Problem 50 - Master Resource Book in JEE Main Physics

Question

An electron (mass \(m\) ) with an initial velocity \(\mathbf{v}=v_{0} \hat{\mathbf{i}}\left(\mathbf{v}_{0}>0\right)\) is in an electric field \(\mathbf{E}=E_{0} \hat{\mathbf{i}}\left(E_{0}=\right.\) constant \(\left.>0\right)\) field. It's de-Broglie wavelength at time \(t\) is given by (a) \(\frac{\lambda_{0}}{\left(1+\frac{e E_{0}}{m} \frac{t}{v_{0}}\right)}\) (b) \(\lambda_{0}\left(1+\frac{e E_{0} t}{m v_{0}}\right)\) (c) \(\lambda_{0}\) (d) \(\lambda_{0} t\)

Step-by-Step Solution

Answer

Option (a) is correct: \( \frac{\lambda_{0}}{1+\frac{e E_{0}}{m} \frac{t}{v_{0}}} \).

Step 1: Understand the de Broglie Wavelength Formula

The de Broglie wavelength for a particle is given by the formula \( \lambda = \frac{h}{p} \), where \( h \) is Planck's constant, and \( p \) is the momentum of the particle. For a particle such as an electron, \( p = mv \) where \( m \) is the mass and \( v \) is the velocity of the electron.

Step 2: Determine Initial Wavelength

The initial de Broglie wavelength is expressed when the electron is moving with its initial velocity \( v_0 \). So, the initial wavelength \( \lambda_0 = \frac{h}{mv_0} \).

Step 3: Find the Effect of the Electric Field on Velocity

Under the influence of an electric field \( \mathbf{E} = E_0 \hat{\mathbf{i}} \), the electron experiences an acceleration \( a \) given by \( a = \frac{eE_0}{m} \), where \( e \) is the charge of the electron.

Step 4: Express Velocity as a Function of Time

The velocity of the electron after time \( t \) in the electric field is given by \( v = v_0 + \frac{eE_0}{m}t \). This formula accounts for the initial velocity \( v_0 \) and the increase in velocity due to the electric field.

Step 5: Calculate De Broglie Wavelength at Time t

The de Broglie wavelength at time \( t \) can be calculated using the updated velocity: \[ \lambda(t) = \frac{h}{m(v_0 + \frac{eE_0}{m}t)}. \]Simplifying:\[ \lambda(t) = \frac{h}{mv_0 (1 + \frac{eE_0}{mv_0} t)}. \]This becomes \[ \lambda(t) = \frac{\lambda_0}{1 + \frac{eE_0}{mv_0} t} \], which matches option (a).

Key Concepts

Momentum

When you hear the word "momentum," you might think of a moving car or a rolling ball. In physics, momentum is a measure of how much "motion" is packed into an object and is given by the formula \( p = mv \). Here, \( m \) represents the mass of the object, and \( v \) is its velocity. This concept is crucial because it helps us calculate the de Broglie wavelength, which links a particle's momentum to its wave properties. For an electron with mass \( m \) and velocity \( v \), momentum becomes \( p = mv \). This relationship is essential when substituting into the de Broglie wavelength equation, \( \lambda = \frac{h}{p} \). The greater the momentum, the shorter the wavelength, meaning fast-moving particles like electrons have a shorter de Broglie wavelength.
Understanding momentum aids us in grasping other concepts like electron velocity changes under an electric field.

Electric Field

An electric field is a force field that surrounds electric charges, affecting other charges within its reach. Imagine it like an invisible force that pushes or pulls charged particles. This field is represented as \( \mathbf{E} \) and exerts a force \( F = eE \) on a charge \( e \). In our scenario, the electric field \( \mathbf{E} = E_0 \hat{\mathbf{i}} \) is constant and helps accelerate the electron.

• The electron, being negatively charged, moves under the influence of this field.
• The field causes the electron's velocity to change with time.

As the electric field applies its force, it is why the electron's velocity does not remain constant. Instead, it accelerates according to the formula \( a = \frac{eE_0}{m} \). Understanding the role of the electric field is key to predicting how an electron's motion and properties, such as de Broglie wavelength, evolve over time.

Electron Velocity

Velocity refers to the speed of an electron in a specific direction. An electron initially traveling in a direction indicated by \( \hat{\mathbf{i}} \) with a velocity \( v_0 \) will experience changes in speed when exposed to an electric field. This change is expressed by the formula \( v = v_0 + \frac{eE_0}{m}t \).

• \( v_0 \) is the initial speed before any force is applied.
• \( \frac{eE_0}{m}t \) accounts for the acceleration influence from the electric field.

As time progresses, this change in velocity affects the momentum and, subsequently, the de Broglie wavelength. Being able to compute the velocity at any given point in time is crucial for understanding how forces like electric fields alter particle motion in real-time. By determining the updated velocity, we can deduce the evolving nature of other physical properties.

Planck's Constant

Planck's constant \( h \) is one of the fundamental constants in the field of quantum mechanics. It bridges the gap between the wave and particle nature of matter. At the core, it allows us to calculate the de Broglie wavelength through the equation \( \lambda = \frac{h}{p} \), where \( p \) is momentum.

• Its value is approximately \( 6.626 \, \times 10^{-34} \, \text{Js} \).
• It highlights the quantized nature of energy and action, aspects the classical physics doesn't account for.

It is this constant that plays a pivotal role in the quantum description of particles. Whether you are calculating wavelengths of photons or considering the wave-like behavior of electrons, Planck's constant is indispensable. Its small magnitude signifies that quantum mechanical effects are significant only on very small scales, like atomic or subatomic particles.