You are provided with a triangular glass prism, four optical pins, and other necessary materials. Place the triangular glass prism on a drawing paper and dr...

Refraction through a triangular glass prism

The refracting angle of the prism (the angle at U) is measured as \(A = U = 60^{\circ}\).

A ray is directed at the face UM so that it makes the chosen angle of incidence \(\phi\) with the normal UN. It refracts into the glass, strikes the second face UR, and emerges making an angle of emergence \(e\) with the normal XY. The angle between the incident direction produced and the emergent ray is the angle of deviation \(d\). The complete ray path traced on the drawing paper is shown below.

Table of readings

For each setting of \(\phi\) the corresponding angle of emergence \(e\) and angle of deviation \(d\) are measured with the protractor and recorded.

S/N	\(\phi\,/^{\circ}\)	\(e\,/^{\circ}\)	\(d\,/^{\circ}\)
1	60.0	39.0	39.0
2	55.0	42.5	37.5
3	50.0	47.0	37.0
4	40.0	58.5	38.5
5	35.0	66.0	41.0

Graph of \(d\) against \(e\)

Plotting \(d\) on the vertical axis against \(e\) on the horizontal axis (both axes starting from the origin) and joining the points with a smooth curve gives a shallow U-shaped curve. The lowest point of the curve gives the minimum deviation.

From the graph the turning point (lowest point of the curve) gives:

\[d_{m} = 37^{\circ}, \qquad e_{m} = 48^{\circ}\]

Refractive index

Using the formula with \(U = A = 60^{\circ}\) and \(d_{m} = 37^{\circ}\):

\[n = \frac{\sin\left(\dfrac{d_{m}+U}{2}\right)}{\sin\left(\dfrac{U}{2}\right)} = \frac{\sin\left(\dfrac{37^{\circ}+60^{\circ}}{2}\right)}{\sin\left(\dfrac{60^{\circ}}{2}\right)}\]\[n = \frac{\sin 48.5^{\circ}}{\sin 30^{\circ}} = \frac{0.7490}{0.5000}\]\[\boxed{n = 1.50}\]

Precautions

• The optical pins were fixed vertically (upright) and reasonably spaced apart before the sighting lines were drawn.
• Parallax error was avoided when reading the protractor by placing the eye directly above each mark.

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(b) Total internal reflection

(i) Conditions necessary for total internal reflection:

• Light must travel from an optically denser medium into an optically less dense (rarer) medium.
• The angle of incidence in the denser medium must be greater than the critical angle.

(ii) The refractive index is related to the critical angle \(c\) by:

\[n = \frac{1}{\sin c}\]

With \(c = 39^{\circ}\):

\[n = \frac{1}{\sin 39^{\circ}} = \frac{1}{0.6293}\]\[\boxed{n = 1.59}\]