If the lower slit has been covered by the sheet, what will be the direction of shift of the pattern?

Answer

To solve the problem of determining the direction of the shift of the interference pattern when the lower slit is covered by a sheet, we can follow these steps:

### Step 1: Understand the Setup
In a double-slit interference experiment, we have two slits (S1 and S2) and a screen where the interference pattern is observed. When the lower slit (S2) is covered by a sheet, it introduces a change in the optical path for the light coming from that slit.

**Hint:** Visualize the setup and identify the slits and the screen.

### Step 2: Introduce the Parameters
Let:
- \( T \) = thickness of the sheet
- \( \mu \) = refractive index of the sheet
- \( d \) = distance between the slits
- \( D \) = distance from the slits to the screen

### Step 3: Determine the Initial Path Difference
The initial path difference \( \Delta p \) between the two slits for a point \( P \) on the screen can be expressed as:
\[ \Delta p = S2P - S1P \]

**Hint:** Remember that path difference is crucial for determining the interference pattern.

### Step 4: Calculate the New Path Difference
When the lower slit (S2) is covered with the sheet, the new path difference \( \Delta p' \) becomes:
\[ \Delta p' = S2P' - S1P' + \mu T \]
where \( P' \) is the new position on the screen due to the covering of the slit.

**Hint:** The term \( \mu T \) accounts for the additional optical path introduced by the sheet.

### Step 5: Relate the Path Difference to the Shift
For constructive interference (maxima), the path difference must satisfy:
\[ \Delta p' = n\lambda \]
where \( n \) is an integer and \( \lambda \) is the wavelength of light.

### Step 6: Analyze the Shift
When the sheet is added, it effectively increases the optical path length for light coming from S2. This means that the interference pattern will shift. The direction of the shift can be determined by analyzing whether the new path difference \( \Delta p' \) is greater or less than the original \( n\lambda \).

Since the sheet increases the optical path from S2, the new maxima will occur at a position that is lower on the screen compared to the original maxima.

**Hint:** Consider how an increase in path length affects the position of the maxima.

### Step 7: Conclusion
The interference pattern will shift downward on the screen due to the covering of the lower slit with the sheet.

**Final Answer:** The direction of the shift of the pattern will be downward.

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