A particular type of fundamental particle decays by...

A particular type of fundamental particle decays by transforming into an electron $\mathrm{e}^{-}$ and a positron $\mathrm{e}^{+} .$ Suppose the decaying particle is at rest in a uniform magnetic field $\vec{B}$ of magnitude $3.53 \mathrm{mT}$ 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

Symmetry in Particle Decay Kinematics

In a decay occurring from rest, conservation of momentum ensures that the decay products have equal and opposite momenta. When the products are oppositely charged and subject to the same magnetic field, they follow symmetric, mirror-image circular trajectories. This symmetry implies that under appropriate conditions, such particles will reconvene after traversing complementary segments of their circular paths, leading to their eventual collision.

Cyclotron Period and Collision Timing

The cyclotron period, which is the time a charged particle takes to complete one full circle in a magnetic field, is given by T = 2?m/(qB). In problems involving symmetric decay into oppositely charged particles moving in circular paths, the geometry of their motion often implies that they collide after a specific fraction of the cyclotron period, typically after they have completed a complementary arc (such as a half-turn) in their trajectories.

Cyclotron Motion

Cyclotron motion refers to the circular motion of charged particles in a uniform magnetic field resulting from the Lorentz force. The key parameters include the cyclotron radius, which depends on the particle's speed, mass, charge, and the magnetic field strength, and the cyclotron frequency, given by ? = qB/m. This frequency determines the period of the circular motion.

Lorentz Force

The Lorentz force is the force exerted on a charged particle moving in a magnetic field. It is given by the equation F = q(v × B), where q is the charge, v is the velocity, and B is the magnetic field. This force is always perpendicular to the velocity of the particle, causing it to undergo circular motion when moving in a plane perpendicular to the magnetic field.

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