Problem 1 - Master Resource Book in JEE Main Physics

Question

Two charged particles traverse identical helical paths in a completely opposite sense in a uniform magnetic field \(\mathbf{B}=B_{0} \hat{\mathbf{k}} . \quad\) [NCERT Exemplar] (a) They have equal z-components of momenta (b) They must have equal charges (c) They necessarily represent a particle-antiparticle pair (d) The charge to mass ratio satisfy : \(\left(\frac{e}{m}\right)_{1}+\left(\frac{e}{m}\right)_{2}=0\)

Step-by-Step Solution

Answer

Correct options are (a) and (d).

Step 1: Analyze the Helical Path

In a magnetic field, a charged particle will move in a helical path if it has a velocity component along the field and perpendicular to it. The motion in the perpendicular plane is circular due to the Lorentz force, while the velocity component along the magnetic field results in forward motion, creating a helical path.

Step 2: Determine the Effect of Identical Helical Paths in Opposite Directions

If two particles traverse identical helical paths in opposite directions in the same magnetic field, their perpendicular circular motions are in opposite senses. This implies that they have opposite charges because the magnetic force direction is reversed for opposite charges (\( q imes ext{velocity} imes B \) results in opposite force directions).

Step 3: Equal Z-Components of Momentum and its Implications

Given that both particles have equal z-components of momentum, it means their mass-to-charge ratios are related since \( p_z = m \cdot v_z \). As \( p_z \) is the same, both particles must have the same mass to velocity ratio for their respective charge signs.

Step 4: Investigate Charge to Mass Ratio Condition

For two particles with opposite charges moving identically except in opposite helical paths, their charge-to-mass ratios are negations of one another (i.e., \( (e/m)_1 = -(e/m)_2 \)). Thus, when summed, they satisfy:\( (e/m)_1 + (e/m)_2 = 0 \).

Step 5: Identify Which Statements are Correct

(a) is correct as the question states they have equal z-components of momenta. (b) is incorrect since they have opposite charges. (c) is also incorrect as identical paths don't imply they are a particle-antiparticle pair, although opposite charges suggest such a relation. (d) is correct as we derived that the charge-to-mass ratios sum to zero.

Key Concepts

z-component of momentum

The z-component of momentum describes a particle's movement along the direction of a magnetic field. When a charged particle moves in such a field, its motion can be split into two parts:

• Perpendicular circular motion due to the Lorentz force.
• Linear motion along the field creating a helical path.

The z-component is crucial because it helps determine how the particle progresses along the field's axis. Unlike the circular component, the z-momentum remains unaffected by the magnetic force since the force acts perpendicular to the motion.
The expression for the z-component of momentum is given by: \[ p_z = m \cdot v_z \]where \( p_z \) is the momentum in the z-direction, \( m \) is the mass, and \( v_z \) is the velocity along the z-axis.
For two particles moving through the magnetic field, having the same \( p_z \) implies they have a relationship between their mass and charge. It suggests that despite potentially different charges and velocities, the product remains equal, showing symmetry in their helical paths but opposite directions.

charge-to-mass ratio

The charge-to-mass ratio, represented as \( \frac{e}{m} \), is a fundamental property that characterizes a charged particle's behavior in a magnetic field. It tells us how much charge a particle has in relation to its mass.
In the context of helical motion in magnetic fields, this ratio sheds light on the particle's specific path and how it spirals due to the Lorentz force. An opposite charge-to-mass ratio for two particles indicates that while their paths are identical, they move in opposite directions.
The key equation here is:\[ \left(\frac{e}{m}\right)_1 + \left(\frac{e}{m}\right)_2 = 0 \]This equation shows that for particles in identical helical paths yet opposite senses, their charge-to-mass ratios cancel each other out. Such conditions could signal situations where particles exhibit symmetry, like those in antiparticle physics. The charges must be opposite, allowing one particle to trace a mirror image path of the other.

particle-antiparticle pair

A particle-antiparticle pair consists of two particles with opposite electric charge and other quantum numbers but identical masses. These pairs provide insight into various physical phenomena, especially in high-energy physics.
Though two particles with identical helical motions in reverse directions might suggest a particle-antiparticle pair, identical paths alone do not confirm such an identity.

• The key feature of a particle-antiparticle pair is their charge being equal in magnitude but opposite in sign.
• They annihilate upon contact, releasing energy in the form of photons or other particles.

In magnetic fields, a particle-antiparticle pair will follow symmetrical paths but only if they meet the mathematical and physical criteria like equal masses and opposite charge-to-mass ratios. Identifying these pairs rests on closely analyzing their trajectory behavior, lifetimes, and interaction outputs.