JEE Main & Advanced Physics Wave Optics / तरंग प्रकाशिकी Interference of Light

When two waves of exactly same frequency (coming from two coherent sources) travels in a medium, in the same direction simultaneously then due to their superposition, at some points intensity of light is maximum while at some other points intensity is minimum. This phenomenon is called Interference of light. It is of following two types

(1) Constructive interference : When the waves meets a point with same phase, constructive interference is obtained at that point (i.e. maximum light)

(i) Phase difference between the waves at the point of observation \[\varphi ={{0}^{o}}\,\text{or}\,2n\pi \]

(ii) Path difference between the waves at the point of observation \[\Delta =n\lambda \](i.e. even multiple of \[\lambda /2\])

(iii) Resultant amplitude at the point of observation will be maximum  \[{{A}_{\max }}={{a}_{1}}+{{a}_{2}}\]

If   \[{{a}_{1}}={{a}_{2}}={{a}_{0}}\Rightarrow {{A}_{\max }}=2{{a}_{0}}\]

(iv) Resultant intensity at the point of observation will be maximum   \[{{I}_{\max }}={{I}_{1}}+{{I}_{2}}+2\sqrt{{{I}_{1}}{{I}_{2}}}\] \[={{\left( \sqrt{{{I}_{1}}}+\sqrt{{{I}_{2}}} \right)}^{2}}\]

If \[{{I}_{1}}={{I}_{2}}={{I}_{0}}\Rightarrow {{I}_{\max }}=4{{I}_{0}}\]

(2) Destructive interference : When the wave meets a point with opposite phase, destructive interference is obtained at that point (i.e. minimum light)

(i) Phase difference \[\varphi ={{180}^{o}}\text{or}\,(2n-1)\,\pi \,;\,\]\[n=1,\,\,2,\,...\]

or   \[(2n+1)\,\pi \,;\]\[n=0,1,2.....\]

(ii) Path difference \[\Delta =(2n-1)\frac{\lambda }{2}\] (i.e. odd multiple of \[\lambda /2\])

(iii) Resultant amplitude at the point of observation will be minimum  \[{{A}_{\min }}={{a}_{1}}-{{a}_{2}}\]

If   \[{{a}_{1}}={{a}_{2}}\Rightarrow {{A}_{\min }}=0\]

(iv) Resultant intensity at the point of observation will be minimum \[{{I}_{\min }}={{I}_{1}}+{{I}_{2}}-2\sqrt{{{I}_{1}}{{I}_{2}}}\]\[={{\left( \sqrt{{{I}_{1}}}-\sqrt{{{I}_{2}}} \right)}^{2}}\] If  \[{{I}_{1}}={{I}_{2}}={{I}_{0}}\Rightarrow {{I}_{\min }}=0\]

(3) Super position of waves of random phase difference : When two waves (or more waves) having random phase difference between them super impose, then no interference pattern is produced. Then the resultant intensity is just the sum of the two intensities.  \[I={{I}_{1}}+{{I}_{2}}\]