Light of wavelength `lambda` from a point source falls on a small circular rings around a central bright spot are formed on a screen beyond the obstacle The distance between the screen and obstacle is D. Then, the condition for the formation of rings, is

Options

A. `sqrt(lambda)approxd/(4D)`
B. `lambdaapproxd^(2)/(4D)`
C. `d approx lambda^(2)/D`
D. `lambdaapproxD/4`

Explanation

To solve the problem of determining the condition for the formation of circular rings around a central bright spot when light of wavelength \( \lambda \) falls on a small circular obstacle, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Setup**:
- We have a point source of light emitting waves of wavelength \( \lambda \).
- There is a small circular obstacle that causes diffraction, leading to the formation of a central bright spot and surrounding rings on a screen located at a distance \( D \) from the obstacle.

2. **Identify the Condition for Ring Formation**:
- The formation of rings (or fringes) is a result of constructive and destructive interference of the light waves that pass around the obstacle.
- The condition for constructive interference (bright rings) can be derived from the geometry of the setup.

3. **Apply the Formula for Ring Formation**:
- The radius of the \( n^{th} \) ring can be expressed in terms of the wavelength and the distance \( D \) using the formula:
\[ r_n^2 = n \lambda D \]
- Here, \( r_n \) is the radius of the \( n^{th} \) ring, and \( n \) is an integer representing the ring number.

4. **Relate the Radius to the Wavelength**:
- For the first ring (where \( n = 1 \)):
\[ r_1^2 = \lambda D \]
- For higher order rings, the relationship remains similar, with \( n \) increasing.

5. **Express the Wavelength in Terms of the Distance**:
- Rearranging the formula gives:
\[ \lambda = \frac{r_n^2}{D} \]
- This indicates that the wavelength \( \lambda \) is dependent on the square of the radius of the ring and the distance \( D \).

6. **Final Condition for Formation of Rings**:
- To find the specific condition for the formation of rings, we can express it as:
\[ \lambda = \frac{D^2}{4D} \]
- This simplifies to:
\[ \lambda = \frac{D}{4} \]
- Thus, the condition for the formation of rings is that the wavelength \( \lambda \) should be approximately equal to \( \frac{D^2}{4D} \).

### Conclusion:
The condition for the formation of circular rings around a central bright spot is given by:
\[ \lambda = \frac{D^2}{4D} \]