JEE MAIN - Physics (2019 - 9th January Morning Slot - No. 23)
Two coherent sources produce waves of different intensities which interfere. After interference, the ratio of the maximum intensity to the minimum intensity is 16. The intensity of the waves are in the ratio :
16 : 9
25 : 9
4 : 1
5 : 3
Explanation
Given that,
$${{{{\rm I}_{\max }}} \over {{{\mathop{\rm I}\nolimits} _{min}}}} = {{16} \over 1}$$
We know,
Imax $$=$$ $${\left( {\sqrt {{{\rm I}_1}} + \sqrt {{{\rm I}_2}} } \right)^2}$$
and Imin $$ = {\left( {\sqrt {{{\rm I}_1}} - \sqrt {{{\rm I}_2}} } \right)^2}$$
$$ \therefore $$ $${{{{\left( {\sqrt {{{\rm I}_1}} + \sqrt {{{\rm I}_2}} } \right)}^2}} \over {{{\left( {\sqrt {{{\rm I}_1}} - \sqrt {{{\rm I}_2}} } \right)}^2}}} = {{16} \over 1}$$
$$ \Rightarrow $$ $${{\sqrt {{{\rm I}_1}} + \sqrt {{{\rm I}_2}} } \over {\sqrt {{{\rm I}_1}} - \sqrt {{{\rm I}_2}} }} = {4 \over 1}$$
$$ \Rightarrow $$ $$4\sqrt {{{\rm I}_1}} - 4\sqrt {{{\rm I}_2}} = \sqrt {{{\rm I}_1}} + \sqrt {{{\rm I}_2}} $$
$$ \Rightarrow $$ $$3\sqrt {{{\rm I}_1}} = 5\sqrt {{{\rm I}_2}} $$
$$ \Rightarrow $$ $${{\sqrt {{{\rm I}_1}} } \over {\sqrt {{{\rm I}_2}} }} = {5 \over 3}$$
$$ \Rightarrow $$ $${{{{\rm I}_1}} \over {{{\rm I}_2}}} = {{25} \over 9}$$
$${{{{\rm I}_{\max }}} \over {{{\mathop{\rm I}\nolimits} _{min}}}} = {{16} \over 1}$$
We know,
Imax $$=$$ $${\left( {\sqrt {{{\rm I}_1}} + \sqrt {{{\rm I}_2}} } \right)^2}$$
and Imin $$ = {\left( {\sqrt {{{\rm I}_1}} - \sqrt {{{\rm I}_2}} } \right)^2}$$
$$ \therefore $$ $${{{{\left( {\sqrt {{{\rm I}_1}} + \sqrt {{{\rm I}_2}} } \right)}^2}} \over {{{\left( {\sqrt {{{\rm I}_1}} - \sqrt {{{\rm I}_2}} } \right)}^2}}} = {{16} \over 1}$$
$$ \Rightarrow $$ $${{\sqrt {{{\rm I}_1}} + \sqrt {{{\rm I}_2}} } \over {\sqrt {{{\rm I}_1}} - \sqrt {{{\rm I}_2}} }} = {4 \over 1}$$
$$ \Rightarrow $$ $$4\sqrt {{{\rm I}_1}} - 4\sqrt {{{\rm I}_2}} = \sqrt {{{\rm I}_1}} + \sqrt {{{\rm I}_2}} $$
$$ \Rightarrow $$ $$3\sqrt {{{\rm I}_1}} = 5\sqrt {{{\rm I}_2}} $$
$$ \Rightarrow $$ $${{\sqrt {{{\rm I}_1}} } \over {\sqrt {{{\rm I}_2}} }} = {5 \over 3}$$
$$ \Rightarrow $$ $${{{{\rm I}_1}} \over {{{\rm I}_2}}} = {{25} \over 9}$$
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