A particle of charge q enters a region of uniform magnetic field 𝐁 (pointing into the page). The

step 1 Hi there. So for this problem, we have a particle of charge Q that enters a region of uniform ma

A particle of charge q enters a region of uniform magnetic...

A particle of charge $q$ enters a region of uniform magnetic field $\mathbf{B}$ (pointing into the page). The field deflects the particle a distance $d$ above the original line of flight, as shown in Fig. $5.8 .$ Is the charge positive or negative? In terms of $a, d, B$ and $q,$ find the momentum of the particle.

Key Concepts

Magnetic Lorentz Force

This fundamental concept describes the force experienced by a moving charged particle in a magnetic field. The force is given by the vector cross?product F = q(v × B), meaning that it acts perpendicular both to the direction of the particle's velocity and to the magnetic field. This principle is central to understanding how magnetic fields influence charged particles’ trajectories.

Circular Motion in a Magnetic Field

Because the magnetic Lorentz force is always perpendicular to the velocity of the charged particle, it does no work and therefore only changes the direction of the particle, not its speed. This leads to the particle undergoing uniform circular motion. The radius of this motion is determined by balancing the magnetic force with the centripetal force required for circular motion.

Right-Hand Rule and Charge Sign Determination

The right?hand rule is used to determine the direction of the magnetic force on a moving charged particle. By arranging the fingers in the direction of velocity and curling them toward the magnetic field, the thumb indicates the force direction for a positive charge. Thus, if the observed deflection is upward relative to the original path, the sign of the charge can be inferred using this rule.

Relationship between Magnetic Force and Momentum

By equating the magnetic force qvB to the centripetal force mv²/R, one obtains the relation mv = qBR. This relation directly links the momentum of the particle (mv) with the magnetic field strength and the radius of the circular path. It is an essential tool for problems involving charged particles moving in magnetic fields.

Geometric Relationships in Magnetic Deflection

When a charged particle is deflected by a magnetic field, measurable geometric parameters such as the deflection distance and chord length can be related to the radius of the circular path via circle geometry. Techniques involving sagitta (the maximum deflection from a straight-line path) allow one to express the radius (and hence the momentum through mv = qBR) in terms of these measurable quantities. This geometric insight is crucial for converting observable deflections into physical quantities like momentum.

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