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 $B_0$ extends over a width $x = \ell$ as shown in Fig. 20-70. The particle is deflected a distance $d$ in the $+y$ direction as it traverses the field. Determine ($a$) whether $q$ is positive or negative, and ($b$) the magnitude of its momentum $p$ in terms of $q, B_0, l,$ and $d$. FIGURE 20-70 Problem 85. (CANT COPY FIGURE)

Key Concepts

Lorentz Force

The Lorentz force is the fundamental principle describing the force acting on a charged particle moving in a magnetic field. It is given by F = q(v × B), which means that the force is perpendicular both to the velocity of the particle and to the magnetic field. This concept allows one to determine the direction of the force (using the right?hand rule for a positive charge, and the opposite for a negative charge) and is key to understanding the motion of a charged particle in the field.

Circular Motion in a Magnetic Field

When a charged particle moves in a uniform magnetic field, the Lorentz force acts as a centripetal force, causing the particle to move along a circular arc. The radius of the circular path is related to the particle’s momentum and the magnetic field by the relation r = p/(|q|B). This relationship connects the geometric path of the particle with its physical properties such as momentum and charge.

Applying the Right-Hand Rule for Sign Determination

The sign of the charge can be determined by employing the right-hand rule, which predicts the direction of the force for a given velocity and magnetic field. By comparing the expected direction of the force with the observed deflection, one can infer whether the charge is positive or negative. This method is crucial for analyzing the dynamics of charged particles in magnetic fields.

Geometry of Circular Deflection

Understanding the geometry of circular motion helps in relating the measured deflection to the physical parameters of the particle’s motion. The deflection distance and the arc length within the magnetic field can be related through circular geometry, enabling one to derive expressions for quantities like momentum in terms of the field strength, deflection distance, and the dimensions of the magnetized region.

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