A particle with charge 6.40 ×10^-19 C travels in a circular orbit with radius 4.68 mm due to the

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A particle with charge 6.40 ×10^-19 C travels in a circular...

A particle with charge $6.40 \times 10^{-19} \mathrm{C}$ travels in a circular orbit with radius 4.68 $\mathrm{mm}$ due to the force exerted on it by a magnetic field with magnitude 1.65 $\mathrm{T}$ and perpendicular to the orbit. (a) What is the magnitude of the linear momentum $\vec{p}$ of the particle? (b) What is the magnitude of the angular momentum $\vec{L}$ of the particle?

A particle with charge $6.40 \times 10^{-19} \mathrm{C}$ travels in a circular orbit with radius 4.68 $\mathrm{mm}$ due to the force exerted on it by a magnetic field with magnitude 1.65 $\mathrm{T}$ and perpendicular to the orbit. (a) What is the magnitude of the linear momentum $\vec{p}$ of the
particle? (b) What is the magnitude of the angular momentum $\vec{L}$ of the particle?

Key Concepts

Angular Momentum

Angular momentum in circular motion is defined as the cross product of the position vector and linear momentum, or in terms of magnitude, L = r * p when the motion is perpendicular to the radius. It quantifies the rotational inertia of the particle as it moves along its circular path.

Linear Momentum

Linear momentum is the product of an object’s mass and its velocity, defined as p = mv. In the context of a charged particle moving in a magnetic field, determining the speed using the balance between the magnetic force and the centripetal force allows one to compute the particle's linear momentum.

Centripetal Motion

Centripetal motion refers to the movement of an object along a circular path due to a force acting towards the center of the circle. For a charged particle in a magnetic field, the Lorentz force provides the necessary centripetal force, which is mathematically expressed as m(v^2/r), where m is the mass of the particle, v is its speed, and r is the radius of the circular path.

Lorentz Force

This is the force experienced by a charged particle moving in a magnetic field. It is given by the equation F = q(v × B), where q is the charge, v is the velocity of the particle, and B is the magnetic field. When the velocity vector is perpendicular to the magnetic field, the force has a magnitude of F = qvB and acts as a centripetal force, deflecting the particle into a circular path.

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