The ratio of the de Broglie wavelength of a proton and alpha particles will be `1:2` if their

Options

A. velocity are in the ratio 1:8
B. velocity are in the ratio 8:1
C. kinetic energy are in the ratio 1:64
D. kinetic energy are in the ratio 1: 256

Answer

To solve the problem of finding the ratio of the de Broglie wavelengths of a proton and an alpha particle, we will follow these steps:

### Step 1: Understand the de Broglie wavelength formula
The de Broglie wavelength (λ) 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: Express momentum in terms of mass and velocity
The momentum \(p\) can be expressed as:
\[ p = mv \]
where \(m\) is the mass of the particle and \(v\) is its velocity.

### Step 3: Write the ratio of the de Broglie wavelengths
For a proton (subscript \(p\)) and an alpha particle (subscript \(α\)), the ratio of their wavelengths can be expressed as:
\[ \frac{\lambda_p}{\lambda_α} = \frac{h/p_p}{h/p_α} = \frac{p_α}{p_p} = \frac{m_α v_α}{m_p v_p} \]

### Step 4: Substitute known values
We know that the mass of an alpha particle is approximately four times the mass of a proton:
\[ m_α = 4m_p \]
Substituting this into the ratio gives:
\[ \frac{\lambda_p}{\lambda_α} = \frac{4m_p v_α}{m_p v_p} = \frac{4 v_α}{v_p} \]

### Step 5: Set the ratio equal to 1/2
According to the problem, we want the ratio of the wavelengths to be:
\[ \frac{\lambda_p}{\lambda_α} = \frac{1}{2} \]
Setting the two expressions equal gives:
\[ \frac{4 v_α}{v_p} = \frac{1}{2} \]

### Step 6: Solve for the velocity ratio
Cross-multiplying yields:
\[ 4 v_α = \frac{1}{2} v_p \]
This can be rearranged to find the ratio of the velocities:
\[ \frac{v_p}{v_α} = 8 \]
Thus, the ratio of the velocities of the proton to the alpha particle is:
\[ v_p : v_α = 8 : 1 \]

### Conclusion
The correct option is that the velocities are in the ratio of \(8:1\).

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