Q93 / 180 · AIIMS 2010 · Physics

If the energy, $E=G^p h^q c^r$, where $G$ is the universal gravitational constant, $h$ is the Planck's constant and $c$ is the velocity of light, then the values of $p, q$ and $r$ are, respectively

  • Correct answer
    A
    $-1 / 2,1 / 2$ and $5 / 2$
  • B
    $1 / 2,-1 / 2$ and $-5 / 2$
  • C
    $-1 / 2,1 / 2$ and $3 / 2$
  • D
    $1 / 2,-1 / 2$ and $-3 / 2$

Solution

$E=G^p h^q c^r$ $$ \begin{aligned} & {\left[\mathrm{M}^1 \mathrm{~L}^2 \mathrm{~T}^{-2}\right]=\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right]^p\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right]^q\left[\mathrm{LT}^{-1}\right]^r} \\ & =\mathrm{M}^{-p+q} \mathrm{~L}^{3 p+2 q+r} \mathrm{~T}^{-2 p-q-r} \end{aligned} $$ Applying principle of homogeneity of dimensions, we get $$ \begin{aligned} & -p+q=1 \\ & 3 p+2 q+r=2 \\ & -2 p-q-r=-2 \end{aligned} $$ Add (iii) and (iv), $p+q=0$ Add (ii) and (v), we get $q=\frac{1}{2}$ From (ii), we get $p=q-1=\frac{1}{2}-1=-\frac{1}{2}$. Put in (iii), we get $-\frac{3}{2}+1+r=2, \quad r=5 / 2$

Hence, the correct answer is option: 'A'.

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