Question

The dimensions of Planck's constant are the same as those of

• A. energy
• B. power
• C. angular frequency
• D. angular momentum
• E. linear momentum

Hint:

One of the easiest ways to remember this is Bohr's quantization rule: $mvr = \frac{nh}{2\pi}$. Since $n$ and $2\pi$ are dimensionless, $h$ must have the same dimensions as angular momentum ($mvr$).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We must determine the dimensional formula for Planck's constant (\(h\)) and match it with the standard physical quantities given in the options.
Step 2: Key Formula or Approach:
Use the energy equation of a photon: \(E = h\nu\), where \(E\) is energy and \(\nu\) is frequency.
Dimensional formula of \(E\) is \([ML^2T^{-2}]\).
Dimensional formula of \(\nu\) is \([T^{-1}]\).
Step 3: Detailed Explanation:
Derive Planck's constant dimensions:
\[ h = \frac{E}{\nu} = \frac{[ML^2T^{-2}]}{[T^{-1}]} = [ML^2T^{-1}] \] Now, derive the dimensions for the options:
(A) Energy: \([ML^2T^{-2}]\)
(B) Power: \(\frac{\text{Energy}}{\text{Time}} = \frac{[ML^2T^{-2}]}{[T]} = [ML^2T^{-3}]\)
(C) Angular frequency (\(\omega\)): \([T^{-1}]\)
(D) Angular momentum (\(L\)): \(L = mvr = [M][LT^{-1}][L] = [ML^2T^{-1}]\)
(E) Linear momentum (\(p\)): \(p = mv = [M][LT^{-1}]\)
Comparing them, Planck's constant has the same dimensions as angular momentum.
Step 4: Final Answer:
The dimensions are the same as angular momentum.