An electron is moving with a kinetic energy of `4.55 xx 10^-25 J`. What will be Broglie wavelength for this electron ?

Options

A. `5.28 xx 10^-7 m`
B. `7.28 xx 10^-7 m`
C. `2 xx 10^-10 m`
D. `3 xx 10^-5 m`

Explanation

To find the de Broglie wavelength of an electron moving with a given kinetic energy, we can use the de Broglie wavelength formula:

\[ \lambda = \frac{h}{p} \]

where \( \lambda \) is the de Broglie wavelength, \( h \) is Planck's constant, and \( p \) is the momentum of the electron. The momentum \( p \) can be expressed in terms of kinetic energy \( KE \) as follows:

\[ p = \sqrt{2m \cdot KE} \]

where \( m \) is the mass of the electron.

### Step-by-Step Solution:

1. **Identify the given values**:
- Kinetic Energy \( KE = 4.55 \times 10^{-25} \, J \)
- Mass of the electron \( m = 9.1 \times 10^{-31} \, kg \)
- Planck's constant \( h = 6.626 \times 10^{-34} \, J \cdot s \)

2. **Calculate the momentum \( p \)**:
\[ p = \sqrt{2m \cdot KE} \]
Substitute the values:
\[ p = \sqrt{2 \times (9.1 \times 10^{-31} \, kg) \times (4.55 \times 10^{-25} \, J)} \]

3. **Calculate the value inside the square root**:
\[ 2 \times (9.1 \times 10^{-31}) \times (4.55 \times 10^{-25}) = 8.2735 \times 10^{-55} \, kg \cdot J \]

4. **Take the square root**:
\[ p = \sqrt{8.2735 \times 10^{-55}} \approx 9.09 \times 10^{-28} \, kg \cdot m/s \]

5. **Calculate the de Broglie wavelength \( \lambda \)**:
\[ \lambda = \frac{h}{p} \]
Substitute the values:
\[ \lambda = \frac{6.626 \times 10^{-34} \, J \cdot s}{9.09 \times 10^{-28} \, kg \cdot m/s} \]

6. **Perform the division**:
\[ \lambda \approx 7.28 \times 10^{-7} \, m \]

### Final Answer:
The de Broglie wavelength for the electron is approximately \( 7.28 \times 10^{-7} \, m \).

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