The de-Broglie wavelength associated with a particle of mass \( m \) and energy \( E \) is \[\lambda = \frac{h}{\sqrt{2mE}}.\]The dimensional formula for Planck's constant is:
- \([ML^{-1}T^{-2}]\)
- \([ML^2T^{-1}]\)
- \([MLT^{-2}]\)
- \([M^2L^2T^{-2}]\)
The Correct Option is B
Approach Solution - 1
The given problem requires us to find the dimensional formula for Planck's constant \( h \). The de-Broglie wavelength formula given is:
\[\lambda = \frac{h}{\sqrt{2mE}}\]To find the dimensional formula of \( h \), we will utilize the given equation and analyze the dimensions involved. The de-Broglie wavelength formula can be rearranged as:
\(h = \lambda \times \sqrt{2mE}\)
Let's express the dimensions of each component:
- Wavelength \(\lambda\): This is a length, and its dimensional formula is \([L]\).
- Mass \(m\): Its dimensional formula is \([M]\).
- Energy \(E\): Energy has a dimensional formula of \([ML^2T^{-2}]\).
Using these, we calculate the dimensions for \( \sqrt{2mE} \):
\(\sqrt{2mE} = \sqrt{[M][ML^2T^{-2}]} = \sqrt{[M^2L^2T^{-2}]}\)
Simplifying, we get:
\([MLT^{-1}]\)
Substituting back into the formula for \( h \):
\(h = \lambda \times \sqrt{2mE} = [L] \times [MLT^{-1}] = [ML^2T^{-1}]\)
Therefore, the dimensional formula for Planck's constant \( h \) is:
[ML2T-1]
This matches the provided correct answer option, which is:
\([ML^2T^{-1}]\)
To conclude, the correct dimensional formula for Planck's constant is indeed \([ML^2T^{-1}]\), and the selected option is correct. The other options do not match this dimensional analysis.
Approach Solution -2
We are given the equation for the de-Broglie wavelength:
\[ \lambda = \frac{h}{\sqrt{2mE}}. \]
From this equation, rearranging to solve for \( h \):
\[ h = \lambda \sqrt{2mE}. \]
Now, let's find the dimensional formula for \( h \):
- The dimensional formula for \( \lambda \) (wavelength) is \([L]\).
- The dimensional formula for \( m \) (mass) is \([M]\).
- The dimensional formula for \( E \) (energy) is \([ML^2T^{-2}]\).
Now, substitute these into the equation:
\[ [h] = [L] \times \sqrt{[M] \times [ML^2T^{-2}]}. \]
Simplifying the terms inside the square root:
\[ [h] = [L] \times \sqrt{[M] \times [M][L^2][T^{-2}]} = [L] \times \sqrt{[M^2L^2T^{-2}]} = [L] \times [MLT^{-1}]. \]
Thus, the dimensional formula for \( h \) is:
\[ [h] = [ML^2T^{-1}]. \]
Therefore, the correct answer is Option (2).
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