Question

What are the dimensional formula for Planck constant and angular momentum is check whether they are same or not

Answer 1

We consider,

$\text{Planck's Constant, \ h=\dfrac{E}{\nu} }$

where $\text{E}$ is the energy and $\nu$ (nu) is the frequency. The equation is given by Planck's Quantum Theory.

We have to have the dimensions of energy and frequency.

Dimension of energy is $\mathrm{[ML^2T^{-2}]}$ To find this we can use many equations like,

$\bullet\ E=mc^2\\\\\bullet\ E=mgh\\\\\bullet\ E=\dfrac{1}{2}mv^2\\\\\text{etc.}$

Frequency is the reciprocal of time period. So its dimension is $\mathrm{[T^{-1}]}$

Now,

$\mathrm{[h]=\left[\dfrac{E}{\nu}\right]=\left[\dfrac{ML^2T^{-2}}{T^{-1}}\right]=[ML^2T^{-1}]}$

Now, angular momentum.

$\text{Angular momentum, \ L=mvr }\\\\\\\therefore\ \mathrm{[L]=[mvr]=[MLT^{-1}L]=[ML^2T^{-1}]}$

We can see that both have same dimensions.