Question

If 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 $r$ respectively:
A.$-\dfrac{1}{2},\dfrac{1}{2}and\dfrac{5}{2}$
B.$\dfrac{1}{2},-\dfrac{1}{2}and\dfrac{-5}{2}$
C.$\dfrac{-1}{2},\dfrac{1}{2}and\dfrac{3}{2}$
D.$\dfrac{1}{2},\dfrac{-1}{2}and\dfrac{-3}{2}$

Answer 1

HintWrite to both sides dimensions and compare both sides
 dimension of $E=$ dimension of ${{G}^{p}}{{h}^{q}}{{c}^{r}}$
Where $G=$ Universal gravitational constant, $h=$ Plank’s constant and $c=$ Speed of light

Complete step-by-step solution: Dimension:- The dimension of a physical quantity are the power to which the fundamental units are raised in order to obtain the derived unit of that quantity.
To express the dimensions of physical quantities in mechanics, the length, mass and time are denoted by $\left[ L \right]$ $\left[ M \right]$ and $\left[ T \right]$ . The dimensions of physical quantities are$a$in length, $b$ in mass and $c$ in time, then the dimensions of that physical quantity shall be written in the following manner
$\left[ {{L}^{a}}{{M}^{b}}{{C}^{C}} \right]$ [dimension formula]
Given
$E={{G}^{p}}{{h}^{q}}{{c}^{r}}$ ……………(1)
$G=$ Gravitational constant
$h=$ Plank’s constant
$c=$ Speed of light
Dimension of $G=\left[ {{M}^{-1}}{{L}^{3}}T-2 \right]$
Dimension of $h=\left[ M{{L}^{2}}{{T}^{-1}} \right]$
Dimension of $c=\left[ L{{T}^{-1}} \right]=\left[ {{M}^{0}}L{{T}^{-1}} \right]$
Dimension of $E=\left[ M{{L}^{2}}{{T}^{-2}} \right]$
Putt these values in equation (1)
$\left[ M{{L}^{2}}{{T}^{-2}} \right]={{\left[ {{M}^{-1}}{{L}^{-3}}{{T}^{-2}} \right]}^{p}}{{\left[ M{{L}^{2}}{{T}^{-1}} \right]}^{q}}{{\left[ {{M}^{0}}L{{T}^{-1}} \right]}^{r}}$
$\left[ M{{L}^{2}}{{T}^{-2}} \right]=\left[ {{M}^{-p+q+0}}{{L}^{3p+2q+r}}{{T}^{-2p-q-r}} \right]$
Comparing the power of $M$ both sides
$1=-p+q$
$-p+q=1$ ………………….(2)
Comparing the power of $L$ both sides
$2=3p+2q+r$
$3p+2q+r=2$ ………………..(3)
Comparing the power of $T$ both sides
$\begin{align}
  & -2=-2p-q-r \\
 & -\left( 2p+q+r \right)=-2 \\
\end{align}$
$2p+q+r=2$ …………….(4)
Equation (3) substract from equation (4)
$\begin{align}
  & 2p+q+r-3p-2q-r=2-2 \\
 & -p-q=0 \\
\end{align}$
$p=-q$…………..(5)
Put the value of $p$ in equation (1)
$\begin{align}
  & -p+q=1 \\
 & -\left( -q \right)+q=1 \\
 & q+q=1 \\
 & 2q=1 \\
 & q=\dfrac{1}{2} \\
\end{align}$
From equation (5)
$\begin{align}
  & p=-q \\
 & p=-\dfrac{1}{2} \\
\end{align}$
Value of $p$ and $q$ put in equation (3)
$\begin{align}
  & 3\times \left( -\dfrac{1}{2} \right)+2\times \dfrac{1}{2}+r=2 \\
 & -\dfrac{3}{2}+1+r=2 \\
 & -\dfrac{1}{2}+r=2 \\
 & r=2+\dfrac{1}{2} \\
 & r=\dfrac{3}{2} \\
\end{align}$
Putting the value of $p,q$ and $r$ in equation (1)
$E={{G}^{-1}}{{h}^{\dfrac{1}{2}}}{{c}^{\dfrac{3}{2}}}$

Note:Student thought that $G$ is a universal constant, so it has no dimension, but it has the dimension$\left[ {{M}^{-1}}{{L}^{3}}{{T}^{-2}} \right]$