The momentum of a photon having energy equal to the rest energy of an electron is

Options

A. Zero
B. `2.73xx10^(-22) "kg ms"^(-1)`
C. `1.99xx10^(-24) "kg ms"^(-1)`
D. Infinite

Answer

To find the momentum of a photon having energy equal to the rest energy of an electron, we can follow these steps:

### Step 1: Understand the energy of a photon
The energy \( E \) of a photon is given by the equation:
\[ E = \frac{hc}{\lambda} \]
where \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength of the photon.

### Step 2: Relate momentum to energy
We know that the momentum \( p \) of a photon can also be expressed in terms of its wavelength:
\[ \lambda = \frac{h}{p} \]
Substituting this into the energy equation gives:
\[ E = \frac{hc}{\frac{h}{p}} = pc \]
Thus, we have:
\[ E = pc \tag{1} \]

### Step 3: Calculate the rest energy of an electron
The rest energy \( E' \) of an electron is given by:
\[ E' = mc^2 \]
where \( m \) is the mass of the electron. The mass of the electron is approximately \( 9.1 \times 10^{-31} \) kg.

### Step 4: Set the energy of the photon equal to the rest energy of the electron
Since the energy of the photon is equal to the rest energy of the electron, we can write:
\[ pc = mc^2 \tag{2} \]

### Step 5: Cancel \( c \) from both sides
Dividing both sides of equation (2) by \( c \) gives:
\[ p = mc \]

### Step 6: Substitute the values
Now, substituting the known values:
- Mass of electron \( m = 9.1 \times 10^{-31} \) kg
- Speed of light \( c = 3 \times 10^{8} \) m/s

We have:
\[ p = (9.1 \times 10^{-31} \, \text{kg}) \times (3 \times 10^{8} \, \text{m/s}) \]

### Step 7: Calculate the momentum
Calculating this gives:
\[ p = 2.73 \times 10^{-22} \, \text{kg m/s} \]

### Conclusion
Thus, the momentum of a photon having energy equal to the rest energy of an electron is:
\[ \boxed{2.73 \times 10^{-22} \, \text{kg m/s}} \]

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