For a proton accelerated through V volts , de Broglie wavelength is given as ` lambda = `

Options

A. `(12.27)/(sqrt(V)) Å`
B. `(0.101)/(sqrt(V)) Å `
C. ` (0.286 )/(sqrt(V)) Å `
D. `(12400)/(V) Å `

Answer

To find the de Broglie wavelength of a proton accelerated through a potential difference of V volts, we can follow these steps:

### Step 1: Understand the de Broglie Wavelength Formula
The de Broglie wavelength (λ) is given by the formula:
\[ \lambda = \frac{h}{\sqrt{2m e V}} \]
where:
- \( h \) is the Planck's constant,
- \( m \) is the mass of the proton,
- \( e \) is the charge of the proton,
- \( V \) is the potential difference.

### Step 2: Substitute Known Values
We need the following values:
- Planck's constant \( h = 6.63 \times 10^{-34} \, \text{J s} \)
- Mass of a proton \( m = 1.67 \times 10^{-27} \, \text{kg} \)
- Charge of a proton \( e = 1.6 \times 10^{-19} \, \text{C} \)

Now substituting these values into the formula:
\[ \lambda = \frac{6.63 \times 10^{-34}}{\sqrt{2 \times 1.67 \times 10^{-27} \times 1.6 \times 10^{-19} \times V}} \]

### Step 3: Simplify the Expression
Calculate the denominator:
\[ 2 \times 1.67 \times 10^{-27} \times 1.6 \times 10^{-19} = 5.344 \times 10^{-46} \]
Thus, the expression for λ becomes:
\[ \lambda = \frac{6.63 \times 10^{-34}}{\sqrt{5.344 \times 10^{-46} \times V}} \]

### Step 4: Further Simplification
The square root can be simplified:
\[ \lambda = \frac{6.63 \times 10^{-34}}{\sqrt{5.344} \times 10^{-23} \sqrt{V}} = \frac{6.63 \times 10^{-34}}{2.31 \times 10^{-23} \sqrt{V}} \]
Calculating the numerical coefficient:
\[ \lambda \approx \frac{2.87 \times 10^{-11}}{\sqrt{V}} \text{ meters} \]

### Step 5: Convert to Angstroms
Since \( 1 \text{ meter} = 10^{10} \text{ angstroms} \):
\[ \lambda \approx \frac{0.286 \times 10^{-10}}{\sqrt{V}} \text{ angstroms} \]

### Final Result
Thus, the final expression for the de Broglie wavelength of a proton accelerated through V volts is:
\[ \lambda = \frac{0.286}{\sqrt{V}} \text{ angstroms} \]