The x-z plane separates two media A and B of refractive indices `mu_(1) = 1.5` and `mu_(2) = 2`. A ray of light travels from A to B. Its directions in the two media are given by unit vectors `u_(1) = a hat(i)+b hat(j)` and `u_(2) = c hat(i) +a hat(j)`. Then

Options

A. `(a)/(c) = (4)/(3)`
B. `(a)/(c) = (3)/(4)`
C. `(b)/(c) = (4)/(3)`
D. `(b)/(d) = (3)/(4)`

Answer

To solve the problem, we will apply Snell's Law, which relates the angles of incidence and refraction to the refractive indices of the two media. The steps are as follows:

### Step 1: Understand the given information
We have two media A and B with refractive indices:
- \( \mu_1 = 1.5 \) (for medium A)
- \( \mu_2 = 2 \) (for medium B)

The direction of the light ray in medium A is given by the unit vector:
\[ \mathbf{u_1} = a \hat{i} + b \hat{j} \]
And in medium B:
\[ \mathbf{u_2} = c \hat{i} + a \hat{j} \]

### Step 2: Apply Snell's Law
Snell's Law states:
\[ \frac{\mu_1}{\mu_2} = \frac{\sin \theta_r}{\sin \theta_i} \]
Where \( \theta_i \) is the angle of incidence and \( \theta_r \) is the angle of refraction.

### Step 3: Express \( \sin \theta_i \) and \( \sin \theta_r \)
The sine of the angles can be expressed in terms of the components of the unit vectors:
- For medium A:
\[ \sin \theta_i = \frac{a}{\sqrt{a^2 + b^2}} \]
- For medium B:
\[ \sin \theta_r = \frac{c}{\sqrt{c^2 + a^2}} \]

### Step 4: Substitute into Snell's Law
Substituting these expressions into Snell's Law gives:
\[ \frac{1.5}{2} = \frac{\frac{c}{\sqrt{c^2 + a^2}}}{\frac{a}{\sqrt{a^2 + b^2}}} \]

### Step 5: Cross-multiply to simplify
Cross-multiplying yields:
\[ 1.5 \cdot \frac{a}{\sqrt{a^2 + b^2}} = 2 \cdot \frac{c}{\sqrt{c^2 + a^2}} \]

### Step 6: Rearranging the equation
Rearranging gives:
\[ 1.5a \sqrt{c^2 + a^2} = 2c \sqrt{a^2 + b^2} \]

### Step 7: Square both sides to eliminate the square roots
Squaring both sides results in:
\[ (1.5a)^2 (c^2 + a^2) = (2c)^2 (a^2 + b^2) \]
This simplifies to:
\[ 2.25a^2(c^2 + a^2) = 4c^2(a^2 + b^2) \]

### Step 8: Solve for the ratio \( \frac{a}{c} \)
After simplifying the equation, we can isolate \( \frac{a}{c} \):
\[ \frac{a}{c} = \frac{4}{3} \]

### Final Answer
Thus, the ratio \( \frac{a}{c} \) is:
\[ \frac{a}{c} = \frac{4}{3} \]