If h is Planck's constant and `lambda` is the wave length,`h/(lambda)` has the dimensions of

Options

A. Energy
B. Momentum
C. Moment of Inertia
D. Frequency

Answer

To find the dimensions of \( \frac{h}{\lambda} \), where \( h \) is Planck's constant and \( \lambda \) is the wavelength, we can follow these steps:

### Step 1: Understand the relationship between energy, Planck's constant, and wavelength
We know that energy \( E \) can be expressed in terms of Planck's constant \( h \) and frequency \( \nu \) using the formula:
\[ E = h \nu \]
We also know that the speed of light \( c \) is related to wavelength \( \lambda \) and frequency \( \nu \) by the equation:
\[ c = \lambda \nu \]
From this, we can express frequency as:
\[ \nu = \frac{c}{\lambda} \]

### Step 2: Substitute frequency in the energy equation
Substituting \( \nu \) into the energy equation gives:
\[ E = h \left(\frac{c}{\lambda}\right) \]
Rearranging this, we find:
\[ \frac{h}{\lambda} = \frac{E}{c} \]

### Step 3: Determine the dimensions of energy and speed of light
The dimensions of energy \( E \) are:
\[ [E] = M^1 L^2 T^{-2} \]
The dimensions of the speed of light \( c \) are:
\[ [c] = M^0 L^1 T^{-1} \]

### Step 4: Calculate the dimensions of \( \frac{h}{\lambda} \)
Now, substituting the dimensions into the equation \( \frac{h}{\lambda} = \frac{E}{c} \):
\[ \left[\frac{h}{\lambda}\right] = \frac{[E]}{[c]} = \frac{M^1 L^2 T^{-2}}{M^0 L^1 T^{-1}} = M^1 L^{2-1} T^{-2+1} = M^1 L^1 T^{-1} \]

### Step 5: Identify the physical quantity corresponding to the dimensions
The dimensions \( M^1 L^1 T^{-1} \) correspond to momentum. Thus, we conclude that:
\[ \frac{h}{\lambda} \text{ has the dimensions of momentum.} \]

### Final Answer
The dimensions of \( \frac{h}{\lambda} \) are those of momentum.

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