Question

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

Answer 1

From energy of a photon, E = hv
Here E is the energy of photon and v is the frequency.
dimension of plank's constant , h = dimension of {E/v}
dimension of h = [ML²T⁻²]/[T⁻¹] = [ML²T⁻¹]

similarly, dimension of speed of light, c = [LT⁻¹]
Gravitational force, F = GMm/r²
So, dimension of G = dimension of {Fr²/Mm}
= [MLT⁻²][L²]/[M²] = [M⁻¹L³T⁻²]

Now, suppose length = $\bold{h^xc^yG^z}\\\bold{[L]=[ML^2T^{-1}]^x[LT^{-1}]^y[M^{-1}L^3T^{-2}]^z}}$
$\bold{[L]=[M^{x-z}][L^{2x+ y+3z}][T^{-x-y-2z}]}$
Compare both sides,
x - z = 0 ⇒x = z ----(1)
2x + y + 3z = 1 ⇒ y + 5z = 1 -----(2)
-x - y - 2z = 0 ⇒x + y + 2z = 0
⇒y + 3z = 0 [ from equation (1)
⇒y + 3z = 0 ----(3)

From equations (2) and (3),
z = 1/2 and x = 1/2 and y = -3/2

Hence, dimension of length = $\bold{\frac{\sqrt{hG}}{c^{\frac{3}{2}}}}$
Hence, option (a) is correct

Answer 2

Answer:see the attachment

Good luck

Explanation: