When a picture drawn on paper is seen through a slab of a transparent material of thickness 5 cm, it appears to be raised by 1.5 cm. The critical angle at the boundary of this transparent material and air is -

Options

A. `sin^(-1) (2/3)`
B. `sin^(-1) (5/7)`
C. `sin^(-1) (6/11)`
D. `sin^(-1) (7/10)`

Answer

To solve the problem, we need to find the critical angle at the boundary of the transparent material and air. Let's break down the solution step by step.

### Step 1: Understand the problem
We know that a picture appears to be raised when viewed through a slab of transparent material. The thickness of the slab is given as 5 cm, and the apparent shift (or raise) of the image is 1.5 cm.

### Step 2: Use the formula for apparent shift
The formula for the apparent shift \(D\) when viewing through a slab of thickness \(T\) and refractive index \(\mu\) is given by:
\[ D = T \left(1 - \frac{1}{\mu}\right) \]
Here, \(D\) is the apparent shift, \(T\) is the thickness of the slab, and \(\mu\) is the refractive index of the material.

### Step 3: Substitute the known values
We know:
- \(D = 1.5 \, \text{cm}\)
- \(T = 5 \, \text{cm}\)

Substituting these values into the formula:
\[ 1.5 = 5 \left(1 - \frac{1}{\mu}\right) \]

### Step 4: Solve for \(\mu\)
Rearranging the equation:
\[ 1 - \frac{1}{\mu} = \frac{1.5}{5} \]
Calculating the right side:
\[ 1 - \frac{1}{\mu} = 0.3 \]
Now, solving for \(\frac{1}{\mu}\):
\[ \frac{1}{\mu} = 1 - 0.3 = 0.7 \]
Thus, we find:
\[ \mu = \frac{1}{0.7} = \frac{10}{7} \]

### Step 5: Find the critical angle
The critical angle \(\theta_c\) can be calculated using Snell's law. When light travels from a denser medium (the slab) to a rarer medium (air), the critical angle is given by:
\[ \mu \sin(\theta_c) = 1 \]
Substituting \(\mu = \frac{10}{7}\):
\[ \frac{10}{7} \sin(\theta_c) = 1 \]
Rearranging gives:
\[ \sin(\theta_c) = \frac{7}{10} \]

### Step 6: Calculate \(\theta_c\)
Now, we find the critical angle:
\[ \theta_c = \sin^{-1}\left(\frac{7}{10}\right) \]

### Final Answer
Thus, the critical angle at the boundary of the transparent material and air is:
\[ \theta_c = \sin^{-1}\left(\frac{7}{10}\right) \]

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