Problem 33 - Master Resource Book in JEE Main Physics

Question

An electron and a proton enter a magnetic field perpendicularly. Both have same kinetic energy. Which of the following is true ? (a) Trajectory of electron is less curved (b) Trajectory of proton is less curved (c) Both trajectories are equally curved (d) Both more on straight line path

Step-by-Step Solution

Answer

(b) Trajectory of proton is less curved.

Step 1: Understanding the Forces

When a charged particle enters a magnetic field perpendicularly, it experiences a force due to the Lorentz force law, which causes it to move in a circular path. The force is given by the equation: \( F = qvB \) where \( q \) is the charge, \( v \) is the velocity, and \( B \) is the magnetic field strength. The force acts as the centripetal force.

Step 2: Relating Force to Radius

The centripetal force needed for circular motion is given by \( F = \frac{mv^2}{r} \), where \( m \) is the mass, \( v \) is the velocity, and \( r \) is the radius of curvature of the trajectory. By equating the magnetic force to the centripetal force, we have: \( qvB = \frac{mv^2}{r} \). Thus, \( r = \frac{mv}{qB} \).

Step 3: Using Kinetic Energy

Since both the electron and proton have the same kinetic energy, \( KE = \frac{1}{2}mv^2 \). Therefore, \( v = \sqrt{\frac{2KE}{m}} \). Substituting this into the radius equation, \( r = \frac{m \sqrt{\frac{2KE}{m}}}{qB} = \frac{\sqrt{2mKE}}{qB} \).

Step 4: Comparing Electron and Proton

The electron and proton both have the same kinetic energy. However, the proton is much more massive than the electron, \( m_p \gg m_e \). Plugging this into the expression for \( r \), we see that \( r_p = \frac{\sqrt{2m_pKE}}{q_pB} \) and \( r_e = \frac{\sqrt{2m_eKE}}{q_eB} \). Because \( m_p \gg m_e \), \( r_p \gg r_e \), meaning the proton's trajectory is less curved.

Step 5: Conclusion

Given the relation between mass and radius derived, the trajectory of the proton is less curved compared to the electron because of its much larger mass at the same kinetic energy.

Key Concepts

Lorentz Force

When a charged particle, such as an electron or proton, moves through a magnetic field, it experiences a specific force known as the Lorentz Force. This force is essential in electromagnetism and describes the impact of magnetic fields on moving charged particles.
The Lorentz Force is the force that acts on a charged particle in a magnetic field and is directly responsible for the particle's trajectory change. The equation for the Lorentz Force is:

• \( F = qvB \)

where:

• \( F \) is the force acting on the particle,
• \( q \) is the charge of the particle,
• \( v \) is the velocity of the particle,
• \( B \) is the magnetic field strength.

This force is perpendicular to both the velocity of the particle and the magnetic field, which causes the particle to move in a circular path rather than a straight line. The Lorentz Force is crucial for understanding how particles move in fields and underpins much of classical electromagnetism.

Centripetal Motion

Centripetal Motion refers to the curved path an object follows as it moves around a circle. For a charged particle in a magnetic field, this occurs because the Lorentz Force acts as a centripetal force.
To keep a particle moving in a circle, it must continuously change direction, which requires a centripetal force. This force is what keeps the particle rotating around a central point. It's quantified by:

• \( F = \frac{mv^2}{r} \)

where:

• \( F \) is the centripetal force,
• \( m \) is the mass of the particle,
• \( v \) is the velocity,
• \( r \) is the radius of curvature.

In cases where the Lorentz Force is the acting force, it meets the requirement for this centripetal force. By equating these forces, we can derive the radius of the circle the particle travels in. Thus, the radius is given by:

• \( r = \frac{mv}{qB} \)

This indicates how the momentum of charged particles determines their path in a magnetic field, highlighting the interplay between mass, charge, and field strength.

Kinetic Energy

Kinetic Energy is the energy a particle possesses as it moves due to its mass and velocity. In this context, it explains why particles like electrons and protons behave differently in a magnetic field.
The Kinetic Energy (KE) of a particle is given by:

• \( KE = \frac{1}{2}mv^2 \)

When electrons and protons have the same kinetic energy, they maintain different velocities due to their different masses.
By substituting into the radius formula:

• \( v = \sqrt{\frac{2KE}{m}} \)

we can see how mass impacts motion as substitutes into the radius equation:

• \( r = \frac{\sqrt{2mKE}}{qB} \)

Because the proton has much more mass than an electron, its higher mass affects its velocity and trajectory. Consequently, at equal kinetic energy, the proton moves in a less curved path. This result showcases the relationship between particle mass and trajectory in magnetic fields, especially when kinetic energy is kept constant.