Light is reflecting off a wedge-shaped thin piece of glass producing bright and dark interference fringes. If a certain location has a bright fringe, a nearby point will have a dark fringe if the thickness of the glass increases by

Options

A. `1 // 8` of a wavelength of the light in glass
B. `1 // 4` of a wavelength of the light in glass
C. `1 // 2` of a wavelength of the light in glass
D. 1 wavelength of the light in glass

Answer

To solve the problem of determining how much the thickness of the wedge-shaped thin glass must increase for a nearby point to change from a bright fringe to a dark fringe, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Interference Fringes**:
- Bright fringes occur when the path difference between the two reflected rays is an integral multiple of the wavelength (nλ), while dark fringes occur when the path difference is an odd multiple of half the wavelength ((n + 1/2)λ).

2. **Thickness Condition for Bright and Dark Fringes**:
- For a bright fringe at thickness \( t \), the condition can be expressed as:
\[ 2 \mu t = n \lambda \]
- For a dark fringe at a nearby point, the condition can be expressed as:
\[ 2 \mu t' = (n + 1/2) \lambda \]
where \( t' \) is the new thickness.

3. **Setting Up the Equation**:
- We want to find the change in thickness \( \Delta t = t' - t \) that results in a transition from a bright fringe to a dark fringe.
- Rearranging the equations gives:
\[ t' = \frac{(n + 1/2) \lambda}{2 \mu} \]
- Therefore, the change in thickness can be expressed as:
\[ \Delta t = t' - t = \frac{(n + 1/2) \lambda}{2 \mu} - \frac{n \lambda}{2 \mu} \]

4. **Simplifying the Expression**:
- Simplifying the equation for \( \Delta t \):
\[ \Delta t = \frac{(n + 1/2 - n) \lambda}{2 \mu} = \frac{\lambda/2}{2 \mu} = \frac{\lambda}{4 \mu} \]

5. **Considering the Refractive Index**:
- If the glass is air (with a refractive index \( \mu = 1 \)), then:
\[ \Delta t = \frac{\lambda}{4} \]

### Final Answer:
Thus, the thickness of the glass must increase by \( \frac{\lambda}{4} \) for a nearby point to change from a bright fringe to a dark fringe.

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