Question

Given energy, E=g¹hmCn, where G= universal gravitational constant, h= Planck's constant, C= velocity of light. Calculate the values of 1, m and n?

Answer 1

l = 1/8, m = 9/8, and n = -5/8

Answer 2

The energy E is given by the relation $E = G^l h^m C^n$. To find the values of l, m, and n, we equate the dimensions on both sides of the equation.

The dimensions of the quantities are:

• Energy (E): $[ML^2T^{-2}]$
• Universal Gravitational Constant (G): $[M^{-1}L^3T^{-2}]$
• Planck's constant (h): $[ML^2T^{-1}]$
• Velocity of light (C): $[LT^{-1}]$

Substituting these dimensions into the given relation: $[ML^2T^{-2}] = ([M^{-1}L^3T^{-2}])^l ([ML^2T^{-1}])^m ([LT^{-1}])^n$ $[ML^2T^{-2}] = [M^{-l+m} L^{3l+2m+n} T^{-2l-m-n}]$

Equating the powers of M, L, and T on both sides: For M: $1 = -l + m$ (1) For L: $2 = 3l + 2m + n$ (2) For T: $-2 = -2l - m - n$ (3)

From (1), $m = l + 1$. Substitute m into (3): $-2 = -2l - (l+1) - n$ $-2 = -3l - 1 - n$ $n = 3l - 1$ (4)

Substitute m and n into (2): $2 = 3l + 2(l+1) + (3l-1)$ $2 = 3l + 2l + 2 + 3l - 1$ $2 = 8l + 1$ $1 = 8l \implies l = \frac{1}{8}$

Now find m and n: $m = l + 1 = \frac{1}{8} + 1 = \frac{9}{8}$ $n = 3l - 1 = 3\left(\frac{1}{8}\right) - 1 = \frac{3}{8} - 1 = -\frac{5}{8}$

The values are $l = \frac{1}{8}$, $m = \frac{9}{8}$, and $n = -\frac{5}{8}$.