A particle with charge q and momentum p, initially moving...

A particle with charge $q$ and momentum $p$, initially moving along the $x$ axis, enters a region where a uniform magnetic field $\overrightarrow{\mathbf{B}}=B_{0} \hat{\mathbf{k}}$ extends over a width $x=\ell$ as shown in Fig. $27-45$. The particle is deflected a distance $d$ in the $+y$ as it traverses the field. Determine (a) whether $q$ is positive or negative, and (b) the magnitude of its momentum $p$.

A particle with charge $q$ and momentum $p$, initially moving along the $x$ axis, enters a region where a uniform magnetic field $\overrightarrow{\mathbf{B}}=B_{0} \hat{\mathbf{k}}$ extends over a width $x=\ell$ as shown in Fig. $27-45$. The particle is deflected
a distance $d$ in the $+y$ as it traverses the field. Determine (a) whether $q$ is positive or negative, and (b) the magnitude of its momentum $p$.

Key Concepts

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving in a magnetic field, expressed by the equation F = q(v × B). This force is perpendicular to both the velocity of the particle and the magnetic field, causing the charged particle to follow a curved path. It is fundamental in analyzing the motion of charged particles in magnetic fields.

Right-Hand Rule and Charge Sign Determination

The right-hand rule is used to establish the direction of the force experienced by a charged particle in a magnetic field. For a positive charge, the force direction follows the curled fingers when the thumb points along the velocity; for a negative charge, the force is in the opposite direction. This concept is crucial in determining whether the charge is positive or negative based on the observed deflection.

Circular Motion in Magnetic Fields

When a charged particle enters a uniform magnetic field perpendicular to its velocity, it undergoes circular motion due to the constant perpendicular force. The radius of this motion is given by the balance between the magnetic force and the required centripetal force, leading to the relation p = qBr, where p is the momentum. This relationship is essential in calculating the momentum of the particle from the measured deflection.

Geometric Analysis of Particle Deflection

The deflection of the particle in a finite magnetic field region can be analyzed through the geometry of its circular trajectory. The lateral displacement (deflection) is related to the arc length traveled and the curvature of the path. By utilizing geometric relationships between chords, arcs, and sagitta in a circle, one can derive the necessary expressions to relate the observed deflection distance to the particle’s momentum.

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