In double slit experiment, the angular width of the fringes is `0.20^@` for the sodium light `(lambda=5890Å)`. In order to increase the angular width of the fringes by `10%`, the necessary change in the wavelength is

Options

A. zero
B. increased by `6749 Å`
C. decreased by 589 `Å`
D. increased by `589 Å`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the relationship between angular width and wavelength
In the double slit experiment, the angular width of the fringe (θ) is directly proportional to the wavelength (λ). The relationship can be expressed as:
\[ \theta \propto \lambda \]
This means that if we want to increase the angular width of the fringes, we need to increase the wavelength.

### Step 2: Calculate the current angular width
The current angular width of the fringes is given as:
\[ \theta = 0.20^\circ \]

### Step 3: Determine the desired increase in angular width
We want to increase the angular width by 10%. Therefore, the new angular width (θ') will be:
\[ \theta' = \theta + 0.10 \times \theta = 0.20^\circ + 0.10 \times 0.20^\circ = 0.20^\circ + 0.02^\circ = 0.22^\circ \]

### Step 4: Calculate the percentage increase in wavelength
Since the angular width is directly proportional to the wavelength, a 10% increase in angular width will also require a 10% increase in wavelength (λ).

### Step 5: Calculate the current wavelength
The current wavelength of sodium light is given as:
\[ \lambda = 5890 \, \text{Å} = 5890 \times 10^{-10} \, \text{m} \]

### Step 6: Calculate the new wavelength
To find the new wavelength (λ'), we will increase the current wavelength by 10%:
\[ \lambda' = \lambda + 0.10 \times \lambda = 5890 \, \text{Å} + 0.10 \times 5890 \, \text{Å} = 5890 \, \text{Å} + 589 \, \text{Å} = 6479 \, \text{Å} \]

### Step 7: Calculate the necessary change in wavelength
The necessary change in wavelength (Δλ) is:
\[ \Delta \lambda = \lambda' - \lambda = 6479 \, \text{Å} - 5890 \, \text{Å} = 589 \, \text{Å} \]

### Final Answer
The necessary change in wavelength to increase the angular width of the fringes by 10% is:
\[ \Delta \lambda = 589 \, \text{Å} \]