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NEET-XII-Physics

17: Light Waves

with Solutions - page 7
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  • #29
    In a Young’s double slit experiment, the separation between the slits = 2⋅0 mm, the wavelength of the light = 600 nm and the distance of the screen from the slits = 2⋅0 m. If the intensity at the centre of the central maximum is 0⋅20 W m-2, what will be the intensity at a point 0⋅5 cm away from this centre along the width of the fringes?
    Ans : Given:
    Separation between the slits, `` d=2\,\mathrm{\,mm\,}=2\times {10}^{-3}\,\mathrm{\,m\,}``
    `` ``
    Wavelength of the light, `` \lambda =600\,\mathrm{\,nm\,}=6\times {10}^{-7}\,\mathrm{\,m\,}``
    Distance of the screen from the slits, D = 2⋅0 m
    `` {I}_{\,\mathrm{\,max\,}}=0.20\,\mathrm{\,W\,}/{\,\mathrm{\,m\,}}^{2}``
    `` \,\mathrm{\,For\,}\,\mathrm{\,the\,}\,\mathrm{\,point\,}\,\mathrm{\,at\,}\,\mathrm{\,a\,}\,\mathrm{\,position\,}y=0.5\,\mathrm{\,cm\,}=0.5\times {10}^{-2}\,\mathrm{\,m\,},``
    `` \text{p}\,\mathrm{\,ath\,}\,\mathrm{\,difference\,},∆x=\frac{yd}{D}.``
    `` \Rightarrow ∆x=\frac{0.5\times {10}^{-2}\times 2\times {10}^{-3}}{2}``
    `` =5\times {10}^{-6}\,\mathrm{\,m\,}``
    So, the corresponding phase difference is given by
    `` ∆\varphi =\frac{2\pi ∆x}{\lambda }=\frac{2\pi \times 5\times {10}^{-6}}{6\times {10}^{-7}}``
    `` =\frac{50\pi }{3}=16\pi +\frac{2\pi }{3}``
    `` \,\mathrm{\,or\,}∆\,\mathrm{\,\varphi \,}=\frac{2\pi }{3}``
    So, the amplitude of the resulting wave at point y = 0.5 cm is given by
    `` A=\sqrt{{a}^{2}+{a}^{2}+2{a}^{2}\,\mathrm{\,cos\,}\left(\frac{2\pi }{3}\right)}``
    `` =\sqrt{{a}^{2}+{a}^{2}-{a}^{2}}=a``
    `` ``
    Similarly, the amplitude of the resulting wave at the centre is 2a.
    Let the intensity of the resulting wave at point y = 0.5 cm be I.
    `` \,\mathrm{\,Since\,}\frac{I}{{I}_{\,\mathrm{\,max\,}}}=\frac{{A}^{2}}{{\left(2a\right)}^{2}},\,\mathrm{\,we\,}\,\mathrm{\,have\,}:``
    `` \frac{I}{0.2}=\frac{{A}^{2}}{4{a}^{2}}=\frac{{a}^{2}}{4{a}^{2}}``
    `` \Rightarrow I=\frac{0.2}{4}=0.05\,\mathrm{\,W\,}/{\,\mathrm{\,m\,}}^{2}``
    Thus, the intensity at a point 0.5 cm away from the centre along the width of the fringes is 0.05 W/m2.
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