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AMU Medical AMU Solved Paper-1996

  • question_answer
    The radiation emitted per unit time by unit area of a black body at temperature \[T\]is \[\sigma {{T}^{4}}\] where a is the Stefan-Boltzmann constant. The constant\[\sigma \]can also be expressed in terms of Boltmanns constant (k). Plancks constant (h) and speed of light  as\[\sigma =A{{k}^{a}}{{h}^{\beta }}{{c}^{\gamma }},\]where \[A,\alpha ,\beta \]and \[\gamma \]are dimensionless constants. The set \[(\alpha ,\beta ,\gamma )\]is given by

    A)  \[(4,-3,-2)\]        

    B) \[(-4,-3,\text{ }2)\]

    C)  \[(4,3,2)\]

    D)  \[(-4,-3,-2)\]

    Correct Answer: A

    Solution :

    : Equate dimensions/powers of M, L, T and\[\theta \]. Dimensions of\[\sigma =[\sigma ]=[M{{L}^{-3}}{{\theta }^{-4}}]\] \[[k]=[M{{L}^{2}}{{T}^{-2}}{{\theta }^{-1}}]\] \[[h]=[M{{L}^{2}}{{T}^{-1}}]\] \[[c]=[L{{T}^{-1}}]\] Given: \[\sigma =A{{k}^{\alpha }}{{h}^{\beta }}{{c}^{\gamma }}\] \[[M{{T}^{-3}}{{\theta }^{-4}}]={{[M{{L}^{2}}{{T}^{-2}}{{\theta }^{-1}}]}^{\alpha }}{{[M{{L}^{2}}{{T}^{-1}}]}^{\beta }}{{[L{{T}^{-1}}]}^{\gamma }}\] From M, \[1=\alpha +\beta \]                         ...(i) From L, \[0=2\alpha +2\beta +\gamma \]                 ...(ii) From T, \[-3=-2\alpha -\beta -\gamma \]               ...(iii) From \[\theta ,\]\[-4=-\alpha \] ...(iv) From (iv),              \[\alpha =4\]             ?.(v) From (i) and (v),       \[\beta =-3\]            ...(vi) From (ii), (v), (vi),     \[\gamma =-2\]         ...(vii) The set\[(\alpha ,\beta ,\gamma )\]is     \[(4,-3,-2)\]


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