You are provided with a glass block, plane mirror, and optical pins. Place the glass block on a drawing sheet and trace its outline ABCD as shown in the dia...

Answer Details

Refraction through a glass block with a plane mirror on face AB

The block is traced as ABCD, the width is measured as \(W = 5.0\ \text{cm}\). For each angle of incidence the emergent ray is fixed by no-parallax pins \(P_3\) and \(P_4\), and the angles \(\theta\) and \(e\) together with the lateral displacement \(d\) are measured directly from the traces. The full set of readings and the derived quantities \(m = \sin e\) and \(n = \cos\!\left(\dfrac{\theta}{2}\right)\) are tabulated below.

Table of readings

\(i/^{\circ}\)	\(\theta/^{\circ}\)	\(e/^{\circ}\)	\(d/\text{cm}\)	\(m = \sin e\)	\(n = \cos\left(\frac{\theta}{2}\right)\)
10	10.4	10.0	3.00	0.174	0.996
20	19.0	20.4	3.90	0.349	0.986
30	20.0	30.0	6.00	0.500	0.985
40	30.0	40.0	7.00	0.643	0.966
50	30.0	50.0	7.50	0.766	0.966

Sample evaluations: for \(i = 30^{\circ}\), \(m = \sin 30.0^{\circ} = 0.500\) and \(n = \cos\!\left(\tfrac{20.0^{\circ}}{2}\right) = \cos 10.0^{\circ} = 0.985\). For \(i = 50^{\circ}\), \(m = \sin 50.0^{\circ} = 0.766\) and \(n = \cos 15.0^{\circ} = 0.966\).

Graph of \(m\) against \(n\)

Slope of the graph

Two points on the line of best fit are \((n_1, m_1) = (0.996,\ 0.174)\) and \((n_2, m_2) = (0.966,\ 0.766)\).

\[ s = \frac{m_2 - m_1}{n_2 - n_1} = \frac{0.766 - 0.174}{0.966 - 0.996} = \frac{0.592}{-0.030} = -19.7 \]

The slope is \(s = -19.7\) (magnitude \(19.7\)).

Evaluation of \(q = 2Ws\)

With \(W = 5.0\ \text{cm}\) and \(s = -19.7\):

\[ q = 2Ws = 2 \times 5.0\ \text{cm} \times (-19.7) = -197\ \text{cm} \]

Hence \(|q| = 197\ \text{cm}\).

Two precautions

• The optical pins were fixed vertically, well separated, and the outline of the block was traced neatly with a sharp pencil so that the rays could be drawn accurately.
• Parallax error was avoided when reading the protractor and metre rule, and zero error on the metre rule was checked before measuring the width \(W\).

(b)(i) Refractive index

The refractive index is the ratio of the velocity of light in air (vacuum) to the velocity of light in a material medium as light waves pass from air into the medium. In terms of wavelength, since the frequency is unchanged on refraction,

\[ n = \frac{\lambda_{1}}{\lambda_{2}} \]

where \(\lambda_{1}\) is the wavelength in air, \(\lambda_{2}\) is the wavelength in the material, and \(n\) is the refractive index of the material.

(b)(ii) Conditions for total internal reflection

• The light must be travelling from a denser medium to a less dense medium.
• The angle of incidence in the denser medium must be greater than the critical angle.