An electron of mass `m_e` and a proton of mass `m_p` are accelerated through the same potential difference. The ratio of the de Broglie wavelength associated with an electron to that associated with proton is

Options

A. 1
B. `(m_(p))/(m_(e))`
C. `(m_(e))/(m_(p))`
D. `sqrt((m_(p))/(m_(e))`

Explanation

To solve the problem of finding the ratio of the de Broglie wavelength associated with an electron to that associated with a proton when both are accelerated through the same potential difference, we can 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. The momentum \( p \) can also be expressed in terms of mass (\( m \)) and velocity (\( v \)):
\[ p = mv \]
Thus, the de Broglie wavelength can be rewritten as:
\[ \lambda = \frac{h}{mv} \]

### Step 2: Calculate the kinetic energy of the particles
When a charged particle is accelerated through a potential difference \( V \), its kinetic energy (\( KE \)) is given by:
\[ KE = qV \]
where \( q \) is the charge of the particle. For an electron and a proton, the charge \( q \) is the elementary charge \( e \).

### Step 3: Relate kinetic energy to velocity
The kinetic energy can also be expressed in terms of mass and velocity:
\[ KE = \frac{1}{2} mv^2 \]
Setting the two expressions for kinetic energy equal gives:
\[ qV = \frac{1}{2} mv^2 \]
From this, we can solve for \( v \):
\[ v = \sqrt{\frac{2qV}{m}} \]

### Step 4: Substitute velocity into the de Broglie wavelength formula
Now, we can substitute the expression for \( v \) back into the de Broglie wavelength formula:
\[ \lambda = \frac{h}{m \sqrt{\frac{2qV}{m}}} = \frac{h}{\sqrt{2mqV}} \]

### Step 5: Calculate the de Broglie wavelengths for the electron and proton
For the electron:
\[ \lambda_e = \frac{h}{\sqrt{2m_e e V}} \]
For the proton:
\[ \lambda_p = \frac{h}{\sqrt{2m_p e V}} \]

### Step 6: Find the ratio of the de Broglie wavelengths
Now, we can find the ratio of the de Broglie wavelengths of the electron to that of the proton:
\[ \frac{\lambda_e}{\lambda_p} = \frac{\frac{h}{\sqrt{2m_e e V}}}{\frac{h}{\sqrt{2m_p e V}}} = \frac{\sqrt{2m_p e V}}{\sqrt{2m_e e V}} = \sqrt{\frac{m_p}{m_e}} \]

### Final Result
Thus, the ratio of the de Broglie wavelength associated with the electron to that associated with the proton is:
\[ \frac{\lambda_e}{\lambda_p} = \sqrt{\frac{m_p}{m_e}} \]