Question

When a ray of light enters a medium of refractive index $\mu $. It is observed that the angle of refraction is half of the angle of incidence is then angle of incidence is?
A. ${\cos ^{ - 1}}\mu $
B. $2{\cos ^{ - 1}}\dfrac{\mu }{2}$
C. $2{\cos ^{ - 1}}\mu $
D. ${\cos ^{ - 1}}\dfrac{\mu }{3}$

Answer 1

Hint: The ratio of the speed of light in a vacuum to the speed of light in a second medium of greater density is used to measure the Refractive Index (Index of Refraction). In descriptive text and mathematical equations, the refractive index variable is most commonly represented by the letter n or n'.

Complete step by step answer:
According to Snell's law, the ratio of the sines of the angles of incidence and refraction is equal to the ratio of phase velocities in the two media, or the reciprocal of the indices of refraction:
$\dfrac{{\sin {\theta _2}}}{{\sin {\theta _1}}} = \dfrac{{{n_1}}}{{{n_2}}}$
$\theta $ is the angle determined from the boundary's normal. n denotes the medium's refractive index (which is not unitless) or we can also say that
$\mu = \dfrac{{\sin i}}{{\sin r}}$
Here $\mu = $ Refractive index of medium, $\sin i = $ Angle of incidence and $\sin r = $ Angle of refraction.

Given: Angle of refraction is half of the angle of incidence
$r = \dfrac{i}{2}$
$\Rightarrow \mu = \dfrac{{\sin i}}{{\sin \dfrac{i}{2}}}$
$\Rightarrow \mu = \dfrac{{2\sin \dfrac{i}{2}\cos \dfrac{i}{2}}}{{\sin \dfrac{i}{2}}}$
$\Rightarrow \mu = 2\cos \dfrac{i}{2}$
If we simplify it then we will get:
$\cos \dfrac{i}{2} = \dfrac{\mu }{2}$
$\therefore i = 2{\cos ^{ - 1}}\dfrac{\mu }{2}$

So the option B is correct.

Note: Snell's law used in things like eyeglasses, contact lenses, cameras, and rainbows. The refractive index of liquids is calculated using Snell's law by a device called a refractometer. It's also used in the candy-making industry.