Question

The ratio of de-Broglie wavelengths for electrons accelerated through \(200\,V\) and \(50\,V\) is:

• A. \(1:2\)
• B. \(2:1\)
• C. \(3:10\)
• D. \(10:3\)

Hint:

Higher voltage \(\Rightarrow\) smaller wavelength.

Answer 1

The Correct Option is A

To find the ratio of the de-Broglie wavelengths for electrons accelerated through two different voltages, we need to use the de-Broglie wavelength formula:

\(\lambda = \frac{h}{\sqrt{2meV}}\)

where:

• \(\lambda\) is the de-Broglie wavelength,
• \(h\) is Planck's constant,
• \(m\) is the mass of the electron,
• \(e\) is the charge of the electron,
• \(V\) is the accelerating voltage.

The relationship shows that the wavelength \(\lambda\) is inversely proportional to the square root of the voltage \(V\):

\(\lambda \propto \frac{1}{\sqrt{V}}\)

Given two voltages \(200\,V\) and \(50\,V\), the wavelengths can be related as follows:

\(\frac{\lambda_1}{\lambda_2} = \sqrt{\frac{V_2}{V_1}}\)

Substituting the given voltages:

\(\frac{\lambda_1}{\lambda_2} = \sqrt{\frac{50}{200}}\)

\(\frac{\lambda_1}{\lambda_2} = \sqrt{\frac{1}{4}}\)

\(\frac{\lambda_1}{\lambda_2} = \frac{1}{2}\)

Thus, the correct ratio of de-Broglie wavelengths is \(1:2\).

The correct answer is: \(1:2\)