Question

When a ray of light enters in a transperent medium of refractive index n, then it is observed that the angle of refraction is half of the angle of incidence. The value of angle of incidence will be​

Answer 1

AnswEr :

• Value of the angle of incidence will be $\sf i=2cos^{-1}(\dfrac{\mu}{2})$ .

GivEn :

• The angle of refraction is half of the angle of incidence .

i. e.,

• $\rm r = \dfrac{i}{2}$ where, r is angle of refraction and i is angle of incidence!!

To find :

• Value of angle of incidence = ?

⠀⠀━━━━━━━━━━━━

★ Concept Used :

You need to know the concept of Snells Law before proceeding to the question !!

Snell's Law states that the ratio of the sin of the angle of incidence is equal to the ratio of the refractive index of the materials at the interface .

$\dag \ \ {\boxed{\rm{\large{n_1 \sin \theta_1 = n_2 \sin \theta_2}}}}$

Where,

• $\rm n_1$ = Incident index

• $\rm n_2$ = Refracted index

• $\rm \theta_1$ = Incident angle

• $\rm \theta_2$ = Refracted angle

So, by using this concept let's solve the question!!

⠀⠀━━━━━━━━━━━━

Solution :

As we know the Snells Law,

$\rm\mu = \dfrac{ \sin i }{ \sin r }$

Here,

• i is the angle of incidence [Given]
• r is the angle of refraction.[Given]

Now,

Substitute $\rm r = \dfrac{i}{2}$ in the above eqn!!

So,

$\sf\dashrightarrow \ \ \ \mu = \dfrac{ \sin i }{ \sin \frac{i}{2} }$

$\sf\dashrightarrow \ \ \ \mu = \dfrac{ 2\sin\frac{i}{2} \ \ 2 \cos \frac{i}{2}}{ \sin \frac{i}{2}} \ \ \ \ \ \ \ \ \bigg (\sin i = 2 \sin \dfrac{i}{2}2 \cos \dfrac{i}{2} \bigg )$

$\sf\dashrightarrow \ \ \ \mu = 2 \cos\frac{i}{2}$

$\sf\dashrightarrow \ \ \ {\underline{\boxed{\sf{\orange{i = 2 { \cos}^{ - 1}( \dfrac{ \mu}{2})}}}}} \ \bigstar$

Therefore, the angle of incidence is $\rm i=2cos^{-1}(\dfrac{\mu}{2})$

⠀⠀━━━━━━━━━━━━