What is the momentum of a photon having frequency `1.5 xx 10^(13) Hz` ?

Options

A. `3.3xx10^(-29) " kg ms "^(-1)`
B. `3.3xx10^(-34) " kg ms "^(-1)`
C. `6.6xx10^(-34) " kg ms "^(-1)`
D. `6.6xx10^(-30) " kg ms "^(-1)`

Answer

To find the momentum of a photon with a frequency of \(1.5 \times 10^{13} \, \text{Hz}\), we can follow these steps:

### Step 1: Understand the relationship between momentum, wavelength, and frequency
The momentum \(P\) of a photon can be expressed using the formula:
\[ P = \frac{h}{\lambda} \]
where \(h\) is Planck's constant and \(\lambda\) is the wavelength of the photon.

### Step 2: Relate wavelength to frequency
The wavelength \(\lambda\) can be calculated using the speed of light \(c\) and frequency \(f\):
\[ \lambda = \frac{c}{f} \]
where \(c\) is the speed of light, approximately \(3 \times 10^8 \, \text{m/s}\).

### Step 3: Calculate the wavelength
Substituting the frequency into the wavelength formula:
\[ \lambda = \frac{3 \times 10^8 \, \text{m/s}}{1.5 \times 10^{13} \, \text{Hz}} \]
Calculating this gives:
\[ \lambda = \frac{3 \times 10^8}{1.5 \times 10^{13}} = 2 \times 10^{-5} \, \text{m} \]

### Step 4: Substitute the wavelength into the momentum formula
Now, using the value of \(\lambda\) in the momentum formula:
\[ P = \frac{h}{\lambda} \]
Substituting \(h = 6.63 \times 10^{-34} \, \text{Js}\) and \(\lambda = 2 \times 10^{-5} \, \text{m}\):
\[ P = \frac{6.63 \times 10^{-34}}{2 \times 10^{-5}} \]

### Step 5: Calculate the momentum
Calculating the above expression:
\[ P = 3.315 \times 10^{-29} \, \text{kg m/s} \]
Rounding this to two significant figures gives:
\[ P \approx 3.3 \times 10^{-29} \, \text{kg m/s} \]

### Final Answer
The momentum of the photon is approximately:
\[ P \approx 3.3 \times 10^{-29} \, \text{kg m/s} \]

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