Question

Planck’s constant (h), speed of light in vacuum (c) and Newton’s gravitational constant (G) are three fundamental constants. Which one of the following combinations of these has the dimension of length?

• A. $\sqrt{\frac{hG}{C^3}}$
• B. $\sqrt{\frac{hG}{C^{5/2}}}$
• C. $\sqrt{\frac{hC}{G}}$
• D. $\sqrt{\frac{GC}{h^{3/2}}}$

Answer 1

Answer: (A)

Step 1: Given that:

Fundamental constants are; Planck's constant(h), speed of light in vacuum(c) and Newton's gravitational constant(G)

Step 2: Determination of the dimension of length in terms of given fundamental constants:

Since Planck constants is given as; $E=h\text{ν}$

h = $\frac{E}{\text{ν}}$

where E is energy and ν is the frequency.

The dimension of Planck's constant is given as;

[h] = $\frac{\left[M{L}^2{T}^{-2}\right]}{\left[T^{-1}\right]}=\left[M{L}^2{T}^{-2+1}\right]=\left[M{L}^2{T}^{-1}\right]$

The dimension of the speed of light in vacuum is; [c]= $\left[L{T}^{-1}\right]$

Newton's gravitational;l constant is given as;

$F=G\frac{m_{1}m{}_2}{r{}^2}$

$\Rightarrow G=\frac{F{r}^2}{m_1{m}_2}$

The dimension of G is given as;

[G] = $\frac{\left[ML{T}^{-2}\right]\left[L^2\right]}{\left[M\right]\left[M\right]}=\frac{\left[M{L}^3{T}^{-2}\right]}{\left[M^2\right]}=\left[M^{-1}{L}^3{T}^{-2}\right]$
h=λ×mvh = \lambda \times mv

Now,

[hG] = $\left[M{L}^2{T}^{-1}\right]\times \left[M^{-1}{L}^3{T}^{-2}\right]=\left[M^0{L}^5{T}^{-3}\right]=\left[L^5{T}^{-3}\right]$

[hG] = $\left[L^5{T}^{-3}\right]=\left[L^2\right]\left[L^3{T}^{-3}\right]=\left[L^2\right]{\left[L{T}^{-1}\right]}^3=\left[L^2\right]\left[c^3\right]$

Thus;

$\left[hG\right]=\left[L^2\right]\left[c^3\right]$

$\left[L^2\right]\left[c^3\right]=\left[hG\right]$

$\left[L^2\right]=\frac{\left[hG\right]}{\left[c^3\right]}$

$\left[L\right]=\sqrt{\frac{\left[hG\right]}{\left[c^3\right]}}$

Thus,

Option a) $\sqrt{\frac{\left[hG\right]}{\left[c^3\right]}}$ is the correct option.