Question

An electron (mass m) with an initial velocity
\(\vec{v}=v_0\hat{i}(v_0>0)\)
is moving in an electric field
\(\vec{E}=E_0\hat{i}(E_0>0)\)
where E₀ is constant. If at t = 0 de Broglie wavelength is
\(λ_0=\frac{ℎ}{mv_0}\)
, then its de Broglie wavelength after time t is given by

• A. \(λ_0\)
• B. \(λ_0\left(1+\frac{eE_0t}{mv_0}\right)\)
• C. \(λ_0t\)
• D. \(\frac{λ_0}{\left(1+\frac{eE_0t}{mv_0}\right)}\)

Answer 1

The Correct Option is D

To solve this problem, we need to analyze the motion of an electron in a given electric field and determine how its de Broglie wavelength changes over time.

The initial de Broglie wavelength \(\lambda_0\) is given as:

\(\lambda_0 = \frac{h}{mv_0}\)

where:

• \(h\) is Planck's constant.
• \(m\) is the mass of the electron.
• \(v_0\) is the initial velocity of the electron.

The electron is subjected to a constant electric field \(\vec{E} = E_0 \hat{i}\). The force experienced by the electron due to this electric field is given by:

\(\vec{F} = e \vec{E} = e E_0 \hat{i}\)

where \(e\) is the charge of the electron.

The acceleration \(\vec{a}\) of the electron is:

\(\vec{a} = \frac{\vec{F}}{m} = \frac{eE_0}{m} \hat{i}\)

Starting with an initial velocity \(\vec{v} = v_0 \hat{i}\), the velocity of the electron at time \(t\) can be calculated using the equation of motion:

\(\vec{v} = v_0 \hat{i} + \vec{a} t\)

\(\vec{v} = v_0 \hat{i} + \frac{eE_0 t}{m} \hat{i}\)

\(\vec{v} = \left(v_0 + \frac{eE_0 t}{m}\right) \hat{i}\)

The de Broglie wavelength \(\lambda\) at time \(t\) is given by:

\(\lambda = \frac{h}{m \left(v_0 + \frac{eE_0 t}{m}\right)}\)

Simplifying the expression, we have:

\(\lambda = \frac{h}{m v_0 \left(1 + \frac{eE_0 t}{mv_0}\right)}\)

Substituting the expression for \(\lambda_0\), we get:

\(\lambda = \frac{\lambda_0}{1 + \frac{eE_0 t}{mv_0}}\)

This confirms the correct answer is:

\(\frac{λ_0}{\left(1+\frac{eE_0t}{mv_0}\right)}\)