## Note on Refraction through Prisms

• Note
• Things to remember

A prism is a transparent refracting medium bounded by two plane surfaces meeting each other along a straight edge.

The two inclined plane surfaces APOB and APRC are called refracting faces and the line AP where they meet is called the refracting edge of the prism. The angle between the two refracting faces is called the angle of the prism, A. any plane in the prism which is perpendicular to AP is called the principle section of the prism. The plane XYZ in the figure is a principle section. We draw this plane to represent a prism for simplicity.

#### Refraction through a prism

Let XYZ be a prism with refracting angle A. A ray of light PQ is incident on face XY at angle i. XY is reflected along QR in the prism and emerges out along RS. Here r1 and r2 be the angle of reflection of the first face and angle of incidence for the second face. The angle d between the direction of the incident ray and emergent ray is called angle of deviation. So ∠STL =d is the angle of deviation shown in the figure. MN and ON are normally drawn on face XY and YZ, respectively.

\begin{align*} \text {The angle of deviation produced at first face is } \\ \delta _1 &= \angle TQR \\ &= \angle TQN -\angle RQN \\ &= i –r_i \\ \text {and that produced at the second face} \\\delta _2 &= \angle TRQ \\ &= \angle TRN -\angle QRN \\ &= e –r_2 \\ \end{align*}Since the deviations are in the same direction at two faces, the net deviation produced by the prism is \begin{align*}\delta = &= \angle RTL = \delta _1 + \delta _2 \\ &= i –r_1 + e –r_2 \\ &= i + e –(r_1+r_2) \\ \text {Since} \\ \angle TQR &= i – r_1 \\ \angle TRQ &= e –r_2 \\ \angle QRT &= 180-\delta \\ \text {so in} \Delta QRT, \text {we have} \\ \angle TQR + \angle TRQ + \angle QRT &=180^o \\ \text {or,} \: i –r_1 + e –r_2 + 180 -\delta &= 180^o \\ \text {or,} \: (i+e) – (r_1+r_2)&= \delta \dots (i) \\ \text {In} \Delta QRN, \\ r_1+r_2 + \angle RNQ = 180^o \dots (ii) \\ \text {In quadrilateral} \: QXRN, \\ \angle QXR + \angle XRN + \angle RNQ + \angle NQX =360^o \\ \text {or,} \: A + 90^o + \angle RQN + 90^o = 360^o \\ \text {or,} \: A + \angle RNQ = 180^o \dots (iii) \\ \text {From equation} \:(ii) \text {and} (iii) \\ A &= r_1 + r_2 \dots (iii) \\ \text {Putting this value in equation} (i), \text {we get} \delta &= i + e – A \dots (v) \\ \text {or,}\: \delta + A &= i + e \\ \end{align*}

Thus, when a ray is refracted through a prism, the sum of the angle of incidence and angle of emergence is equal to the sum of deviation and angle of prism.

#### Minimum Deviation

Angle of deviation depends on different factors such as angle of incidence, angle of prism and material of prism etc. When the angle of incidence is increased from zero, angle of deviation first increases, becomes minimum and then increases. When the prism is in minimum deviation position, the light ray passes symmetrically through the prism. In this position angle of incident is equal to angle of emergence and angle of refraction of first phase is equal to angle of incident of second phase. In such case, the refracted ray passes parallel to the base of equilateral prism. So in minimum deviation position,

\begin{align*} \delta = \delta _m , \: i = e \: \text {and} \: r_1 =r_2 =r_3 \\ \text {From equation} \: (iv), \\ A &= r_1+r_2 = r+r = 2r \\ \text {or,} r \frac A2 \dots (i) \\\text {From equation} \: (v), \\ \delta _m &= i + I + A \\ \text {or,} \: 2i &= A + \delta _m \\ \text {or,} \: i &= \frac {A+\delta _m}{2} \dots (iii) \\ \text {From Snell’s law} \\ _q\mu _g &= \frac {\sin i}{\sin r} \\ &= \frac {\sin \left (\frac {A+\delta _m}{2}\right )}{\sin \frac A2} \dots (iii) \\ \therefore \: _a\mu g &= \frac {\sin \left (\frac {A+\delta _m}{2}\right )}{\sin \frac A2} \end{align*}

#### Deviation through a Small Angled Prism

Consider a prism ABC of small angle A about 10o-12o as shown in the figure. When a ray of light PQ making angle of incidence I incident on face AB, it is refracted along QR and finally emerges out along RS. Here, e is angle of emergence, r1 is angle of refraction in first face, r2 is angle of incidence on second face and d is the angle of deviation. The angle of deviation of the ray PQ is given by

\begin{align*}\delta &= (i-r_1) + (e-r_2) \\ &= (i+e)-(r_1+r_2) \dots (i) \\ \text {At face AB,} \\ \mu &= \frac {\sin i}{\sin r_1} \dots (ii) \\ \end{align*}Since angle of incident is small so that angle of refraction is also small, we have \begin{align*}\: \sin \approx i \: \text {and} \: \sin r_1 \approx r_1, \text {and equation} \: (ii) \text {becomes,} \\ \mu &= \frac {i}{r_1} \\ \text {or,} \: I &= \mu r_1 \\ \text {Similarly at face AC,} \: e &= \mu r_2 \\\end{align*} Substituting vale of I and e in equation\begin{align*} \: (i), \text {we get} \\ \delta &= \mu r_1 + \mu r_2 –(r_1+r_2) \\ &= \mu (r_1+r_2) –(r_1+r_2) \\ &= (\mu -1)(r_1+r_2) &= (\mu-1)A \: [\therefore r_1+r_2 =A] \\ \therefore \delta &= A(\mu -1) \\ \end{align*}

#### Grazing Incidence and Grazing Emergence

Grazing Incidence

When a ray light on a face of a prism with an angle of incidence 900, the ray lies on the surface and is refracted through the prism. This refraction of the prism is called the grazing incidence. Since the maximum value of the angle of incidence is 900, the angle of deviation of prism is maximum. From figure ÐNQP = 900, angle of refraction ÐMQR = critical angle, c.

Then the maximum angle of deviation

\begin{align*} \delta _{max} &= (i-r_1)+(e-r_2) &= 90^0-c+e-r \\ &= 90^0+e-(c+r) \\ \end{align*}

From the principle of reversibility of light, it follows that a ray making angle of incidence e on one face of prism, emerges out with the angle of emergence 900 and e. it can be shown that in each case

$$_a\mu _g = \frac {\sqrt {1+ (\cos A + \sin e)}}{\sin A}$$

Grazing Emergence

When a ray PQ is at grazing incidence on face XY of the prism XYZ and angle of Prism A is gradually increased, the refracted ray QR on the face XZ will make a bigger and bigger angle of incidence. At a particular angle of the prism, the emergent ray RS will graze along the surface XZ as shown in the figure. At this condition, we have, ÐQRT = critical angle, c. Again if A is increased further, the rays on the face XZ strikes at an angle of incidence greater than C and hence no ray emerges out of the face XZ. The figure shows the largest angle of prism for grazing emergence. This angle is called limiting angle of the prism. So the limiting angle,

\begin{align*} A &= c+c \\ \text {or,} \: A &= 2c \\ \end{align*}

Hence limiting angle of the prism is twice the critical angle.

• A prism is a transparent refracting medium bounded by two plane surfaces meeting each other along a straight edge.
•  when a ray is refracted through a prism, the sum of the angle of incidence and angle of emergence is equal to the sum of deviation and angle of the prism.
• the limiting angle of the prism is twice the critical angle.
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##### rimjhim

A light ray incident normally on one of the faces of the prism of refractive index 2 and the emergent ray grazes the second face of the prism then the angle of deviation is