Light of wavelength `lamda` is incident on a slit width d. The resulting diffraction pattern is observed on a screen at a distance D. The linear width of the principal maximum is then equal to the width of the slit if D equals

Options

A. `d/lamda`
B. `(2lamda)/d`
C. `d^2/(2lamda)`
D. `(2lamda^2)/d`

Answer

To solve the problem, we need to determine the condition under which the linear width of the principal maximum in a single-slit diffraction pattern is equal to the width of the slit.

### Step-by-step Solution:

1. **Understanding the Diffraction Pattern**:
The diffraction pattern produced by a single slit can be analyzed using the formula for the width of the principal maximum. The width \( W \) of the principal maximum is given by:
\[ W = \frac{2\lambda D}{d} \]
where \( \lambda \) is the wavelength of the light, \( D \) is the distance from the slit to the screen, and \( d \) is the width of the slit.

2. **Setting the Condition**:
We want to find the condition where the width of the principal maximum \( W \) is equal to the width of the slit \( d \):
\[ W = d \]

3. **Substituting the Expression for Width**:
From the equation for \( W \), we substitute:
\[ \frac{2\lambda D}{d} = d \]

4. **Cross-Multiplying**:
To eliminate the fraction, we cross-multiply:
\[ 2\lambda D = d^2 \]

5. **Solving for \( D \)**:
Now, we can solve for \( D \):
\[ D = \frac{d^2}{2\lambda} \]

6. **Identifying the Correct Option**:
The expression we derived for \( D \) matches with one of the given options. The correct option is:
\[ D = \frac{d^2}{2\lambda} \]
This corresponds to option number 3.

### Final Answer:
The value of \( D \) for which the linear width of the principal maximum is equal to the width of the slit is:
\[ D = \frac{d^2}{2\lambda} \]