An electron and a proton have same kinetic energy Ratio of their respective de-Broglie wavelength is about

Options

A. `(12.27)/(0.266)`
B. `(0.101)/(0.266) `
C. `(0.266)/(12.27)`
D. `(0.101)/(12.27)`

Answer

To find the ratio of the de Broglie wavelengths of an electron and a proton that have the same kinetic energy, we will follow these steps:

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

### Step 2: Relate kinetic energy to momentum
The kinetic energy (\( KE \)) of a particle is related to its momentum by the equation:
\[ KE = \frac{p^2}{2m} \]
From this, we can express momentum in terms of kinetic energy:
\[ p = \sqrt{2m \cdot KE} \]

### Step 3: Substitute momentum into the de Broglie wavelength formula
Substituting the expression for momentum into the de Broglie wavelength formula, we get:
\[ \lambda = \frac{h}{\sqrt{2m \cdot KE}} \]

### Step 4: Write the de Broglie wavelengths for the electron and proton
Let \( \lambda_e \) be the wavelength of the electron and \( \lambda_p \) be the wavelength of the proton. Since both particles have the same kinetic energy, we can write:
\[ \lambda_e = \frac{h}{\sqrt{2m_e \cdot KE}} \quad \text{and} \quad \lambda_p = \frac{h}{\sqrt{2m_p \cdot KE}} \]

### Step 5: Find the ratio of the de Broglie wavelengths
Now, we can find the ratio of the de Broglie wavelengths:
\[ \frac{\lambda_e}{\lambda_p} = \frac{\frac{h}{\sqrt{2m_e \cdot KE}}}{\frac{h}{\sqrt{2m_p \cdot KE}}} = \frac{\sqrt{2m_p \cdot KE}}{\sqrt{2m_e \cdot KE}} = \frac{\sqrt{m_p}}{\sqrt{m_e}} = \sqrt{\frac{m_p}{m_e}} \]

### Step 6: Substitute the masses of the electron and proton
The mass of the proton (\( m_p \)) is approximately \( 1.67 \times 10^{-27} \) kg and the mass of the electron (\( m_e \)) is approximately \( 9.1 \times 10^{-31} \) kg. Thus, we can calculate:
\[ \frac{\lambda_e}{\lambda_p} = \sqrt{\frac{1.67 \times 10^{-27}}{9.1 \times 10^{-31}}} \]

### Step 7: Calculate the ratio
Calculating the ratio:
\[ \frac{1.67}{9.1} \approx 0.1835 \]
Taking the square root:
\[ \sqrt{0.1835} \approx 0.428 \]

### Conclusion
Thus, the ratio of the de Broglie wavelengths of the electron to the proton is approximately:
\[ \frac{\lambda_e}{\lambda_p} \approx 0.428 \]