Two coherent sources of intensity ratio α interfere. In interference pattern, I ma x + I min I ma x − I min is

Question

Two coherent sources of intensity ratio α interfere. In interference pattern, I ma x ​ + I min ​ I ma x ​ − I min ​ ​ is

TestHub · Accepted Answer

Let the intensities of the two coherent sources be $I_{1}$ and $I_{2}$ .

Given the intensity ratio $\frac{I _{1}}{I _{2}} = α$ .

The maximum intensity in the interference pattern is given by:

$I_{ma x} = (I_{1} + I_{2})^{2} = I_{1} + I_{2} + 2 I_{1} I_{2}$

The minimum intensity is given by:

$I_{min} = (I_{1} - I_{2})^{2} = I_{1} + I_{2} - 2 I_{1} I_{2}$

Calculating the difference and sum of the maximum and minimum intensities:

$I_{ma x} - I_{min} = 4 I_{1} I_{2}$

$I_{ma x} + I_{min} = 2 (I_{1} + I_{2})$

Taking the ratio of these two quantities:

$\frac{I _{ma x} - I _{min}}{I _{ma x} + I _{min}} = \frac{4 I _{1} I _{2}}{2 ( I _{1} + I _{2} )} = \frac{2 I _{1} I _{2}}{I _{1} + I _{2}}$

Dividing the numerator and the denominator by $I_{2}$ :

$\frac{I _{ma x} - I _{min}}{I _{ma x} + I _{min}} = \frac{2 \frac{I _{1}}{I _{2}}}{\frac{I _{1}}{I _{2}} + 1}$

Substituting $\frac{I _{1}}{I _{2}} = α$ :

$\frac{I _{ma x} - I _{min}}{I _{ma x} + I _{min}} = \frac{2 α}{α + 1}$

Answer: $\frac{2 α}{1 + α}$

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