दो कलासंबद्ध प्रकाश स्रोत, जिनकी तीव्रताओं का अनुपात 2x है, व्यतिकरण पैटर्न उत्पन्न करते हैं। अनुपात \(\frac{I_{\max}− I_{\min}}{I_{\max}+ I_{\min}}\) का मान होगा :

CONCEPT:

• For the maximum intensity we write;

           \(I_{max}=({\sqrt {I_1}+\sqrt {I_2}})^2\)

           Here I_max is the maximum intensity, and I₁ and I₂ are the intensity in 1 source and 2nd source.

• For the minimum intensity we write;

           \(I_{min}=({\sqrt {I_1}-\sqrt {I_2}})^2\)

           Here Imin is the minimum intensity, and I1 and I2 are the intensity in 1 source and 2nd source.

CALCULATION:

Given: The two coherent light sources having the intensity in the ratio of 2x which is written as;

\(\frac{I_1}{I_2} = 2x\)

⇒ \(\frac{I_1}{I_2} = \frac {2x}{1}\)

Let, \(I_1 = 2x\), and \(I_2 = 1\)

The maximum intensity, I_max = \(({\sqrt {I_1}+\sqrt {I_2}})^2\)

= \(({\sqrt {2x}+\sqrt {1}})^2\)

= \(({\sqrt {2x}+{1}})^2\)

The minimum intensity, Imin = \(({\sqrt {I_1}-\sqrt {I_2}})^2\)

= \(({\sqrt {2x}-\sqrt {1}})^2\)

= \(({\sqrt {2x}-{1}})^2\)

To calculate: \(\frac{I_{max}-I_{min}}{I_{max}+I_{min}}\)    ----(1)

Now, on putting the values in equation (1) we have;

⇒ \(\frac{I_{max}-I_{min}}{I_{max}+I_{min}}\) = \(\frac{ (\sqrt{2x} +1)^2- (\sqrt{2x} -1)^2}{ (\sqrt{2x} +1)^2+ (\sqrt{2x} -1)^2}\)

⇒ \(\frac{I_{max}-I_{min}}{I_{max}+I_{min}}\) = \(\frac{ 2x^2 +1+2\sqrt 2x -( 2x^2+1-2 \sqrt 2x)}{2x^2 +1+2 \sqrt 2x +( 2x^2+1-2 \sqrt 2x)}\)

⇒ \(\frac{I_{max}-I_{min}}{I_{max}+I_{min}}\) = \(\frac{2\sqrt{2x}}{2x+1}\)

Hence, option 2) is the correct answer.