If the energy is expressed as E = G^p h^q c^r, where G is the universal gravitational constant, h is Planck's constant, and c is the speed of light, find the values of p, q, r respectively.

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NEET Physics QUESTION #7218

Question 1

If the energy is expressed as $E = G^p h^q c^r$, where $G$ is the universal gravitational constant, $h$ is Planck's constant, and $c$ is the speed of light, find the values of $p$, $q$, $r$ respectively.

$-\frac{1}{2},\ \frac{1}{2}$ and $\frac{5}{2}$✔️
$\frac{1}{2},\ -\frac{1}{2}$ and $-\frac{5}{2}$
$-\frac{1}{2},\ \frac{1}{2}$ and $\frac{3}{2}$
$\frac{1}{2},\ \frac{1}{2}$ and $-\frac{3}{2}$

Correct Answer Explanation

Write dimensions of each quantity:

$[E] = ML^2T^{-2}$
$[G] = M^{-1}L^3T^{-2}$
$[h] = ML^2T^{-1}$
$[c] = LT^{-1}$

So: $ML^2T^{-2} = (M^{-1}L^3T^{-2})^p (ML^2T^{-1})^q (LT^{-1})^r$

Comparing powers of M: $1 = -p + q \Rightarrow q = 1+p$

Comparing powers of T: $-2 = -2p - q - r$

Comparing powers of L: $2 = 3p + 2q + r$

Solving these simultaneously gives: $p = -\dfrac{1}{2}$, $q = \dfrac{1}{2}$, $r = \dfrac{5}{2}$

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