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JEE Main & Advanced Sample Paper JEE Main - Mock Test - 34

  • question_answer
    A charge of magnitude Q is placed at O, the centre of an imaginary sphere, which can be considered over any radius as shown in figure. Consider parts of the spheres over a solid angle \[\Omega =\pi \]steradian through which three areas \[{{A}_{1}},{{A}_{2}},{{A}_{3}}\] are considered at a distance \[r,2r,3r\]from the centre respectively as shown. Then study the following statements:
    (i) The flux passing through \[{{A}_{1}}\] is given by \[\frac{Q}{4{{\varepsilon }_{0}}}\].
    (ii) The flux passing through \[{{A}_{2}}\] is given by \[\frac{Q}{9{{\varepsilon }_{0}}}\](iii) The flux passing through \[{{A}_{2}}\] within the area equal to \[{{A}_{1}}\] is given by\[\frac{Q}{16{{\varepsilon }_{0}}}\].
    (iv) The flux passing through \[{{A}_{3}}\] within the area equal to \[{{A}_{2}}\] is given by \[\frac{Q}{4{{\varepsilon }_{0}}}\]
    The correct statements are

    A) (i) and (iii)     

    B)        (ii) and (iii)

    C) (i) and (iv)          

    D)        (iii) and (iv)

    Correct Answer: A

    Solution :

    [a] We have \[{{A}_{1}}=4\pi {{r}^{2}}\] \[{{A}_{2}}=4\pi {{(2r)}^{2}}=4{{A}_{1}}\] \[{{A}_{3}}=4\pi {{(3r)}^{2}}=9{{A}_{1}}=\frac{9}{4}{{A}_{2}}\] \[{{\phi }_{1}}={{\phi }_{2}}={{\phi }_{3}}\]   through same solid angle \[{{\phi }_{1}}=\int{EdA=\int{\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{Q}{{{r}^{2}}}}dA=\frac{Q}{4\pi {{\varepsilon }_{0}}}\Omega =\frac{Q}{4\pi {{\varepsilon }_{0}}}\pi =\frac{Q}{4{{\varepsilon }_{0}}}}\] Flux passing through \[{{A}_{1}}\] within area                                     \[{{A}_{1}}=\frac{1}{4}\frac{Q}{4{{\varepsilon }_{0}}}=\frac{Q}{16{{\varepsilon }_{0}}}\] Flux passing through \[{{A}_{2}}\] within area                                     \[{{A}_{2}}=\frac{1}{9}\frac{4Q}{4{{\varepsilon }_{0}}}=\frac{Q}{9{{\varepsilon }_{0}}}\]


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