Question

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

• 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 $-3/2$

Answer 1

Answer: (A)

$E=G^p{h}^q{c}^r\dots \left(i\right)$

$\left[M^1{L}^2{T}^{-2}\right]={\left[M^{-1}{L}^3{T}^{-2}\right]}^p{\left[M{L}^2{T}^{-1}\right]}^q{\left[L{T}^{-1}\right]}^r$

$=\left[M^{-p+q}{L}^{3p+2q+r}{T}^{-2p-q-r}\right]$

Applying principle of homogeneity of dimensions, we get

$-p+q=1\dots \left(ii\right)$

$3p+2q+r=2\dots \left(iii\right)$

$-2p-q-r=-2\dots \left(iv\right)$

Adding $\left(iii\right)$ and $\left(iv\right)$,

we get $p+q=0\dots \left(v\right)$

Adding $\left(ii\right)$ and $\left(v\right)$,

we get $q=\frac{1}{2}$

From $\left(ii\right)$, we get $p=q-1$

$=\frac{1}{2}-1=-\frac{1}{2}$

From $\left(ii\right)$, we get $-\frac{3}{2}+1+r=2$, $r=\frac{5}{2}$