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NEET-XII-Physics

17: Light Waves

with Solutions - page 7
Qstn# iv-30-a Prvs-QstnNext-Qstn
  • #30-a
    half the maximum, (b) one-fourth the maximum.
    Ans : When the intensity is half the maximum:
    Let Imax be the maximum intensity and I be the intensity at the required point at a distance y from the central point.
    So, `` I={a}^{2}+{a}^{2}+2{a}^{2}\,\mathrm{\,cos\,}\varphi ``
    Here, `` \varphi `` is the phase difference in the waves coming from the two slits.
    So, `` I=4{a}^{2}{\,\mathrm{\,cos\,}}^{2}\left(\frac{\varphi }{2}\right)``
    `` \Rightarrow \frac{I}{{I}_{\,\mathrm{\,max\,}}}=\frac{1}{2}``
    `` \Rightarrow \frac{4{a}^{2}{\,\mathrm{\,cos\,}}^{2}\left({\displaystyle \frac{\varphi }{2}}\right)}{4{a}^{2}}=\frac{1}{2}``
    `` \Rightarrow {\,\mathrm{\,cos\,}}^{2}\left(\frac{\varphi }{2}\right)=\frac{1}{2}``
    `` \Rightarrow \,\mathrm{\,cos\,}\left(\frac{\varphi }{2}\right)=\frac{1}{\sqrt{2}}``
    `` \Rightarrow \frac{\,\mathrm{\,\varphi \,}}{2}=\frac{\,\mathrm{\,\pi \,}}{4}``
    `` \Rightarrow \varphi =\frac{\,\mathrm{\,\pi \,}}{2}``
    `` \,\mathrm{\,Corrosponding\,}\,\mathrm{\,path\,}\,\mathrm{\,difference\,},∆x=\frac{\lambda }{4}``
    `` \Rightarrow y=\frac{∆xD}{d}=\frac{\lambda D}{4d}`` (b) When the intensity is one-fourth of the maximum:
    `` \frac{I}{{I}_{\,\mathrm{\,max\,}}}=\frac{1}{4}``
    `` \Rightarrow 4{a}^{2}{\,\mathrm{\,cos\,}}^{2}\left(\frac{\varphi }{2}\right)=\frac{1}{4}``
    `` \Rightarrow {\,\mathrm{\,cos\,}}^{2}\left(\frac{\varphi }{2}\right)=\frac{1}{4}``
    `` \Rightarrow \,\mathrm{\,cos\,}\left(\frac{\varphi }{2}\right)=\frac{1}{2}``
    `` \Rightarrow \frac{\varphi }{2}=\frac{\pi }{3}``
    `` \,\mathrm{\,So\,},\,\mathrm{\,corrosponding\,}\,\mathrm{\,path\,}\,\mathrm{\,difference\,},∆x=\frac{\lambda }{3}``
    `` \,\mathrm{\,and\,}\,\mathrm{\,position\,},y=\frac{∆xD}{d}=\frac{\lambda D}{3d}.``
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  • #30-b
    one-fourth the maximum.
    Ans : When the intensity is one-fourth of the maximum:
    `` \frac{I}{{I}_{\,\mathrm{\,max\,}}}=\frac{1}{4}``
    `` \Rightarrow 4{a}^{2}{\,\mathrm{\,cos\,}}^{2}\left(\frac{\varphi }{2}\right)=\frac{1}{4}``
    `` \Rightarrow {\,\mathrm{\,cos\,}}^{2}\left(\frac{\varphi }{2}\right)=\frac{1}{4}``
    `` \Rightarrow \,\mathrm{\,cos\,}\left(\frac{\varphi }{2}\right)=\frac{1}{2}``
    `` \Rightarrow \frac{\varphi }{2}=\frac{\pi }{3}``
    `` \,\mathrm{\,So\,},\,\mathrm{\,corrosponding\,}\,\mathrm{\,path\,}\,\mathrm{\,difference\,},∆x=\frac{\lambda }{3}``
    `` \,\mathrm{\,and\,}\,\mathrm{\,position\,},y=\frac{∆xD}{d}=\frac{\lambda D}{3d}.``
    Page No 382: