Neutral Particle A neutral particle is at rest in a uniform...

Neutral Particle A neutral particle is at rest in a uniform magnetic field $\vec{B}$. At time $t=0$ it decays into two charged particles, each of mass $m$. (a) If the charge of one of the particles is $+q$, what is the charge of the other? (b) The two particles move off in separate paths, both of which lie in the plane perpendicular to $\vec{B}$. At a later time the particles collide. Express the time from decay until collision in terms of $m,|\vec{B}|$, and $|q|$

Key Concepts

Charged Particle Motion in a Uniform Magnetic Field

When a charged particle moves in a uniform magnetic field, it experiences a Lorentz force that acts perpendicular to both its velocity and the magnetic field. This force causes the particle to undergo circular motion at a constant speed, with the radius of the circle determined by the balance between the Lorentz force and the required centripetal force. The frequency of this circular motion, known as the cyclotron frequency, is given by ? = |q|B/m, and it is instrumental in determining the period and timing of phenomena such as collisions between charged particles moving in circular paths.

Charge Conservation

Charge conservation is a fundamental principle in physics stating that the total charge in an isolated system remains constant over time. In the context of particle decay, if a neutral particle decays into charged products, the algebraic sum of the charges of the resulting particles must equal the original charge, which is zero. This principle directly determines that if one particle has a positive charge of +q, the other must carry an equal magnitude of negative charge, -q, to preserve the overall neutrality.

Conservation of Momentum

Conservation of momentum is a basic law of physics that asserts the total momentum of an isolated system remains unchanged if no external forces are acting. In the decay of a particle initially at rest, the resulting charged particles must have momenta that are equal in magnitude but opposite in direction. This ensures that the momentum before and after the decay remains zero, leading to symmetry in the speed and trajectories of the decay products within the plane perpendicular to the magnetic field.

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