The interference pattern is obtained with two coherent light sources of intensity ( I 1 and I 2 ) ratio ' β '. In the interference pattern, the ratio ( I ma x − I min I ma x + I mi

Question

The interference pattern is obtained with two coherent light sources of intensity ( I 1 ​ and I 2 ​ ) ratio ' β '. In the interference pattern, the ratio ( I ma x ​ − I min ​ I ma x ​ + I min ​ ​ ) will be ( β = I 1 ​ I 2 ​ ​ )

TestHub · Accepted Answer

The maximum and minimum intensities in an interference pattern are given by:

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

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

Adding the two equations:

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

Subtracting the two equations:

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

Taking the ratio:

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

Dividing the numerator and the denominator by $I_{1}$ :

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

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

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

Answer: $\frac{β + 1}{2 β}$

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