Answer
64.8k+ views
Hint:In this question, we know that if the angle between two lines is the same as the angle between their perpendiculars and one angle of right angles prism is always $90^\circ $. The angle of incidence should be minimum for total internal reflection.
Complete step by step solution:
In this question we have given that a right-angled prism has the refractive index $\mu $. We need to calculate the maximum value of $\theta $ up to which this incident ray necessarily undergoes total internal reflection at the face AC of the prism.
In this question let us assume that the angle of incidence is $i$ and the angle of incidence is incident on $AC$.
The angle between two lines is same as the angle between their perpendiculars so we can write,
$i = A$
Since the prism is right angled, we can write,
\[ \Rightarrow \theta = \dfrac{\pi }{2} - A\]
Since $i = A$ we can write,
$ \Rightarrow \theta = \dfrac{\pi }{2} - i......\left( 1 \right)$
We know that refractive index is expressed as,
$\mu = \dfrac{1}{{\sin i}}$
As we know that the angle of incidence should be minimum for the total internal reflection,
${i_c} = {\sin ^{ - 1}}\dfrac{1}{\mu }......\left( 2 \right)$
Here, ${i_c}$ is a critical angle.
Now the maximum value of $\theta $ for total internal expression can be calculated by substituting the expression of ${i_c}$ to the equation (1) as,
$ \Rightarrow \theta = \dfrac{\pi }{2} - {i_C}$
Now we substitute the value of critical angle as expressed in equation (2) in above equation to get the maximum value of $\theta $.
$\therefore {\theta _{\max }} = \dfrac{\pi }{2} - {\sin ^{ - 1}}\dfrac{1}{\mu }$
Therefore, the maximum value will be ${\theta _{\max }} = \dfrac{\pi }{2} - {\sin ^{ - 1}}\dfrac{1}{\mu }$.
Note:As we know that the minimum angle for total internal reflection is the critical angle. The angle of the right-angled prism is $90^\circ $. The maximum value of $\theta $ is calculated corresponding to the minimum value of $i$.
Complete step by step solution:
In this question we have given that a right-angled prism has the refractive index $\mu $. We need to calculate the maximum value of $\theta $ up to which this incident ray necessarily undergoes total internal reflection at the face AC of the prism.
In this question let us assume that the angle of incidence is $i$ and the angle of incidence is incident on $AC$.
The angle between two lines is same as the angle between their perpendiculars so we can write,
$i = A$
Since the prism is right angled, we can write,
\[ \Rightarrow \theta = \dfrac{\pi }{2} - A\]
Since $i = A$ we can write,
$ \Rightarrow \theta = \dfrac{\pi }{2} - i......\left( 1 \right)$
We know that refractive index is expressed as,
$\mu = \dfrac{1}{{\sin i}}$
As we know that the angle of incidence should be minimum for the total internal reflection,
${i_c} = {\sin ^{ - 1}}\dfrac{1}{\mu }......\left( 2 \right)$
Here, ${i_c}$ is a critical angle.
Now the maximum value of $\theta $ for total internal expression can be calculated by substituting the expression of ${i_c}$ to the equation (1) as,
$ \Rightarrow \theta = \dfrac{\pi }{2} - {i_C}$
Now we substitute the value of critical angle as expressed in equation (2) in above equation to get the maximum value of $\theta $.
$\therefore {\theta _{\max }} = \dfrac{\pi }{2} - {\sin ^{ - 1}}\dfrac{1}{\mu }$
Therefore, the maximum value will be ${\theta _{\max }} = \dfrac{\pi }{2} - {\sin ^{ - 1}}\dfrac{1}{\mu }$.
Note:As we know that the minimum angle for total internal reflection is the critical angle. The angle of the right-angled prism is $90^\circ $. The maximum value of $\theta $ is calculated corresponding to the minimum value of $i$.
Recently Updated Pages
Write a composition in approximately 450 500 words class 10 english JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Arrange the sentences P Q R between S1 and S5 such class 10 english JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
What is the common property of the oxides CONO and class 10 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
What happens when dilute hydrochloric acid is added class 10 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
If four points A63B 35C4 2 and Dx3x are given in such class 10 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
The area of square inscribed in a circle of diameter class 10 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Other Pages
A boat takes 2 hours to go 8 km and come back to a class 11 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Electric field due to uniformly charged sphere class 12 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
In the ground state an element has 13 electrons in class 11 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
According to classical free electron theory A There class 11 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Excluding stoppages the speed of a bus is 54 kmph and class 11 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)