What is the ratio of the de Broglie wavelength for electrons accelerated through 50 volts and 200 volts?

Options

A. `3 : 10`
B. `10 : 3`
C. `1:2`
D. `2 : 1`

Answer

To find the ratio of the de Broglie wavelength for electrons accelerated through 50 volts and 200 volts, 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:
\[ \lambda = \frac{h}{mv} \]
where \( h \) is Planck's constant, \( m \) is the mass of the electron, and \( v \) is its velocity.

### Step 2: Relate the kinetic energy to the voltage
When an electron is accelerated through a potential difference \( V \), it gains kinetic energy equal to the work done on it by the electric field:
\[ KE = eV \]
where \( e \) is the charge of the electron. This kinetic energy can also be expressed as:
\[ KE = \frac{1}{2} mv^2 \]
Setting these two expressions for kinetic energy equal gives:
\[ eV = \frac{1}{2} mv^2 \]

### Step 3: Solve for velocity \( v \)
Rearranging the equation to solve for \( v \):
\[ v^2 = \frac{2eV}{m} \]
Taking the square root:
\[ v = \sqrt{\frac{2eV}{m}} \]

### Step 4: Substitute \( v \) into the de Broglie wavelength formula
Substituting the expression for \( v \) back into the de Broglie wavelength formula:
\[ \lambda = \frac{h}{m\sqrt{\frac{2eV}{m}}} = \frac{h}{\sqrt{2emV}} \]

### Step 5: Analyze the ratio of wavelengths for different voltages
Let \( \lambda_1 \) be the wavelength for \( V = 50 \) volts and \( \lambda_2 \) be the wavelength for \( V = 200 \) volts:
\[ \lambda_1 = \frac{h}{\sqrt{2em \cdot 50}} \quad \text{and} \quad \lambda_2 = \frac{h}{\sqrt{2em \cdot 200}} \]

### Step 6: Find the ratio \( \frac{\lambda_1}{\lambda_2} \)
Now, we can find the ratio of the two wavelengths:
\[ \frac{\lambda_1}{\lambda_2} = \frac{\frac{h}{\sqrt{2em \cdot 50}}}{\frac{h}{\sqrt{2em \cdot 200}}} = \frac{\sqrt{200}}{\sqrt{50}} = \sqrt{\frac{200}{50}} = \sqrt{4} = 2 \]

### Step 7: Conclusion
Thus, the ratio of the de Broglie wavelengths for electrons accelerated through 50 volts and 200 volts is:
\[ \frac{\lambda_1}{\lambda_2} = 2:1 \]