Question

say with a proper explanation

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 =

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}}}}