The de-Broglie wavelength associated with a particle of mass m and energy E is h/2mE. The dimensional formula for Planck's constant is :

 

The de-Broglie wavelength associated with a particle of mass $m$ and energy $E$ is $h/\sqrt{2mE}$. The dimensional formula for Planck's constant is :

Here is your right answer

To determine the dimensional formula for Planck's constant, we will start by analyzing the given de-Broglie wavelength equation:

$\lambda =\frac{h}{\sqrt{2mE}}$

Here, $\lambda$ is the wavelength, $h$ is the Planck's constant, $m$ is the mass of the particle, and $E$ is the energy of the particle.

First, let's derive the dimensional formula for each term involved:

1. Wavelength $\lambda$ has the dimensional formula of length $[L]$.

2. Mass $m$ has the dimensional formula $[\text{M}]$.

3. Energy $E$ has the dimensional formula of work, which is force times distance:

$[E]=[F][L]=[{\text{MLT}}^{-2}][L]=[{\text{ML}}^2{\text{T}}^{-2}]$.

Now, let's rewrite the equation in terms of the dimensions:

$[L]=\frac{[h]}{\sqrt{2[\text{M}][{\text{ML}}^2{\text{T}}^{-2}]}}$

Simplifying inside the square root:

$[L]=\frac{[h]}{\sqrt{2[\text{M}][\text{M}][{\text{L}}^2{\text{T}}^{-2}]}}$

$[L]=\frac{[h]}{\sqrt{2[{\text{M}}^2][{\text{L}}^2{\text{T}}^{-2}]}}$

Since the constants like 2 do not affect the dimensional formula, we can simplify further:

$[L]=\frac{[h]}{[{\text{M L T}}^{-1}]}$

Cross multiplying to solve for the dimensional formula of $h$:

$[h]=[L][{\text{M L T}}^{-1}]$

$[h]=[{\text{M L}}^2{\text{T}}^{-1}]$

Therefore, the dimensional formula for Planck's constant $h$ is:

$\left[{\text{ML}}^2{\text{T}}^{-1}\right]$

Hence, the right answer is  $\left[{\mathrm{ML}}^2{\mathrm{T}}^{-1}\right]$