Question

The x-z plane separates two media A and B of refractive indices µ1 = 1.5 and µ2 = 2. A ray of light travels from A to B. its directions in the two media are given by unit vectors u1 = ai + bj and u2 = ci + dj.

Answer 1

Given that,

Refractive index of A = 1.5

Refractive index of B = 2

Suppose, we need to find the value of ratio a  and c

The unit vectors are

$u_{1}=ai+bj$

$u_{2}=ci+dj$

We need to calculate the incident angle

Using formula of vector

$\cos\theta_{i}=\dfrac{u\cdot j}{|u||j|}$

Put the value into the formula

$\cos\theta_{i}=\dfrac{(ai+bj)\cdot j}{\sqrt{a^2+b^2}\times1}$

Here, $\sqrt{a^2+b^2}=1$

$\cos\theta_{i}=b$

Similarly,

$\sin\theta_{i}=a$

We need to calculate the reflection angle

Using formula of vector

$\cos\theta_{r}=\dfrac{u\cdot i}{|u||j|}$

Put the value into the formula

$\cos\theta_{r}=\dfrac{(ci+dj)\cdot i}{\sqrt{c^2+d^2}\times1}$

Here, $\sqrt{c^2+d^2}=1$

$\cos\theta_{r}=d$

Similarly,

$\sin\theta_{r} =c$

We need to calculate the value of ratio of a and c

Using snell's law

$\mu_{1}\sin\theta_{i}=\mu_{2}\sin\theta_{r}$

Put the value into the formula

$1.5\times a=2\times c$

$\dfrac{a}{c}=\dfrac{20}{15}$

$\dfrac{a}{c}=\dfrac{4}{3}$

Hence, The ratio of a and c is 4:3