Dimensional Analysis Using G h and c JEE Physics Question

 

❓ Question

If

$$F=kG^x h^y c^z$$

where $G$ is the gravitational constant, $h$ is Planck's constant, $c$ is the speed of light and $k$ is a dimensionless constant, find the values of $x,y,z$.

---

✍️ Solution

Given,

$$F=kG^x h^y c^z$$

Dimensional formula of gravitational constant:

$$[G]=[M^{-1}L^3T^{-2}]$$

Dimensional formula of Planck's constant:

$$[h]=[ML^2T^{-1}]$$

Dimensional formula of speed of light:

$$[c]=[LT^{-1}]$$

Also,

$$[F]=[MLT^{-2}]$$

Therefore,

$$[MLT^{-2}] = [M^{-1}L^3T^{-2}]^x [ML^2T^{-1}]^y [LT^{-1}]^z$$

Comparing powers of $M,L,T$:

$$-x+y=1$$
$$3x+2y+z=1$$
$$-2x-y-z=-2$$

Solving these equations, we get

$$x=-\frac12,\qquad y=\frac12,\qquad z=-\frac52$$

✅ Final Answer

$$\boxed{x=-\frac12,\quad y=\frac12,\quad z=-\frac52}$$