∙A proton traveling at 3.60 km / s suddenly enters a uniform magnetic field of 0.750 T , traveli

step 1 For the given problem, we have a following situation. A particle proton is moving in a J -KAP di

∙A proton traveling at 3.60 km / s suddenly enters a uniform...

$\bullet$ A proton traveling at 3.60 $\mathrm{km} / \mathrm{s}$ suddenly enters a uniform magnetic field of 0.750 $\mathrm{T}$ , traveling at an angle of $55.0^{\circ}$ with the field lines (Figure 20.57$)$ . (a) Find the magnitude and direction of the force this magnetic field exerts on the proton. (b) If you can vary the direction of the proton's velocity, find the magnitude of the maximum and minimum forces you could achieve, and show how the velocity should be oriented to achieve these forces. (c) What would the answers to part (a) be if the proton were replaced by an electron traveling in the same way as the proton?

$\bullet$ A proton traveling at 3.60 $\mathrm{km} / \mathrm{s}$ suddenly enters a uniform magnetic
field of 0.750 $\mathrm{T}$ , traveling at an angle of $55.0^{\circ}$ with the field lines (Figure 20.57$)$ . (a) Find the magnitude and direction of the force this magnetic field exerts on the proton. (b) If you can vary the direction of the proton's velocity, find the magnitude of the maximum and minimum forces you could achieve, and show how the velocity should be oriented to achieve these forces. (c) What would the answers to part (a) be if the proton were replaced by an electron traveling in the same way as the proton?

Key Concepts

Lorentz Force

The Lorentz force is the fundamental force experienced by a charged particle moving in a magnetic field. It is given by the vector cross product F = q(v × B), where q is the charge, v is the velocity, and B is the magnetic field. This principle explains how moving charged particles are deflected in magnetic fields and is central to many areas such as electromagnetism and particle physics.

Vector Cross Product

The cross product operation is used to determine both the magnitude and direction of the force acting on the charged particle. The magnitude of F is given by qvB sin(?), where ? is the angle between the velocity and the magnetic field vectors, and its direction is perpendicular to both. This mathematical operation encapsulates the dependence of the force on the relative orientation of v and B.

Dependence of Force on Angle of Incidence

The magnitude of the magnetic force depends on the sine of the angle between the moving charge’s velocity and the magnetic field. When the velocity is perpendicular to the field (? = 90°), sin ? is maximized and so is the force; conversely, when the velocity is parallel or antiparallel (? = 0° or 180°), sin ? is zero and no force is exerted. Recognizing this relationship is key to understanding the maximum and minimum forces achievable in a given magnetic field.

Charge Sign and Particle Type Effects

The nature of the charged particle, including the magnitude and sign of its charge, directly influences the force’s magnitude and the direction indicated by the cross product. For example, while a proton with a positive charge follows the right-hand rule, an electron, which has a negative charge, will experience a force in the opposite direction. This sign difference is crucial when analyzing the behavior of different particles in magnetic fields.

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