Question 81: Units , Dimensions and Error Analysis

Question

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

Accepted Answer

To determine the dimensional formula for Planck's constant, we will start by analyzing the given de-Broglie wavelength equation:
$\lambda = \frac{h}{\sqrt{2 m E}}$
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 $[	ext{M}]$.
3. Energy $E$ has the dimensional formula of work, which is force times distance:
$[E] = [F][L] = [	ext{MLT}^{-2}][L] = [	ext{ML}^{2}	ext{T}^{-2}]$.
Now, let's rewrite the equation in terms of the dimensions:
$[L] = \frac{[h]}{\sqrt{2 [	ext{M}] [	ext{ML}^{2}	ext{T}^{-2}]}}$
Simplifying inside the square root:
$[L] = \frac{[h]}{\sqrt{2 [	ext{M}] [	ext{M}] [	ext{L}^{2}	ext{T}^{-2}]}}$
$[L] = \frac{[h]}{\sqrt{2 [	ext{M}^{2}] [	ext{L}^{2}	ext{T}^{-2}]}}$
Since the constants like 2 do not affect the dimensional formula, we can simplify further:
$[L] = \frac{[h]}{[	ext{M L T}^{-1}]}$
Cross multiplying to solve for the dimensional formula of $h$:
$[h] = [L] [	ext{M L T}^{-1}]$
$[h] = [	ext{M L}^{2} 	ext{T}^{-1}]$
Therefore, the dimensional formula for Planck's constant $h$ is:
$\left[	ext{ML}^{2}	ext{T}^{-1}ight]$
Hence, the correct option is:
Option C $\left[\mathrm{ML}^2 \mathrm{~T}^{-1}ight]$

Answer

$\left[\mathrm{M}^2 \mathrm{~L}^2 \mathrm{~T}^{-2}\right]$

Answer

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

Answer

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

Answer

$\left[\mathrm{MLT}^{-2}\right]$

Units , Dimensions and Error Analysis

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

Choose the correct answer: