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:
Physical quantities are said to have the same dimensions if they have the same power of fundamental units (Mass, Length, Time). We can find the dimensions using standard formulas.

Step 2: Detailed Explanation:
1. Planck's constant ($h$):
From $E = h\nu \implies h = E/\nu$, where $E$ is energy and $\nu$ is frequency ($1/T$).
Dimension of $E = [ML^2T^{-2}]$.
Dimension of $\nu = [T^{-1}]$.
Dimension of $h = \frac{[ML^2T^{-2}]}{[T^{-1}]} = [ML^2T^{-1}]$.
2. Angular momentum ($L$):
From $L = mvr$, where $m$ is mass, $v$ is velocity, and $r$ is radius.
Dimension of $m = [M]$.
Dimension of $v = [LT^{-1}]$.
Dimension of $r = [L]$.
Dimension of $L = [M][LT^{-1}][L] = [ML^2T^{-1}]$.
The dimensions match.

Step 3: Final Answer:
Planck's constant has the same dimensions as angular momentum.