A particular type of fundamental particle decays by...

A particular type of fundamental particle decays by transforming into an electron $e^{-}$and a positron $e^{+} .$Suppose the decaying particle is at rest in a uniform magnetic field $\vec{B}$ of magnitude $9.57 \mathrm{mT}$ and the $\mathrm{e}^{-}$and $\mathrm{e}^{+}$move away from the decay point in paths lying in a plane perpendicular to $\vec{B}$. How long after the decay do the $\mathrm{e}^{-}$and $\mathrm{e}^{+}$collide?

Key Concepts

Circular Motion of Charged Particles in Magnetic Fields

When a charged particle moves in a uniform magnetic field, the constant perpendicular Lorentz force causes circular motion. The radius of the circle (radius of curvature) depends on the particle’s momentum, charge, and the magnetic field strength, while the frequency of the motion (cyclotron frequency) depends on the charge-to-mass ratio and the field strength.

Collision Dynamics in Circular Paths

In systems where decay products follow circular trajectories due to a magnetic field, understanding the geometry of circular motion is crucial for analyzing collision times. The equality and symmetry in the charged particles' motion can be exploited to determine when their paths intersect, leading to their collision.

Lorentz Force

The Lorentz force is the force exerted on a charged particle moving through a magnetic field. It acts perpendicular to both the direction of the particle’s velocity and the magnetic field, causing the particle to follow a curved, typically circular, trajectory when no electric field is present.

Particle Decay and Conservation Laws

In particle decay, a parent particle transforms into other particles with the process governed by conservation of energy and momentum. This principle ensures that the vector sum of the momenta of the decay products equals the momentum of the original particle, and it plays a significant role in determining how the decay products move immediately after the decay.

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