The de Broglie wavelenght of an electron whose speed is half that of light is:

Options

A. `3.6 xx 10^(-12)m`
B. `4.8 xx 10^(-12)m`
C. `8.4 xx 10^(-12)`
D. `0.12 xx 10^(-12)m`

Answer

To find the de Broglie wavelength of an electron moving at half the speed of light, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the de Broglie Wavelength Formula**:
The de Broglie wavelength (\( \lambda \)) is given by the formula:
\[ \lambda = \frac{h}{p} \]
where \( h \) is Planck's constant and \( p \) is the momentum of the particle.

2. **Express Momentum**:
The momentum (\( p \)) of an electron can be expressed as:
\[ p = mv \]
where \( m \) is the mass of the electron and \( v \) is its velocity.

3. **Substitute Momentum into the Wavelength Formula**:
Substituting the expression for momentum into the de Broglie wavelength formula gives:
\[ \lambda = \frac{h}{mv} \]

4. **Insert Known Values**:
- Planck's constant \( h = 6.626 \times 10^{-34} \, \text{Js} \)
- Mass of the electron \( m = 9.11 \times 10^{-31} \, \text{kg} \)
- Speed of light \( c = 3 \times 10^8 \, \text{m/s} \)
- Since the speed of the electron is half the speed of light:
\[ v = \frac{c}{2} = \frac{3 \times 10^8}{2} = 1.5 \times 10^8 \, \text{m/s} \]

5. **Plug Values into the Wavelength Formula**:
Now substituting the values into the wavelength formula:
\[ \lambda = \frac{6.626 \times 10^{-34}}{(9.11 \times 10^{-31})(1.5 \times 10^8)} \]

6. **Calculate the Denominator**:
First, calculate the denominator:
\[ 9.11 \times 10^{-31} \times 1.5 \times 10^8 = 1.3665 \times 10^{-22} \]

7. **Calculate the Wavelength**:
Now calculate \( \lambda \):
\[ \lambda = \frac{6.626 \times 10^{-34}}{1.3665 \times 10^{-22}} \approx 4.85 \times 10^{-12} \, \text{m} \]

8. **Final Result**:
The de Broglie wavelength of the electron is approximately:
\[ \lambda \approx 4.85 \times 10^{-12} \, \text{m} \]