what should be the velocity of an electron so that its momentum becomes equal to that of a photon of wavelength `5200 Å`

Options

A. 700 m/s
B. 1000 m/s
C. 1400 m/s
D. 2800 m/s

Answer

To find the velocity of an electron such that its momentum is equal to that of a photon with a wavelength of \(5200 \, \text{Å}\), we can follow these steps:

### Step 1: Understand the relationship between momentum and wavelength for a photon
The momentum \(p\) of a photon is given by the equation:
\[ p = \frac{h}{\lambda} \]
where:
- \(h\) is Planck's constant (\(6.63 \times 10^{-34} \, \text{Js}\))
- \(\lambda\) is the wavelength of the photon

### Step 2: Calculate the momentum of the photon
Given the wavelength \(\lambda = 5200 \, \text{Å} = 5200 \times 10^{-10} \, \text{m}\), we can substitute this into the momentum equation:
\[ p = \frac{6.63 \times 10^{-34}}{5200 \times 10^{-10}} \]

### Step 3: Simplify the momentum calculation
Calculating the momentum:
\[ p = \frac{6.63 \times 10^{-34}}{5.2 \times 10^{-7}} \approx 1.275 \times 10^{-27} \, \text{kg m/s} \]

### Step 4: Relate the electron's momentum to its velocity
The momentum of an electron is given by:
\[ p = m \cdot v \]
where:
- \(m\) is the mass of the electron (\(9.1 \times 10^{-31} \, \text{kg}\))
- \(v\) is the velocity of the electron

### Step 5: Set the momentum of the electron equal to the momentum of the photon
To find the velocity of the electron, we set the two momenta equal:
\[ m \cdot v = \frac{h}{\lambda} \]
Rearranging this gives:
\[ v = \frac{h}{m \cdot \lambda} \]

### Step 6: Substitute the known values into the equation
Substituting the known values:
\[ v = \frac{6.63 \times 10^{-34}}{9.1 \times 10^{-31} \cdot 5200 \times 10^{-10}} \]

### Step 7: Calculate the velocity
Calculating the velocity:
\[ v \approx \frac{6.63 \times 10^{-34}}{4.572 \times 10^{-40}} \approx 1.45 \times 10^{6} \, \text{m/s} \]

### Step 8: Final Result
Thus, the velocity of the electron should be approximately:
\[ v \approx 1.45 \times 10^{6} \, \text{m/s} \]