To find the de Broglie wavelength of a neutron with a given energy, we will follow these steps:
### Step 1: Write down the given data
- Mass of neutron, \( m = 1.7 \times 10^{-27} \, \text{kg} \)
- Energy of neutron, \( E = 3 \, \text{eV} \)
- Planck's constant, \( h = 6.6 \times 10^{-34} \, \text{J.s} \)
### Step 2: Convert energy from eV to Joules
To convert the energy from electron volts to joules, we use the conversion factor:
\[ 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \]
Thus,
\[
E = 3 \, \text{eV} = 3 \times 1.6 \times 10^{-19} \, \text{J} = 4.8 \times 10^{-19} \, \text{J}
\]
### Step 3: Use the de Broglie wavelength formula
The de Broglie wavelength \( \lambda \) is given by:
\[
\lambda = \frac{h}{p}
\]
where \( p \) is the momentum. The momentum can be expressed in terms of energy and mass:
\[
p = \sqrt{2mE}
\]
### Step 4: Substitute for momentum in the de Broglie wavelength formula
Substituting the expression for momentum into the de Broglie wavelength formula gives:
\[
\lambda = \frac{h}{\sqrt{2mE}}
\]
### Step 5: Substitute the known values
Now, substituting the known values into the equation:
\[
\lambda = \frac{6.6 \times 10^{-34}}{\sqrt{2 \times (1.7 \times 10^{-27}) \times (4.8 \times 10^{-19})}}
\]
### Step 6: Calculate the denominator
First, calculate the product in the square root:
\[
2 \times (1.7 \times 10^{-27}) \times (4.8 \times 10^{-19}) = 1.632 \times 10^{-45}
\]
Now, take the square root:
\[
\sqrt{1.632 \times 10^{-45}} \approx 1.278 \times 10^{-22}
\]
### Step 7: Calculate the de Broglie wavelength
Now substitute this back into the equation for \( \lambda \):
\[
\lambda = \frac{6.6 \times 10^{-34}}{1.278 \times 10^{-22}} \approx 5.16 \times 10^{-12} \, \text{m}
\]
### Step 8: Final result
Thus, the de Broglie wavelength of the neutron is approximately:
\[
\lambda \approx 1.6 \times 10^{-11} \, \text{m}
\]
