An electron of mass `m_(e )` and a proton of mass `m_(p) = 1836 m_(e )` are moving with the same speed. The ratio of their de Broglie wavelength `(lambda_("electron"))/(lambda_("proton"))` will be :

Options

A. 1836
B. 1
C. 918
D. `(1)/(1836)`

Answer

To find the ratio of the de Broglie wavelengths of an electron and a proton moving with the same speed, we can follow these steps:

### Step 1: Write the formula for de Broglie wavelength
The de Broglie wavelength (\( \lambda \)) is given by the formula:
\[ \lambda = \frac{h}{mv} \]
where \( h \) is Planck's constant, \( m \) is the mass of the particle, and \( v \) is its velocity.

### Step 2: Write the expression for the de Broglie wavelength of the electron
For the electron, the de Broglie wavelength (\( \lambda_e \)) is:
\[ \lambda_e = \frac{h}{m_e v} \]
where \( m_e \) is the mass of the electron and \( v \) is its velocity.

### Step 3: Write the expression for the de Broglie wavelength of the proton
For the proton, the de Broglie wavelength (\( \lambda_p \)) is:
\[ \lambda_p = \frac{h}{m_p v} \]
where \( m_p \) is the mass of the proton.

### Step 4: Find the ratio of the de Broglie wavelengths
Now, we need to find the ratio \( \frac{\lambda_e}{\lambda_p} \):
\[ \frac{\lambda_e}{\lambda_p} = \frac{\frac{h}{m_e v}}{\frac{h}{m_p v}} = \frac{m_p}{m_e} \]
Here, \( h \) and \( v \) cancel out.

### Step 5: Substitute the mass of the proton
Given that the mass of the proton \( m_p = 1836 m_e \), we can substitute this into our ratio:
\[ \frac{\lambda_e}{\lambda_p} = \frac{m_p}{m_e} = \frac{1836 m_e}{m_e} \]

### Step 6: Simplify the expression
The \( m_e \) cancels out:
\[ \frac{\lambda_e}{\lambda_p} = 1836 \]

### Final Answer
Thus, the ratio of the de Broglie wavelengths of the electron to the proton is:
\[ \frac{\lambda_e}{\lambda_p} = 1836 \]

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