Question

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:

• A. \([ML^{-1}T^{-2}]\)
• B. \([ML^2T^{-1}]\)
• C. \([MLT^{-2}]\)
• D. \([M^2L^2T^{-2}]\)

Answer 1

The Correct Option is B

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.

Answer 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).