Problem 49 - Master Resource Book in JEE Main Physics

Question

An electron is moving with an initial velocity \(\mathbf{v}=v_{0} \hat{\mathbf{i}}\) and is in a magnetic field \(\mathbf{B}=B_{0} \hat{\mathbf{j}}\).Then it's de-Broglie wavelength \(\quad\) (a) remains constant (b) increases with time (c) decreases with time (d) increases and decreases periodically

Step-by-Step Solution

Answer

The de-Broglie wavelength remains constant.

Step 1: Understanding the Problem

We need to find out how the de-Broglie wavelength of an electron changes over time when it moves in a magnetic field. We are given that the electron initially has a velocity of \( \vec{v} = v_0 \hat{i} \), and the magnetic field is \( \vec{B} = B_0 \hat{j} \).

Step 2: Determine Electron Motion Under Magnetic Field

When an electron moves with velocity \( v \) perpendicular to a magnetic field \( B \), it experiences a force that causes it to move in a circular path. Since \( \vec{v} \) is along \( \hat{i} \) and \( \vec{B} \) is along \( \hat{j} \), they are perpendicular. The electron will move in a circular path with radius \( r = \frac{mv}{qB} \), where \( m \) is the mass and \( q \) is the charge of the electron.

Step 3: Applying de-Broglie Wavelength Formula

The de-Broglie wavelength \( \lambda \) of a particle is given by \( \lambda = \frac{h}{p} \), where \( h \) is Planck's constant and \( p \) is the momentum of the particle. For the electron, \( p = mv \).

Step 4: Evaluate Impact on de-Broglie Wavelength

Since the magnetic field only changes the direction of the velocity but not its magnitude, the speed \( v \) of the electron remains constant. Therefore, the momentum \( p = mv \) remains constant, which means the de-Broglie wavelength \( \lambda = \frac{h}{mv} \) remains constant over time.

Step 5: Conclusion

The de-Broglie wavelength of the electron remains constant since neither the mass \( m \) nor the velocity magnitude \( v \) changes in a magnetic field. Hence, the correct option is (a) remains constant.

Key Concepts

Electron Motion

Understanding the motion of an electron in a magnetic field requires exploring how its path is influenced by the forces acting on it. Electrons have charge and mass, and their movement is dictated by electromagnetic forces. When an electron with an initial velocity, v, interacts with a magnetic field, it begins to move in a circular path.

• This occurs because the force exerted by the magnetic field is perpendicular to the electron's velocity.
• This perpendicular force changes the direction of the electron's motion, causing it to turn and eventually form a circular trajectory.
• Importantly, as this force only alters the direction, not the speed or magnitude of velocity, the speed remains constant.

The concept of electron motion in magnetic fields is crucial for understanding how particle accelerators and other electronic devices function in practical scientific and technological applications.

Magnetic Field

Magnetic fields are vector fields that exert forces on moving charged particles like electrons. A stationary electron does not experience a force due to a magnetic field, but once it moves, the interaction becomes significant.

• The magnitude of force experienced by an electron in a magnetic field is given by the Lorentz force equation: \( F = qvB \sin(\theta) \), where \( \theta \) is the angle between velocity \( v \) and magnetic field \( B \).
• In our exercise scenario, since the electron velocity vector and the magnetic field are perpendicular, \( \sin(90^\circ) = 1 \), thus maximizing the effect of the force.
• This effective force leads to circular motion, as it is continually perpendicular to the velocity, never increasing or decreasing the electron's speed.

Understanding magnetic field interactions is key for fields like electromagnetism, astrophysics, and the development of motors and generators.

Circular Motion

When a charged particle like an electron moves through a magnetic field perpendicular to its path, it experiences what is known as circular motion. This is because the magnetic force acts as a centripetal force, continually changing the direction of the electron's velocity and keeping its path circular.

• The radius of this circular path can be determined by the equation \( r = \frac{mv}{qB} \), where \( m \) is mass, \( v \) is velocity, \( q \) is charge, and \( B \) is the magnetic field intensity.
• This formula shows that the path radius depends directly on the velocity and is inversely proportional to the magnetic field strength and electron charge.
• Because the velocity's magnitude remains constant, the path remains a consistent circle until magnetic or other external forces change.

Circular motion underpins many technologies, including cyclotrons and synchrotrons, which are essential in particle physics research.

Momentum Conservation

In the context of the exercise and the de-Broglie wavelength, momentum conservation plays a pivotal role. Momentum, defined as the product of an object's mass and its velocity (\( p = mv \)), is a crucial factor when examining particle motion.

• In a magnetic field, an electron's speed remains unchanged due to perpendicular force application, thus its momentum does not vary.
• The constant momentum ensures that the de-Broglie wavelength \( \lambda = \frac{h}{p} \), where \( h \) is Planck's constant, also remains consistent over time as both mass and velocity are constant.
• This constancy means there is no change in the wavelength, highlighting an important application of momentum conservation in wave-particle duality.

Momentum conservation is fundamental across physics, especially in quantum mechanics where it provides insights into particle behavior linked with de-Broglie wavelengths.