Answer
399.9k+ views
Hint : We can find the refractive index corresponding to minimum deviation by using a prism formula.
$n=\dfrac{\sin \dfrac{(A+{{S}_{m}})}{2}}{\sin \dfrac{A}{2}}$
Here, n is refractive index of material,
A is the angle of the prism.
${{S}_{m}}$ is the angle of minimum deviation
Use the condition, refractive index is minimum when angle of prism is \[\text{9}0{}^\circ \]
Refractive index is maximum when the angle of the prism is \[0{}^\circ \].
Complete step by step answer
The prism formula is given by
$n=\dfrac{\sin \dfrac{(A+{{S}_{m}})}{2}}{\sin \dfrac{A}{2}}$
Here n is a refractive index.
We know deviation will be minimum, angle of incidence is equal to angle of prism.
${{S}_{m}}=2i-A$
${{S}_{m}}=\text{2}AA=A$
From prism formula, put the value of minimum deviation,
n = $n=\dfrac{\sin \dfrac{(A+A)}{2}}{\sin \dfrac{A}{2}}=\dfrac{\sin \text{ }A}{\sin \dfrac{A}{2}}$
Use$\left[ \text{sin A=2 sin}\dfrac{A}{2}\text{cos }\dfrac{A}{2} \right]$
The refractive index is given by,
$n=\dfrac{2\operatorname{Sin}\dfrac{A}{2}\operatorname{Cos}\dfrac{A}{2}}{\operatorname{Sin}\dfrac{A}{2}}$
$n=2\operatorname{Cos}\dfrac{A}{2}$
This is the formula for prism,
Now, we have to find a minimum refractive index.
Angle of prism A varies from \[0{}^\circ \]to \[\text{9}0{}^\circ \]
For ${{n}_{\min }}$, minimum value of refractive index is given by
Put \[\text{A}=\text{9}0{}^\circ \]
$n=2\operatorname{Cos}\dfrac{90{}^\circ }{2}$ Use$\left[ \text{Cos 45}{}^\circ =\dfrac{1}{\sqrt{2}} \right]$
= $2\operatorname{Cos}45{}^\circ $
${{n}_{\min }}=2\times \dfrac{1}{\sqrt{2}}\text{ = }\sqrt{2}$
For${{n}_{\max }}$, maximum value of refractive index
A= 0°
\[\text{n}=\text{2 Cos }0{}^\circ =\text{2}\] (1) \[~[\text{Cos }0{}^\circ =\text{ 1}]\]
${{n}_{\max }}=2$
Hence, the value of refractive index varies between 2 and $\sqrt{2}$
Note
When refracting angle is small, the deviation is calculated from,
S =\[\left( n-1 \right)\]A
When refraction angle is bigger for the prism the deviation is calculated from, S = $({{i}_{1}}+{{i}_{2}})-A$where ${{i}_{1}}$and ${{i}_{2}}$are angle of incidence on different faces of prism.
$n=\dfrac{\sin \dfrac{(A+{{S}_{m}})}{2}}{\sin \dfrac{A}{2}}$
Here, n is refractive index of material,
A is the angle of the prism.
${{S}_{m}}$ is the angle of minimum deviation
Use the condition, refractive index is minimum when angle of prism is \[\text{9}0{}^\circ \]
Refractive index is maximum when the angle of the prism is \[0{}^\circ \].
Complete step by step answer
The prism formula is given by
$n=\dfrac{\sin \dfrac{(A+{{S}_{m}})}{2}}{\sin \dfrac{A}{2}}$
Here n is a refractive index.
We know deviation will be minimum, angle of incidence is equal to angle of prism.
${{S}_{m}}=2i-A$
${{S}_{m}}=\text{2}AA=A$
From prism formula, put the value of minimum deviation,
n = $n=\dfrac{\sin \dfrac{(A+A)}{2}}{\sin \dfrac{A}{2}}=\dfrac{\sin \text{ }A}{\sin \dfrac{A}{2}}$
Use$\left[ \text{sin A=2 sin}\dfrac{A}{2}\text{cos }\dfrac{A}{2} \right]$
The refractive index is given by,
$n=\dfrac{2\operatorname{Sin}\dfrac{A}{2}\operatorname{Cos}\dfrac{A}{2}}{\operatorname{Sin}\dfrac{A}{2}}$
$n=2\operatorname{Cos}\dfrac{A}{2}$
This is the formula for prism,
Now, we have to find a minimum refractive index.
Angle of prism A varies from \[0{}^\circ \]to \[\text{9}0{}^\circ \]
For ${{n}_{\min }}$, minimum value of refractive index is given by
Put \[\text{A}=\text{9}0{}^\circ \]
$n=2\operatorname{Cos}\dfrac{90{}^\circ }{2}$ Use$\left[ \text{Cos 45}{}^\circ =\dfrac{1}{\sqrt{2}} \right]$
= $2\operatorname{Cos}45{}^\circ $
${{n}_{\min }}=2\times \dfrac{1}{\sqrt{2}}\text{ = }\sqrt{2}$
For${{n}_{\max }}$, maximum value of refractive index
A= 0°
\[\text{n}=\text{2 Cos }0{}^\circ =\text{2}\] (1) \[~[\text{Cos }0{}^\circ =\text{ 1}]\]
${{n}_{\max }}=2$
Hence, the value of refractive index varies between 2 and $\sqrt{2}$
Note
When refracting angle is small, the deviation is calculated from,
S =\[\left( n-1 \right)\]A
When refraction angle is bigger for the prism the deviation is calculated from, S = $({{i}_{1}}+{{i}_{2}})-A$where ${{i}_{1}}$and ${{i}_{2}}$are angle of incidence on different faces of prism.
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
When people say No pun intended what does that mea class 8 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name the states which share their boundary with Indias class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give an account of the Northern Plains of India class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Advantages and disadvantages of science
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Bimbisara was the founder of dynasty A Nanda B Haryanka class 6 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
10 examples of evaporation in daily life with explanations
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you graph the function fx 4x class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)