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: Recall the dimensional formula of energy.
Energy has the dimensional formula: \[ [E]=ML^2T^{-2} \]

Step 2: Recall the dimensional formula of Planck's constant.
Planck’s constant \( h \) has the dimensions of action: \[ [h]=ML^2T^{-1} \]

Step 3: Form the ratio \( \dfrac{E}{h} \).
Now, \[ \frac{E}{h}=\frac{ML^2T^{-2}}{ML^2T^{-1}} \]

Step 4: Simplify the expression.
Canceling common terms \( M \) and \( L^2 \), we get: \[ \frac{E}{h}=T^{-1} \]

Step 5: Interpret the resulting dimension.
The dimension \( T^{-1} \) corresponds to: \[ \text{frequency} \]

Step 6: Verify with physical relation.
From quantum physics: \[ E=h\nu \] Thus, \[ \frac{E}{h}=\nu \] which directly represents frequency.

Step 7: Final conclusion.
Hence, the dimensions correspond to \[ \boxed{\text{frequency}} \] Therefore, the correct option is \[ \boxed{(3)\ \text{frequency}} \]