Proton and electron are moving along circular path with same speed. Find out ratioof debroglie wavelength that is `(lambda_e)/(lambda_p) If `m_p =1836 m_e`

Options

A. 1836
B. 1837
C. `1/1836`
D. `1/1837`

Answer

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

### Step 1: Understand the formula for de Broglie wavelength
The de Broglie wavelength (\( \lambda \)) of a particle 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 expressions for the de Broglie wavelengths of the electron and proton
For the electron:
\[ \lambda_e = \frac{h}{m_e v} \]
For the proton:
\[ \lambda_p = \frac{h}{m_p v} \]

### Step 3: Set up the ratio of the de Broglie wavelengths
We want to find the ratio \( \frac{\lambda_e}{\lambda_p} \):
\[ \frac{\lambda_e}{\lambda_p} = \frac{\frac{h}{m_e v}}{\frac{h}{m_p v}} \]

### Step 4: Simplify the ratio
In the above expression, \( h \) and \( v \) cancel out:
\[ \frac{\lambda_e}{\lambda_p} = \frac{m_p}{m_e} \]

### Step 5: Substitute the given mass relationship
We are given that the mass of the proton \( m_p \) is 1836 times the mass of the electron \( m_e \):
\[ m_p = 1836 m_e \]
Now substituting this into our ratio:
\[ \frac{\lambda_e}{\lambda_p} = \frac{1836 m_e}{m_e} \]

### Step 6: Simplify the ratio further
The \( m_e \) terms cancel out:
\[ \frac{\lambda_e}{\lambda_p} = 1836 \]

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