Question

An electron is moving with initial velocity \(\overset{⃗}{v} = V_0\hat{j}\) in a magnetic field \(\overset{⃗}{B} = B_0\hat{i}\). Its de-Broglie wavelength will

• A. decrease with time.
• B. increase and decrease periodically.
• C. remains constant.
• D. increase with time.

Hint:

A magnetic force does no work, so the speed and hence the wavelength stay the same.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
The de Broglie wavelength is \(\lambda=\dfrac hp=\dfrac h{mv}\). It changes only if the speed changes.

Step 2: Effect of the magnetic field
The magnetic force on a moving charge is \(\vec F=q\vec v\times\vec B\). It is always perpendicular to the velocity, so it does no work. The kinetic energy and the speed stay constant. Only the direction of motion changes, and the electron moves along a circle or a helix.

Step 3: Result
Since \(|v|\) is constant, \(p\) is constant and \(\lambda\) is constant.

Step 4: Check the options
Decrease, increase, or periodic change would all need a changing speed. So the wavelength remains constant, option (C).

Final Answer:
A magnetic force does no work, so the speed and de Broglie wavelength stay constant, option (C). \[ \boxed{\text{Remains constant}} \]