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

Explanation

To solve the problem step by step, we will follow the principles of ray optics and the properties of a thin prism.

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

### Step 2: Calculate the angle of deviation (\(\delta\))
For a thin prism, the angle of deviation can be calculated using the formula:
\[ \delta = (\mu - 1) \times A \]
where \(A\) is the angle of the prism.

Substituting the values:
\[ \delta = (1.5 - 1) \times 5^\circ = 0.5 \times 5^\circ = 2.5^\circ \]

### Step 3: Convert the angle of deviation to radians
To convert degrees to radians, we use the conversion factor \(\frac{\pi}{180}\):
\[ \delta = 2.5^\circ \times \frac{\pi}{180} = \frac{2.5\pi}{180} = \frac{\pi}{72} \, \text{radians} \]

### Step 4: Use the small angle approximation
Since \(\delta\) is a small angle, we can use the small angle approximation:
\[ \tan(\delta) \approx \delta \]
Thus, we can write:
\[ \tan(\delta) \approx \delta \approx \frac{d}{10} \]
where \(d\) is the distance between the object and the image.

### Step 5: Set up the equation
From the small angle approximation:
\[ \frac{d}{10} \approx \frac{\pi}{72} \]

### Step 6: Solve for \(d\)
Now, we can solve for \(d\):
\[ d \approx 10 \times \frac{\pi}{72} = \frac{10\pi}{72} = \frac{5\pi}{36} \]

### Step 7: Conclusion
Thus, the distance of the image from the object is:
\[ d = \frac{5\pi}{36} \, \text{cm} \]

### Final Answer
The distance of the image from the object is \(\frac{5\pi}{36} \, \text{cm}\).

---