, where the thin lines represent these two surfaces.

The three initial sub-steps of Step 1 are carried out except that Sub-step 1 is modified by extending the lines AB and CD (broken lines) as follows:

- extend the incident ray AB forwards so that it cuts the opposite face of the glass slab at point I; this is the path that the incident ray would follow if the glass slab were absent;
- extend the emergent ray CD backwards so that it cuts the opposite face of the glass slab at point J.

If the incident and refracted rays when extended do not cut the surfaces of the glass slab at points I and J extend the lengths representing these surfaces so that points I and J can be found.

BHCF is a rectangle formed by the surfaces of the glass slab and the normals to the surfaces at points B and C; its diagonal BC represents the path followed by the refracted ray through the glass slab.

From the geometry of the various right-angled triangles in the above diagram, note:

- the angle of incidence, angle ABE, is equal to the angle FBI and its sine is equal to the ratio of lengths FI over BI;
- the angle of emergence, angle DCG, is equal to the angle HCJ and its sine is equal to the ratio of lengths HJ over CJ;
- the angle of refraction is angle CBF (or angle BCH) and its sine is equal to the ratio of lengths FC over BC (or the ratio of lengths BH over BC).

Thus, without using a protractor or having to work out the sines of angles using a calculator with trigonometric functions (or trigonometric tables), the sines of the various angles needed for calculating the RI of the glass slab can be simply found from measurements of various length.

## Step 3: Description of Another Procedure for Finding the Sines of the Angles Made by the Incident and Refracted Rays Without Using a Protractor.

Unlike the procedure described in Step 2, here we have to include a full outline of the glass slab as various lengths that need to be measured involve the sides of the glass slab that are at right angles to the surfaces of the slab through which the light rays travel. Two of the corners of the glass slab are labelled K and L in the above diagrams. Extensions of the lengths of lines that have to be drawn are as follows:

- in the left-hand diagram above, the incident and emergent rays are extended so that they cut the right and left surfaces of the glass slab that are at right angles to the surfaces through which the rays travel respectively; these surfaces are cut at points I and J respectively;
- in the right-hand diagram above, the left and right surfaces of the glass slab that are at right angles to the surfaces through which the rays travel are extended to as to cut the incident and emergent rays at points I and J respectively.

Again, from the geometry of the various right-angled triangles in the above diagrams, note the following for both diagrams:

- the angle of incidence, angle ABE, is equal to the angle BIK and its sine is equal to the ratio of lengths BK over BI;
- the angle of emergence, angle DCG, is equal to the angle CJL and its sine is equal to the ratio of lengths CL over CJ;
- the angle of refraction is angle CBF (or angle BCH) and its sine is equal to the ratio of lengths FC over BC (or the ratio of lengths BH over BC).

Thus again, without using a protractor or having to work out the sines of angles using a calculator with trigonometric functions (or trigonometric tables), the sines of the various angles needed for calculating the RI of the glass slab can be simply found from measurements of various lengths.