A thin prism of angle `5^(@)`is placed at a distance of`10cm` from object.What is the distance of the image from object?Given `mu`of prism=`1.5`).

Options

A. `pi/8cm`
B. `pi/12cm`
C. `(5pi)/36cm`
D. `pi/7cm`

Answer

To solve the problem, we need to determine the distance of the image from the object when a thin prism is placed in front of it. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Problem
We have a thin prism with an angle \( A = 5^\circ \) placed at a distance of \( d = 10 \, \text{cm} \) from the object. The refractive index \( \mu \) of the prism is given as \( 1.5 \). We need to find the distance of the image from the object.

### Step 2: Calculate the Angular Deviation
The angular deviation \( \delta \) caused by the prism can be calculated using the formula:
\[ \delta = A \times (\mu - 1) \]
Substituting the values:
\[ \delta = 5^\circ \times (1.5 - 1) = 5^\circ \times 0.5 = 2.5^\circ \]

### Step 3: Convert Angular Deviation to Radians
To use the angular deviation in calculations, we convert degrees to radians:
\[ \delta \text{ (in radians)} = \frac{2.5 \times \pi}{180} = \frac{2.5\pi}{180} = \frac{\pi}{72} \, \text{radians} \]

### Step 4: Calculate the Shift in Position
The shift \( x \) of the image due to the prism can be approximated using the formula:
\[ x = d \times \tan(\delta) \]
For small angles, \( \tan(\delta) \approx \delta \). Therefore:
\[ x \approx d \times \delta \]
Substituting the values:
\[ x \approx 10 \, \text{cm} \times \frac{\pi}{72} \]

### Step 5: Calculate the Final Distance
Now, we can calculate the distance:
\[ x \approx 10 \times \frac{3.14}{72} \approx 10 \times 0.0436 \approx 0.436 \, \text{cm} \]

### Step 6: Conclusion
The distance of the image from the object is approximately \( 0.436 \, \text{cm} \).

### Final Answer
The distance of the image from the object is approximately \( 0.436 \, \text{cm} \).

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