Question

The dimensions of ratio of energy to Planck's constant are those of

• A. time
• B. velocity
• C. frequency
• D. linear momentum
• E. angular momentum

Hint:

Remember the key relation \( E = h\nu \). It directly tells that \( \dfrac{E}{h} \) has the dimension of frequency.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The question asks for the physical quantity that has the same dimensions as the ratio of Energy (E) to Planck's constant (h). We can solve this either by using a known physics formula that connects these three quantities or by performing dimensional analysis.
Step 2: Key Formula or Approach:
Method 1: Using the Planck-Einstein Relation
The energy of a photon (E) is related to its frequency (f or \( \nu \)) by the equation:
\[ E = hf \] where h is Planck's constant. Rearranging this formula will directly give us the answer.
Method 2: Dimensional Analysis
1. Find the dimensions of Energy (E).
2. Find the dimensions of Planck's constant (h).
3. Calculate the dimensions of the ratio E/h.
4. Compare the result with the dimensions of the quantities listed in the options.
Step 3: Detailed Explanation:
Method 1:
Using the relation \( E = hf \). We want to find the dimensions of the ratio E/h.
From the formula, we can write:
\[ \frac{E}{h} = f \] where f is frequency. Therefore, the ratio of energy to Planck's constant has the dimensions of frequency.
Method 2:
1. Dimensions of Energy (E): Energy has the same dimensions as work (Force \( \times \) Distance).
Force \( F = ma \implies [\text{M}][\text{L}][\text{T}]^{-2} \).
Energy \( E = F \times d \implies [\text{M}][\text{L}][\text{T}]^{-2} \times [\text{L}] = [\text{M}][\text{L}]^2[\text{T}]^{-2} \).
2. Dimensions of Planck's constant (h): From \( E=hf \), we have \( h = E/f \).
Frequency \( f \) has dimensions of [T]\(^{-1}\).
So, \( [h] = \frac{[E]}{[f]} = \frac{[\text{M}][\text{L}]^2[\text{T}]^{-2}}{[\text{T}]^{-1}} = [\text{M}][\text{L}]^2[\text{T}]^{-1} \).
(Alternatively, angular momentum \( L = mvr \), which has dimensions \( [\text{M}][\text{L}][\text{T}]^{-1}[\text{L}] = [\text{M}][\text{L}]^2[\text{T}]^{-1} \), same as Planck's constant).
3. Dimensions of the ratio E/h:
\[ \left[\frac{E}{h}\right] = \frac{[\text{M}][\text{L}]^2[\text{T}]^{-2}}{[\text{M}][\text{L}]^2[\text{T}]^{-1}} = [\text{M}]^{1-1}[\text{L}]^{2-2}[\text{T}]^{-2-(-1)} = [\text{M}]^0[\text{L}]^0[\text{T}]^{-1} = [\text{T}]^{-1} \] 4. Comparison: The dimension [T]\(^{-1}\) is the dimension of frequency (cycles per second).
Step 4: Final Answer:
The ratio of energy to Planck's constant has the dimensions of frequency. This corresponds to option (C).