The X-Y plane be taken as the boundary between two transparent media $\text{M}_1$ and $\text{M}_2$. $\text{M}_1$ in $Z \geq 0$ has a refractive index of $\sqrt{2}$ and $\text{M}_2$ with $Z<0$ has a refractive index of $\sqrt{3}$. A ray of light travelling in $\text{M}_1$ along the direction given by the vector $\vec{A}=4\sqrt{3}\hat{i}-3\sqrt{3}\hat{j}-5\hat{k}$, is incident on the plane of separation. The value of difference between the angle of incident in $\text{M}_1$ and the angle of refraction in $\text{M}_2$ will be ______ degree.

JEE Main · 2022 · 29 Jul 2022 · Shift 1 · Geometrical Optics

The X-Y plane be taken as the boundary between two transparent media

M_{1}

and

M_{2}

M_{1}

Z \geq 0

has a refractive index of

2

and

M_{2}

with

Z < 0

has a refractive index of

3

. A ray of light travelling in

M_{1}

along the direction given by the vector

A = 43 \hat{i} - 33 \hat{j} - 5 \hat{k}

, is incident on the plane of separation. The value of difference between the angle of incident in

M_{1}

and the angle of refraction in

M_{2}

will be ______ degree.

Answer: 15

The magnitude of

A

∣ A ∣ = (43)^{2} + (33)^{2} + 5^{2} = 48 + 27 + 25 = 10

. The angle of incidence with the normal (z-axis) is

i = cos^{- 1} (\frac{5}{10}) = 6 0^{\circ}

. Using Snell's law:

2 sin 6 0^{\circ} = 3 sin r

, which gives

2 \times \frac{3}{2} = 3 sin r

, so

sin r = \frac{1}{2}

and

r = 4 5^{\circ}

. The difference is

i - r = 6 0^{\circ} - 4 5^{\circ} = 1 5^{\circ}

JEE Main · 2022 · 29 Jul 2022 · Shift 1 · Integer answer