Two coherent sources of light interfere. The intensity ratio of two sources is $$1 : 4$$. For this interference pattern if the value of $$\frac{I_{max}+I_{min}}{I_{max}-I_{min}}$$ is equal to $$\frac{2\alpha+1}{\beta+3}$$, then $$\frac{\alpha}{\beta}$$ will be

Two coherent sources have an intensity ratio of $$1:4$$, and we are required to determine $$\frac{\alpha}{\beta}$$ given that $$\frac{I_{max}+I_{min}}{I_{max}-I_{min}} = \frac{2\alpha+1}{\beta+3}$$.

Let us denote the intensities by $$I_1 = I$$ and $$I_2 = 4I$$; since amplitude is proportional to the square root of intensity, we set $$a_1 = a$$ and $$a_2 = 2a$$.

For constructive interference, the amplitudes add, giving

$$I_{max} = (\sqrt{I_1} + \sqrt{I_2})^2 = (\sqrt{I} + 2\sqrt{I})^2 = 9I\,. $$

Conversely, destructive interference yields

$$I_{min} = (\sqrt{I_1} - \sqrt{I_2})^2 = (\sqrt{I} - 2\sqrt{I})^2 = I\,. $$

Substituting these expressions into the given ratio gives

$$\frac{I_{max}+I_{min}}{I_{max}-I_{min}} = \frac{9I + I}{9I - I} = \frac{10I}{8I} = \frac{5}{4}\,. $$

Therefore,

$$\frac{2\alpha+1}{\beta+3} = \frac{5}{4}\,. $$

By equating numerators and denominators separately, we obtain

$$2\alpha + 1 = 5 \implies \alpha = 2$$

and

$$\beta + 3 = 4 \implies \beta = 1\,. $$

Hence, the required ratio is

$$\frac{\alpha}{\beta} = \frac{2}{1} = 2\,. $$

Answer: Option B: 2