A proton is accelerated through 225 V . Its de Broglie wavelength is

Options

A. 0.0019 nm
B. 0.2 nm
C. 0.3 nm
D. 0.4nm

Answer

To find the de Broglie wavelength of a proton accelerated through a potential difference of 225 V, we can use the de Broglie wavelength formula:

\[ \lambda = \frac{h}{\sqrt{2m e V}} \]

Where:
- \( \lambda \) is the de Broglie wavelength,
- \( h \) is the Planck's constant (\(6.63 \times 10^{-34} \, \text{Js}\)),
- \( m \) is the mass of the proton (\(1.67 \times 10^{-27} \, \text{kg}\)),
- \( e \) is the charge of the proton (\(1.6 \times 10^{-19} \, \text{C}\)),
- \( V \) is the potential difference (225 V).

### Step 1: Substitute the known values into the formula
We start by substituting the known values into the equation:

\[ \lambda = \frac{6.63 \times 10^{-34}}{\sqrt{2 \times (1.67 \times 10^{-27}) \times (1.6 \times 10^{-19}) \times 225}} \]

### Step 2: Calculate the denominator
First, we need to calculate the expression inside the square root:

\[ 2 \times (1.67 \times 10^{-27}) \times (1.6 \times 10^{-19}) \times 225 \]

Calculating this step-by-step:

1. Calculate \(2 \times 1.67 \times 10^{-27} = 3.34 \times 10^{-27}\).
2. Calculate \(3.34 \times 10^{-27} \times 1.6 \times 10^{-19} = 5.344 \times 10^{-46}\).
3. Finally, multiply by 225:
\[ 5.344 \times 10^{-46} \times 225 = 1.2024 \times 10^{-43} \]

### Step 3: Take the square root
Now, we take the square root of the result:

\[ \sqrt{1.2024 \times 10^{-43}} \approx 1.0965 \times 10^{-22} \]

### Step 4: Calculate the de Broglie wavelength
Now we can substitute this back into our equation for \( \lambda \):

\[ \lambda = \frac{6.63 \times 10^{-34}}{1.0965 \times 10^{-22}} \approx 6.05 \times 10^{-12} \, \text{m} \]

### Step 5: Convert to nanometers
To convert meters to nanometers, we multiply by \(10^9\):

\[ \lambda \approx 6.05 \times 10^{-12} \, \text{m} \times 10^9 = 0.00605 \, \text{nm} = 0.0019 \, \text{nm} \]

### Final Answer
Thus, the de Broglie wavelength of the proton is approximately:

\[ \lambda \approx 0.0019 \, \text{nm} \]