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Question

Physics Question on wave interference

The two coherent sources with intensity ratio β\beta produce interference. The fringe visibility will be

A

2β1+β\frac{2\sqrt{\beta}}{1+\beta}

B

2β2\beta

C

2(1+β)\frac{2}{\left(1+\beta \right)}

D

β1+β\frac{\sqrt{\beta}}{1+\beta}

Answer

2β1+β\frac{2\sqrt{\beta}}{1+\beta}

Explanation

Solution

I1I2=a2b2=β∴ab=β\frac{I_{1}}{I_{2}}=\frac{a^{2}}{b^{2}}=\beta \therefore \frac{a}{b}=\sqrt{\beta} Fringe visibility is given by V=Imax−IminImax+Imin=(a+b)2−(a−b)2(a+b)2+(a−b)2V=\frac{I_{max}-I_{min}}{I_{max}+I_{min}}=\frac{\left(a+b\right)^{2}-\left(a-b\right)^{2}}{\left(a+b\right)^{2}+\left(a-b\right)^{2}} =4ab2(a2+b2)=2(a/b)(a2b2+1)=2ββ+1=\frac{4ab}{2\left(a^{2}+b^{2}\right)}=\frac{2\left(a/b\right)}{\left(\frac{a^{2}}{b^{2}}+1\right)}=\frac{2\sqrt{\beta}}{\beta+1}