Find the image shift if the paperweight is inverted at its place so that the spherical surface touches the paper

Explanation

To solve the problem of finding the image shift when the paperweight is inverted so that the spherical surface touches the paper, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- The paperweight has a spherical surface that is now in contact with the paper. When the spherical surface touches the paper, the light rays reflecting off the surface will create an image that appears shifted.

2. **Identify the Refractive Indices**:
- The refractive index of glass (μg) is typically around 1.5.
- The refractive index of air (μa) is approximately 1.

3. **Determine the Thickness**:
- The thickness (t) of the glass in this case is equal to the radius of the spherical surface. Given that the radius is 3 cm, we have:
\[ t = 3 \, \text{cm} \]

4. **Use the Image Shift Formula**:
- The formula for the image shift (Δt) when light passes through a medium is given by:
\[ \Delta t = t \left(1 - \frac{1}{\mu}\right) \]
- Here, we will use the refractive index of glass (μg = 1.5).

5. **Substituting Values**:
- Substitute the values into the formula:
\[ \Delta t = 3 \, \text{cm} \left(1 - \frac{1}{1.5}\right) \]

6. **Calculating the Shift**:
- First, calculate \(1 - \frac{1}{1.5}\):
\[ 1 - \frac{1}{1.5} = 1 - \frac{2}{3} = \frac{1}{3} \]
- Now, substitute this back into the equation for Δt:
\[ \Delta t = 3 \, \text{cm} \times \frac{1}{3} = 1 \, \text{cm} \]

7. **Conclusion**:
- The image shift when the paperweight is inverted and the spherical surface touches the paper is 1 cm.

### Final Answer:
The image shift is **1 cm**.

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