Problem 29 - Master Resource Book in JEE Main Physics

Question

A cubical region of space is filled with some uniform electric and magnetic fields. An electron enters the cube across one of its faces with velocity \(\mathbf{v}\) and a positron enters via opposite face with velocity - v. At this instant, \(\quad\) [NCERT Exemplar] (a) the electric forces on both the particles cause identical accelerations (b) the magnetic forces on both the particles cause equal accelerations (c) both particles gain or loose energy at the same rate (d) the motion of the centre of mass (CM) is determined by \(B\) alone

Step-by-Step Solution

Answer

Options (a) and (c) are correct.

Step 1: Understand the Electric Force

The electric force on a charged particle is given by \( \mathbf{F}_e = q \mathbf{E} \), where \( q \) is the charge and \( \mathbf{E} \) is the electric field. Electrons and positrons have equal magnitude of charge but opposite signs (electron has \( -e \) and positron has \( +e \)). The force will have the same magnitude but opposite direction for the electron and positron, resulting in equal magnitudes of acceleration since their masses are equal.

Step 2: Analyze the Magnetic Force

The magnetic force is given by \( \mathbf{F}_m = q(abla × \mathbf{v}) \), where \( \mathbf{v} \) is the velocity. The magnetic force depends on both the charge and velocity direction. Electrons and positrons entering the fields experience \( \mathbf{F}_m \) in opposite directions (due to the charge sign difference) but have equal magnitudes. Since they have the same mass, they will experience equal acceleration magnitudes but in opposite directions.

Step 3: Look at Energy Changes

The energy change for a charged particle in an electric field is governed by the work done by the electric field, which is path-independent. As both particles have the same mass and charge magnitudes, they will gain or lose energy at the same rate as they pass through the electric field.

Step 4: Center of Mass Motion Analysis

The total force on the system's center of mass (CM) is the vector sum of the electric and magnetic forces on both particles. Since magnetic forces do no work and cancel each other, they do not affect the rate of energy change of the center of mass or its motion. Therefore, only the electric component of the forces changes the velocity of the CM.

Key Concepts

Electric Force

The electric force is a fundamental interaction that occurs due to electric charges. For particles like the electron and positron, the electric force formula is given by \( \mathbf{F}_e = q \mathbf{E} \).
This means that the force is directly proportional to the charge \( q \) and the strength of the electric field \( \mathbf{E} \).

Both electron and positron have equal but opposite charges:

• Electron has a charge of \( -e \)
• Positron has a charge of \( +e \)

Thus, when they enter the space with an electric field, the electron and positron experience forces of equal magnitude but opposite directions. This opposition comes from their opposite charge signs. Since their masses are the same, they undergo identical accelerations. Understanding this intersection of charge and electric field helps us predict their motion when introduced to an electric field.

Magnetic Force

Magnetic forces act on moving charges like electrons and positrons due to their velocity in a magnetic field. The force acting on a charged particle moving in a magnetic field is expressed by \( \mathbf{F}_m = q(\mathbf{v} \times \mathbf{B}) \), where \( \mathbf{v} \) is the velocity and \( \mathbf{B} \) is the magnetic field.

With electrons and positrons:

• They have the same speed but oppositely directed velocities: the electron with \( \mathbf{v} \) and the positron with \( -\mathbf{v} \)
• The charge of the electron is negative, while that of the positron is positive

Due to opposite charges and velocity directions, the forces are in opposite directions.
Even so, the magnitude of these forces remains the same because of the uniform velocity magnitudes. This results in equal but opposite accelerations for the particles, reflecting the symmetry between the magnetic interactions they experience.

Center of Mass Motion

The concept of center of mass (CM) describes the motion of the combined mass of a system, in this case, the electron and positron. To analyze CM motion, we consider the total forces acting on the system.

The forces acting include:

• Magnetic forces, which do not perform work due to their perpendicular action to the velocity vector
• Electric forces, which result in work done and thus can change the system’s energy

In the scenario, magnetic forces are equal and opposite, thus they effectively cancel each other out in terms of effects on energy change. Therefore, the motion of the center of mass is predominantly influenced by the electric forces when analyzing velocity changes.
The overall CM motion results from a delicate balance of these forces, showing how only specific interactions truly impact momentum and energy on a large scale within electromagnetic fields.