Answer Hint : In this solution, we will use the relation of the refractive index of the prism with the angle of prism and angle of minimum deviation. By taking the assumption that the angle of minimum deviation of a prism is equal to its refracting angle, we will obtain a range of possible refractive index for the prism.Formula used: In this solution, we will use the following formula: $\Rightarrow \mu = \dfrac{{sin\left( {\dfrac{{A + {\delta _m}}}{2}} \right)}}{{sin\left( {\dfrac{A}{2}} \right)}} $ where $ \mu $ is the refractive index, $ A $ is the angle of the prism, and $ {\delta _m} $ is the angle of minimum deviation. Complete step by step answer The minimum deviation $ {\delta _m} $ in a prism occurs when the entering angle and the exiting angle are the same. In this situation, the ray of light when inside the prism is parallel to the base of the prism. The angle of refraction inside the prism on either side of the prism is also the same in this situation. Given that we know that angle of minimum deviation and the angle of the prism, we can find the refractive index of the prism $ \mu $ as: $\Rightarrow \mu = \dfrac{{sin\left( {\dfrac{{A + {\delta _m}}}{2}} \right)}}{{sin\left( {\dfrac{A}{2}} \right)}} $ Since we’ve been told for the angle of minimum deviation of a prism ( $ {\delta _{min}} $ ) to be equal to its refracting angle ( $ A $ ), we can write $ {\delta _m} = A $ . On substituting it in the above equation, we can write $\Rightarrow \mu = \dfrac{{sin\left( {\dfrac{{A + A}}{2}} \right)}}{{sin\left( {\dfrac{A}{2}} \right)}} $ $ \Rightarrow \mu = \dfrac{{sin\left( A \right)}}{{sin\left( {\dfrac{A}{2}} \right)}} $ Now, we know that $ \sin (A) = 2\sin (A/2)\cos (A/2) $ so we can write, $\Rightarrow \mu = \dfrac{{2\sin (A/2)\cos (A/2)}}{{sin\left( {\dfrac{A}{2}} \right)}} $ $ \Rightarrow \mu = 2\cos (A/2) $ The angle of the prism $ A $ can only lie between $ 0^\circ $ and $ 90^\circ $ , so the minimum value of the refractive index will be for $ 90^\circ $ and will be, $\Rightarrow {\mu _{min}} = 2\cos (45^\circ ) $ $ \Rightarrow {\mu _{\min }} = \dfrac{2}{{\sqrt 2 }} = \sqrt 2 $ And the maximum value will be for $ 0^\circ $ which will be $\Rightarrow {\mu _{min}} = 2\cos (0^\circ ) $ $ \Rightarrow {\mu _{\min }} = 2 $So, $ \sqrt 2 < \mu < 2 $ which corresponds to option A. Note While calculating the limits of the refractive index, the limits depend on the angle of the prism which can only lie between $ 0^\circ $ and $ 90^\circ $ and a prism cannot have any angle outside this range. However, the range of refractive index that we obtained is specific for the case where the angle of minimum deviation of a prism is equal to its refracting angle otherwise a prism can have a wide range of refractive index depending on the material it is made up of.
Option 2 : lies between 2 and √2 India's Super Teachers for all govt. exams Under One Roof
CONCEPT:
where r1 and r2 are the refractive angles. The formula for the angle of minimum deviation: δmin = i + e - A where, i = angle of incidence, e = angle of emergence, A = prism angle. EXPLANATION: From the above discussion, for minimum deviation δmin = A then 2A = i + e in case of δ min i = e \(2A = 2i,{r_1} = {r_2} = \frac{A}{2}\) i = A = 90° From snell's law sin i = n sin r1 sin A = n sin A/2 By trigonometry, sinθ = 2 sin θ /2 cos θ /2 2 sin A/2 cos A/2 = n sin A/2 2 cos A/2 = n when A = 90° = imin nmin = √ 2 i = A = 0, nmax = 2 The correct option is b.
Formula: μ = c/v where c = speed of light, v = phase velocity of light. sin i = sin r where, i = angle of incidence, r = angle of refraction India’s #1 Learning Platform Start Complete Exam Preparation
Daily Live MasterClasses
Practice Question Bank
Mock Tests & Quizzes Trusted by 3.3 Crore+ Students In a prism, the angle of deviation (δ) decreases with increase in the angle of incidence (i) up to a particular angle. This angle of incidence where the angle of deviation in a prism is minimum is called the minimum deviation position of the prism and that very deviation angle is known as the minimum angle of deviation (denoted by δmin, Dλ, or Dm). The angle of minimum deviation is related with the refractive index as:
n
21
=
sin
(
A
+
D
m
2
)
sin
(
A
2
)
{\displaystyle n_{21}={\dfrac {\sin \left({\dfrac {A+D_{m}}{2}}\right)}{\sin \left({\dfrac {A}{2}}\right)}}}
This is useful to calculate the refractive index of a material. Rainbow and halo occur at minimum deviation. Also, a thin prism is always set at minimum deviation. Formula
In minimum deviation, the refracted ray in the prism is parallel to its base. In other words, the light ray is symmetrical about the axis of symmetry of the prism.[1][2][3] Also, the angles of refractions are equal i.e. r1 = r2. And, the angle of incidence and angle of emergence equal each other (i = e). This is clearly visible in the graph below. The formula for minimum deviation can be derived by exploiting the geometry in the prism. The approach involves replacing the variables in the Snell's law in terms of the Deviation and Prism Angles by making the use of the above properties. From the angle sum of
△
O
P
Q
{\textstyle \triangle OPQ}
A
+
∠
O
P
Q
+
∠
O
Q
P
=
180
∘
{\displaystyle A+\angle OPQ+\angle OQP=180^{\circ }}
Using the exterior angle theorem in
△
P
Q
R
{\textstyle \triangle PQR}
D
m
=
∠
R
P
Q
+
∠
R
Q
P
{\displaystyle D_{m}=\angle RPQ+\angle RQP}
This can also be derived by putting i = e in the prism formula: i + e = A + δ From Snell's law,
n
21
=
sin
i
sin
r
{\displaystyle n_{21}={\dfrac {\sin i}{\sin r}}}
∴ n 21 = sin ( A + D m 2 ) sin ( A 2 ) {\displaystyle \therefore n_{21}={\dfrac {\sin \left({\dfrac {A+D_{m}}{2}}\right)}{\sin \left({\dfrac {A}{2}}\right)}}} [4][3][1][2][5][excessive citations]∴ D m = 2 sin − 1 ( n sin ( A 2 ) ) − A {\displaystyle \therefore D_{m}=2\sin ^{-1}\left(n\sin \left({\frac {A}{2}}\right)\right)-A} (where n is the refractive index, A is the Angle of Prism and Dm is the Minimum Angle of Deviation.) This is a convenient way used to measure the refractive index of a material(liquid or gas) by directing a light ray through a prism of negligible thickness at minimum deviation filled with the material or in a glass prism dipped in it.[5][3][1][6] Worked out examples:
Also, the variation of the angle of deviation with an arbitrary angle of incidence can be encapsulated into a single equation by expressing e in terms of i in the prism formula using Snell's law: δ = i − A + sin − 1 ( n ⋅ sin ( A − sin − 1 ( sin i n ) ) ) {\displaystyle \delta =i-A+\sin ^{-1}\left(n\cdot \sin \left(A-\sin ^{-1}\left({\frac {\sin i}{n}}\right)\right)\right)} Finding the minima of this equation will also give the same relation for minimum deviation as above. For thin prismIn a thin or small angle prism, as the angles become very small, the sine of the angle nearly equals the angle itself and this yields many useful results. Because Dm and A are very small,
n
≈
A
+
D
m
2
A
2
n
=
A
+
D
m
A
D
m
=
A
n
−
A
{\displaystyle {\begin{aligned}n&\approx {\dfrac {\frac {A+D_{m}}{2}}{\frac {A}{2}}}\\n&={\frac {A+D_{m}}{A}}\\D_{m}&=An-A\end{aligned}}}
∴ D m = A ( n − 1 ) {\displaystyle \therefore D_{m}=A(n-1)} [1][4]Interestingly, using a similar approach with the Snell's law and the prism formula for an in general thin-prism ends up in the very same result for the deviation angle. Because i, e and r are small,
n
≈
i
r
1
,
n
≈
e
r
2
{\displaystyle n\approx {\frac {i}{r_{1}}},n\approx {\frac {e}{r_{2}}}}
From the prism formula,
δ
=
n
r
1
+
n
r
2
−
A
=
n
(
r
1
+
r
2
)
−
A
=
n
A
−
A
=
A
(
n
−
1
)
{\displaystyle {\begin{aligned}\delta &=nr_{1}+nr_{2}-A\\&=n(r_{1}+r_{2})-A\\&=nA-A\\&=A(n-1)\end{aligned}}}
Thus, it can be said that a thin prism is always in minimum deviation. Experimental determination
Minimum deviation can be found manually or with spectrometer. Either the prism is kept fixed and the incidence angle is adjusted or the prism is rotated keeping the light source fixed.[7][8][9][10][11] Minimum angle of dispersion
The minimum angle of dispersion[12] for white light is the difference in minimum deviation angle between red and violet rays of a light ray through a prism.[2] For a thin prism, the deviation of violet light,
δ
v
{\displaystyle \delta _{v}}
ApplicationsOne of the factors that causes a rainbow is the bunching of light rays at the minimum deviation angle that is close to the rainbow angle (42°).[3][13] It is also responsible for phenomena like halos and sundogs, produced by the deviation of sunlight in mini prisms of hexagonal ice crystals in the air bending light with a minimum deviation of 22°.[3][14] See also
References
External links
|