Question

An electron with mass $ m $ with an initial velocity $ (t = 0) \, \vec{v} = \vec{v_0} \, (v_0>0) $ enters a magnetic field $ \vec{B} = B \hat{j} $. If the initial de-Broglie wavelength at $ t = 0 $ is $ \lambda_0 $, then its value after time $ t $ would be:

• A. \( \frac{\lambda_0}{\sqrt{1 - \frac{e^2 B^2 t^2}{m^2}}} \)
• B. \( \frac{\lambda_0}{\sqrt{1 + \frac{e^2 B^2 t^2}{m^2}}} \)
• C. \( \lambda_0 \sqrt{1 + \frac{e^2 B^2 t^2}{m^2}} \)
• D. \( \lambda_0 \)

Hint:

When dealing with an electron in a magnetic field, remember that the magnetic force only changes the direction of motion, not the speed. Thus, the de-Broglie wavelength remains unaffected.

Answer 1

The Correct Option is D

Electron's de-Broglie Wavelength in a Magnetic Field

Given: An electron (mass \(m\), charge \(-e\)) with initial velocity \(\vec{v} = \vec{v_0}\) (where \(v_0 > 0\)) enters a uniform magnetic field \(\vec{B} = B\hat{j}\). The initial de-Broglie wavelength is \(\lambda_0\). Determine its wavelength after time \(t\).

Concept Used:

The magnetic field exerts a Lorentz force on the moving electron:

\[ \vec{F} = -e(\vec{v} \times \vec{B}). \]

This force alters the velocity's direction but not its magnitude, as it is perpendicular to \(\vec{v}\). Consequently, the speed remains constant.

The de-Broglie wavelength is defined as:

\[ \lambda = \frac{h}{p} = \frac{h}{mv}. \]

Since \(v\) (velocity magnitude) is invariant, the de-Broglie wavelength also remains constant over time.

Step-by-Step Solution:

Step 1: Formulate the equation of motion:

\[ m \frac{d\vec{v}}{dt} = -e(\vec{v} \times \vec{B}). \]

The magnetic field induces circular motion at an angular frequency (cyclotron frequency):

\[ \omega = \frac{eB}{m}. \]

Step 2: The velocity magnitude \(v_0\) is constant; only its direction changes. Thus, the momentum magnitude \(p = mv_0\) is conserved.

Step 3: The de-Broglie wavelength at time \(t\) is therefore:

\[ \lambda = \frac{h}{mv_0} = \lambda_0. \]

Final Computation & Result:

\[ \boxed{\lambda(t) = \lambda_0.} \]

The electron's de-Broglie wavelength remains constant over time.