Question

Two coherent sources of different intensities send waves which interfere. The ratio of maximum intensity to the minimum intensity is 25. The intensities of the sources are in the ratio
(a) 25 1
(b) 5 1
(c) 9 4
(d) 625 1

Answer 1

Two coherent sources of different intensities send waves which interfere. The ratio of maximum intensity to the minimum intensity is 25. The intensities of the sources are in the ratio $9: 4$

Explanation:

Step 1:

Two reliable sources of the various $I_{1}$ and $I_{2}$ intensities interfere.

The Maximum Ratio of intensity to the minimum intensity is $25$

$\frac{I_{\max }}{I_{\min }}=25$

For maximum intensity,  

$I_{\max }=(\sqrt{I_{1}}+\sqrt{I_{2}})^{2}$

for minimum intensity

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

According to the question  

Step 2:

Total to minimum intensity ratio shall be determined by-

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

$\frac{25}{1}=\frac{5^{2}}{1^{2}}=\frac{(\sqrt{I_{1}}+\sqrt{I_{2}})^{2}}{(\sqrt{I_{1}}-\sqrt{I_{2}})^{2}}$

$\frac{5}{1}=\frac{\sqrt{I_{1}}+\sqrt{I_{2}}}{\sqrt{I_{1}}-\sqrt{I_{2}}}$

$5(\sqrt{I_{1}}-\sqrt{I_{2}})=\sqrt{I_{1}}+\sqrt{I_{2}}$

$5 \sqrt{I_{1}}-5 \sqrt{I_{2}}=\sqrt{I_{1}}+\sqrt{I_{2}}$

$5 \sqrt{I_{1}}-\sqrt{I_{1}}=\sqrt{I_{2}}+5 \sqrt{I_{2}}$

$4 \sqrt{I_{1}}=6 \sqrt{I_{2}}$

$\frac{\sqrt{I_{1}}}{\sqrt{I_{2}}}=\frac{6}{4}$

$\frac{\sqrt{I_{1}}}{\sqrt{I_{2}}}=\frac{3}{2}$

Step 3:

Squaring on both sides, We get

$\frac{I_{1}}{I_{2}}=\frac{9}{4}$

$I_{1}: I_{2}=9: 4$

Hence source intensity ratio is $9: 4$.