When a transparent parallel plate of uniform thickness `t` and refractive index `mu` is interposed normally in the path of a beam of light, the optical path is

Options

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

Answer

To solve the problem of finding the optical path when a transparent parallel plate of uniform thickness \( t \) and refractive index \( \mu \) is interposed normally in the path of a beam of light, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding 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 the light travels. When light passes through a medium with a refractive index \( \mu \), the optical path length can be expressed as:
\[ \text{OPL} = \text{Physical Path Length} \times \mu \]

2. **Identifying Physical Path Length**:
In this case, the physical path length is simply the thickness of the parallel plate, which is \( t \).

3. **Calculating Optical Path Length**:
Therefore, when light travels through the parallel plate, the optical path length becomes:
\[ \text{OPL} = t \times \mu \]

4. **Considering the Optical Path Change**:
If there were no plate, the optical path length would just be the distance traveled in air (or vacuum), which we can consider as \( t \) (since the light travels through the same thickness in air). The optical path length in air is simply \( t \).

5. **Finding the Change in Optical Path**:
The change in optical path length due to the introduction of the plate is given by:
\[ \Delta \text{OPL} = \text{OPL with plate} - \text{OPL without plate} \]
Substituting the values we have:
\[ \Delta \text{OPL} = (t \cdot \mu) - t \]

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

7. **Final Expression for Optical Path**:
Thus, the optical path change when the transparent parallel plate is introduced is:
\[ \Delta \text{OPL} = t(\mu - 1) \]

### Conclusion:
The optical path when a transparent parallel plate of uniform thickness \( t \) and refractive index \( \mu \) is interposed normally in the path of a beam of light is given by:
\[ \Delta \text{OPL} = t(\mu - 1) \]