Two monochromatic and coherent point sources of light are placed at a certain distance from each other in the horizontal plane. The locus of all those points in the horizontal plane which have constructive interference will be-

Options

A. A hyperbola
B. family of hyperbola
C. family of straight lines
D. family of parabolas

Answer

To solve the problem of finding the locus of points in the horizontal plane that exhibit constructive interference from two coherent point sources of light, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Constructive Interference**:
Constructive interference occurs when the path difference between the two waves arriving at a point is an integral multiple of the wavelength (λ). Mathematically, this condition can be expressed as:
\[ |d_1 - d_2| = n\lambda \]
where \(d_1\) and \(d_2\) are the distances from the point to the two sources, and \(n\) is an integer (0, 1, 2, ...).

2. **Defining the Sources**:
Let the two point sources be \(S_1\) and \(S_2\). We can denote their positions in the horizontal plane. For simplicity, assume \(S_1\) is at the origin (0, 0) and \(S_2\) is at (d, 0), where d is the distance between the two sources.

3. **Choosing a Point in the Plane**:
Consider a point \(P(x, y)\) in the horizontal plane. The distances from this point to the two sources can be calculated as:
\[ d_1 = \sqrt{x^2 + y^2} \quad \text{(distance to } S_1\text{)} \]
\[ d_2 = \sqrt{(x - d)^2 + y^2} \quad \text{(distance to } S_2\text{)} \]

4. **Setting Up the Path Difference Condition**:
According to the condition for constructive interference:
\[ |d_1 - d_2| = n\lambda \]
This leads to two cases:
- Case 1: \(d_1 - d_2 = n\lambda\)
- Case 2: \(d_2 - d_1 = n\lambda\)

5. **Deriving the Equation**:
For Case 1:
\[ \sqrt{x^2 + y^2} - \sqrt{(x - d)^2 + y^2} = n\lambda \]
Squaring both sides and simplifying will yield a hyperbolic equation.

For Case 2:
\[ \sqrt{(x - d)^2 + y^2} - \sqrt{x^2 + y^2} = n\lambda \]
Similarly, squaring and simplifying will also yield a hyperbolic equation.

6. **Conclusion**:
The locus of all points where constructive interference occurs is a family of hyperbolas. Each hyperbola corresponds to a different integer value of \(n\).

### Final Answer:
The locus of all points in the horizontal plane that exhibit constructive interference from the two coherent point sources is a family of hyperbolas.