When a thin transparent plate of thickness t and refractive index `mu` is placed in the path of one the two interfering waves of light, then the path difference changes by

Options

A. `(mu+1)t`
B. `(mu-1)t`
C. `((mu+1))/t`
D. `((mu-1))/t`

Answer

To solve the problem of how the path difference changes when a thin transparent plate of thickness \( t \) and refractive index \( \mu \) is placed in the path of one of the two interfering waves of light, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Optical Path Length**:
- The optical path length (OPL) is defined as the product of the physical path length and the refractive index of the medium through which light travels.
- For a medium with thickness \( t \) and refractive index \( \mu \), the optical path length is given by:
\[ \text{OPL} = \mu \cdot t \]

2. **Identifying the Change in Path**:
- When there is no plate, the light travels a distance \( t \) in air (or vacuum), which has a refractive index of approximately 1. Thus, the optical path length without the plate is simply:
\[ \text{OPL}_{\text{without plate}} = t \]

3. **Calculating the Change in Optical Path Length**:
- When the thin transparent plate is introduced, the optical path length changes to \( \mu \cdot t \).
- The change in optical path length (\( \Delta \text{OPL} \)) due to the introduction of the plate can be calculated as:
\[ \Delta \text{OPL} = \text{OPL}_{\text{with plate}} - \text{OPL}_{\text{without plate}} = \mu t - t \]

4. **Factoring the Change**:
- We can factor out \( t \) from the equation:
\[ \Delta \text{OPL} = t(\mu - 1) \]

5. **Conclusion**:
- Thus, the change in path difference when a thin transparent plate of thickness \( t \) and refractive index \( \mu \) is placed in the path of one of the interfering waves of light is:
\[ \Delta \text{OPL} = t(\mu - 1) \]

### Final Answer:
The path difference changes by \( t(\mu - 1) \).

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