Region A-Region B Figure 29-39 shows a charged particle that...

Region A-Region B Figure $29-39$ shows a charged particle that is moving in the positive $x$ direction when it encounters region A with a uniform magnetic field. Its path is bent in a halfcircle and then moves into region $\mathrm{B}$ also with a uniform magnetic field. The particle undergoes another half revolution. Finally it passes between two charged capacitor plates and is deflected downward in the negative $y$ direction. (a) Is the charge positive or negative? Explain. (b) What is the direction of the magnetic field in region $\mathrm{A}$ ? Explain. (c) What is the direction of the magnetic field in region $\mathrm{B}$ ? Explain. (d) Which region has the larger magnetic field, $A$ or $B$ ? Explain.

Region A-Region B Figure $29-39$ shows a charged particle that is moving in the positive $x$ direction when it encounters region A with a uniform magnetic field. Its path is bent in a halfcircle and then moves into region $\mathrm{B}$ also with a uniform magnetic field. The particle undergoes another half revolution.
Finally it passes between two charged capacitor plates and is deflected downward in the negative $y$ direction.
(a) Is the charge positive or negative? Explain.
(b) What is the direction of the magnetic field in region $\mathrm{A}$ ? Explain.
(c) What is the direction of the magnetic field in region $\mathrm{B}$ ? Explain.
(d) Which region has the larger magnetic field, $A$ or $B$ ? Explain.

Key Concepts

Lorentz Force

The Lorentz force is the fundamental force acting on a charged particle moving in a magnetic field. It is given by the vector cross product F = q(v × B), meaning the force is perpendicular to both the velocity of the particle and the magnetic field. This force is responsible for bending the trajectory of the particle into a curved path, such as a semicircular arc when the field is uniform.

Charged Particle Circular Motion in a Magnetic Field

When a charged particle enters a uniform magnetic field, the perpendicular Lorentz force causes it to undergo uniform circular motion. The radius of this circular motion depends on the particle's speed, its mass, its charge, and the strength of the magnetic field. Understanding this relationship is key to analyzing the particle’s path through regions of different magnetic field strengths.

Charge Sign and Deflection Direction

The sign of the charged particle (positive or negative) affects the direction of the force experienced under a given magnetic field. The direction of the deflection, determined by the cross product in the Lorentz force, can be used to deduce whether the charge is positive or negative, particularly when compared with additional information from electric forces such as those exerted by charged capacitor plates.

Right-Hand and Left-Hand Rules

The right-hand rule (for positive charges) and the left-hand rule (for negative charges) are mnemonic devices used to determine the direction of the magnetic force acting on a moving charged particle. They help in establishing the relationship between the direction of the velocity, the magnetic field, and the resultant force, which is crucial in predicting and explaining the particle's trajectory.

Electric Field Force from Charged Capacitor Plates

When a charged particle passes between two charged capacitor plates, it experiences an electric force due to the electric field present between the plates. This force acts in a direction determined by the sign of the particle and the orientation of the electric field. Observations of the particle’s deflection by the capacitor plates can provide additional insights into the particle’s charge and the nature of the forces acting on it.

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