Problem 52 - Master Resource Book in JEE Main Physics

Question

An electron (mass \(m\) ) with an initial velocity \(\mathbf{v}=v_{0} \hat{\mathbf{i}}\) is in an electric \(\mathbf{E}=E_{0} \hat{\mathbf{j}} .\) If \(\lambda_{0}=h / m v_{0}\), it's de-Broglie wavelength at time \(t\) is given by \(\quad\) (a) \(\lambda_{0}\) (b) \(\lambda_{0} \sqrt{1+\frac{e^{2} E^{2} t^{2}}{m^{2} v_{0}^{2}}}\) (c) \(\frac{\lambda_{\mathrm{D}}}{\sqrt{1+\frac{\mathrm{e}^{2} E_{0^{2}}^{2}}{m^{2} v_{0}^{2}}}}\) (d) \(\frac{\lambda_{0}}{\left(1+\frac{e^{2} E_{0}^{2} t^{2}}{m^{2} v_{0}^{2}}\right)}\)

Step-by-Step Solution

Answer

Option (c) is correct: \( \frac{\lambda_0}{\sqrt{1 + \frac{e^2E_0^2t^2}{m^2v_0^2}}} \).

Step 1: Understand the Scenario

An electron is moving with an initial velocity \( \mathbf{v}=v_0 \hat{\mathbf{i}} \) in an electric field \( \mathbf{E}=E_0 \hat{\mathbf{j}} \). We need to calculate its de-Broglie wavelength, \( \lambda(t) \), at time \( t \).

Step 2: Analyze Force and Motion

The electric field \( \mathbf{E} = E_0 \hat{\mathbf{j}} \) exerts a force \( \mathbf{F} = eE_0 \hat{\mathbf{j}} \) on the electron. Since force equals mass times acceleration \( \mathbf{F} = m \mathbf{a} \), the acceleration \( \mathbf{a} = \frac{eE_0}{m} \hat{\mathbf{j}} \). This affects the velocity in the \( \hat{j} \) direction but not in the \( \hat{i} \) direction.

Step 3: Compute Velocity over Time

The initial velocity is \( v_0 \hat{\mathbf{i}} \). Over time, due to the force, the velocity in the \( \hat{\mathbf{j}} \) direction increases as \( v_j = \frac{eE_0}{m}t \). Thus, the velocity at any time \( t \) is \( \mathbf{v}(t) = v_0 \hat{\mathbf{i}} + \frac{eE_0}{m}t \hat{\mathbf{j}} \).

Step 4: Calculate Speed at Time t

The speed \( v(t) \) is given by the magnitude of \( \mathbf{v}(t) \). Compute it as: \[ v(t) = \sqrt{v_0^2 + \left( \frac{eE_0}{m}t \right)^2}\] This gives the resultant speed of the electron at time \( t \).

Step 5: Determine de-Broglie Wavelength

The de-Broglie wavelength \( \lambda(t) \) is given by \( \lambda(t) = \frac{h}{mv(t)} \). Substituting the expression for speed, we find:\[ \lambda(t) = \frac{h}{m \sqrt{v_0^2 + \left( \frac{eE_0}{m}t \right)^2}}\]Factor \( v_0^2 \) out of the square root to simplify:\[ \lambda(t) = \frac{h}{m v_0} \frac{1}{\sqrt{1 + \frac{e^2E_0^2t^2}{m^2v_0^2}}}\]Substitute \( \lambda_0 = \frac{h}{mv_0} \):\[ \lambda(t) = \frac{\lambda_0}{\sqrt{1 + \frac{e^2E_0^2t^2}{m^2v_0^2}}}\]

Step 6: Match with Given Options

The expression matches option (c): \( \frac{\lambda_0}{\sqrt{1 + \frac{e^2E_0^2t^2}{m^2v_0^2}}} \). This confirms the correct de-Broglie wavelength formula at time \( t \).

Key Concepts

Electron Motion in Electric Field

When an electron moves through an electric field, its motion is influenced by the electric forces acting upon it. In this scenario, the electron initially travels with velocity \( \mathbf{v} = v_0 \hat{\mathbf{i}} \), parallel to the x-axis. Meanwhile, the electric field \( \mathbf{E} = E_0 \hat{\mathbf{j}} \) is directed along the y-axis, perpendicular to the initial motion of the electron.

An electric field consists of lines of force exerting a push or pull on charged particles, like electrons. In our case, the field applies a force in the y-direction. The electron experiences this as a change in velocity along the y-axis, resulting in a change of motion over time.

• The initial motion does not change along the x-axis.
• The electric field adds a component of velocity along the y-axis.

In essence, the electron's path becomes a combination of its original direction and the influence of the electric field.

Velocity Calculation

Velocity is the rate of change of an object's position with respect to time. In this exercise, we start with the electron's initial velocity of \( v_0 \hat{\mathbf{i}} \), directed along the x-axis. As the electric field exerts a force\( \mathbf{F} = eE_0 \hat{\mathbf{j}} \), it leads to an additional velocity component along the y-axis.

The electron's velocity \( \mathbf{v}(t) \) at any time \( t \) becomes a vector with components in both the x and y directions:

• \( v_x(t) = v_0 \), remaining constant in the x-direction.
• \( v_y(t) = \frac{eE_0}{m}t \), developing in the y-direction over time.

Thus, the overall velocity of the electron is represented as \( \mathbf{v}(t) = v_0 \hat{\mathbf{i}} + \frac{eE_0}{m} t \hat{\mathbf{j}} \). The speed of the electron, therefore, integrates both these components into a single magnitude according to the formula:\[ v(t) = \sqrt{v_0^2 + \left(\frac{eE_0}{m} t \right)^2} \]

Acceleration Due to Electric Field

Acceleration is a crucial concept in understanding how forces affect motion. It represents the rate of change of velocity over time. In the context of our exercise, the electric field influences the electron, providing an acceleration along the y-axis.

Given that the force exerted by the electric field is \( \mathbf{F} = eE_0 \hat{\mathbf{j}} \), we use Newton's second law of motion which states \( \mathbf{F} = m \mathbf{a} \). Here, \( e \) is the charge of the electron, \( E_0 \) is the magnitude of the electric field, and \( m \) is the mass of the electron.

• The acceleration \( \mathbf{a} = \frac{eE_0}{m} \hat{\mathbf{j}} \) occurs only in the y-direction.
• There is no acceleration in the x-direction since the electric field is perpendicular to it.

Ultimately, this acceleration impacts how quickly the electron's y-velocity increases with time, as reflected in the earlier velocity computation.

Force and Motion in Physics

The relationship between force and motion is a fundamental element of physics, especially when analysing particle dynamics in electric fields. Newton’s laws of motion describe how forces influence the movement of objects. In this problem, the focus centers on how the electric force impacts an electron's motion, altering its velocity and, consequently, its de-Broglie wavelength over time.

The de-Broglie wavelength \( \lambda \) is intimately linked to a particle’s momentum, given by \( \lambda = \frac{h}{p} \), where \( h \) is Planck’s constant and \( p \) the momentum. As velocity changes, so does momentum; hence the wavelength is influenced.

• The electron begins with a momentum component \( mv_0 \).
• Over time, additional momentum develops due to changes from \( v_y(t) \).

The resultant momentum vector and its magnitude in time translate into the altered de-Broglie wavelength \( \lambda(t) = \frac{h}{mv(t)} \). As shown in the calculations, this change fits with option (c) from the problem statement, demonstrating how forces impact motion and other related physical properties.