A particle with negative charge q and mass m = 2.58 ×10 ^ - 15 kg is traveling through a reg

step 1 For this exercise, we have for A, Q equals negative 2 .45 N divided by 5 times 1 .6 times 10 to

A particle with negative charge q and mass m = 2.58 ×10 ^ -...

A particle with negative charge $q$ and mass $m = 2.58 \times$ $10 ^ { - 15 } \mathrm { kg }$ is traveling through a region containing a uniform magnetic field $\vec { \boldsymbol { B } } = - ( 0.120 \mathrm { T } ) \hat { \boldsymbol { k } }$ . At a particular instant of time the velocity of the particle is $\vec { \boldsymbol { v } } = \left( 1.05 \times 10 ^ { 6 } \mathrm { m } / \mathrm { s } \right) ( - 3 \hat { \imath } + 4 \hat { \jmath } + 12 \hat { \boldsymbol { k } } )$ and the force $\vec { \boldsymbol { F } }$ on the particle has a magnitude of 2.45$\mathrm { N }$ . (a) Determine the charge $q .$ (b) Determine the acceleration $\vec { a }$ of the particle. (c) Explain why the path of the particle is a helix, and determine the radius of curvature $R$ of the circular component of the helical path. (d) Determine the cyclotron frequency of the particle. (c) Explain why the path of the particle is a helix, and determine the radius of curvature $R$ of the circular component of the helical path. (d) Determine the cyclotron frequency of the particle. (e) Although helical motion is not periodic in the full sense of the word, the $x$ - and $y$ -coordinates do vary in a periodic way. If the coordinates of the particle at $t = 0$ are $( x , y , z ) = ( R , 0,0 ) ,$ determine its coordinates at a time $t = 2 T ,$ where $T$ is the period of the motion in the $x y$ -plane.

A particle with negative charge $q$ and mass $m = 2.58 \times$ $10 ^ { - 15 } \mathrm { kg }$ is traveling through a region containing a uniform magnetic
field $\vec { \boldsymbol { B } } = - ( 0.120 \mathrm { T } ) \hat { \boldsymbol { k } }$ . At a particular instant of time the velocity
of the particle is $\vec { \boldsymbol { v } } = \left( 1.05 \times 10 ^ { 6 } \mathrm { m } / \mathrm { s } \right) ( - 3 \hat { \imath } + 4 \hat { \jmath } + 12 \hat { \boldsymbol { k } } )$ and
the force $\vec { \boldsymbol { F } }$ on the particle has a magnitude of 2.45$\mathrm { N }$ . (a) Determine the charge $q .$ (b) Determine the acceleration $\vec { a }$ of the particle.
(c) Explain why the path of the particle is a helix, and determine
the radius of curvature $R$ of the circular component of the helical
path. (d) Determine the cyclotron frequency of the particle. (c) Explain why the path of the particle is a helix, and determine
the radius of curvature $R$ of the circular component of the helical
path. (d) Determine the cyclotron frequency of the particle.
(e) Although helical motion is not periodic in the full sense of the
word, the $x$ - and $y$ -coordinates do vary in a periodic way. If the coordinates of the particle at $t = 0$ are $( x , y , z ) = ( R , 0,0 ) ,$
determine its coordinates at a time $t = 2 T ,$ where $T$ is the period
of the motion in the $x y$ -plane.

Key Concepts

Cyclotron Frequency

The cyclotron frequency, defined as ? = |q|B/m, is the frequency with which a charged particle orbits in a magnetic field. This quantity is independent of the particle's velocity and is a key parameter in describing the periodic motion of charged particles in magnetic confinement and accelerator physics.

Helical Motion

When a charged particle has a velocity component both perpendicular and parallel to a magnetic field, the resulting path is a helix. The circular motion caused by the perpendicular component and the constant velocity along the field direction combine to create a spiral motion along the field lines.

Circular Motion and Radius of Curvature

For the portion of the motion perpendicular to the magnetic field, the magnetic force acts as the centripetal force, which is set equal to mv²/R. This relationship allows the determination of the radius of the circular path or curvature, which is crucial in characterizing the particle's trajectory.

Decomposition of Velocity Components

In problems involving magnetic fields, it is often useful to decompose the velocity of the particle into perpendicular and parallel components with respect to the direction of the magnetic field. The perpendicular component determines circular motion while the parallel component results in linear motion, together forming a helical path.

Uniform Magnetic Field Effects

In a uniform magnetic field, the magnetic force is constant in magnitude and always perpendicular to the velocity of the charged particle. This perpendicular force causes the particle to undergo circular motion in the plane perpendicular to the field, while any velocity component parallel to the field remains unaffected, leading to combined motions.

Lorentz Force

This concept describes the force experienced by a charged particle in the presence of electric and magnetic fields, given by the equation F = q(v × B). It is fundamental for understanding how magnetic fields influence the motion, trajectory, and acceleration of charged particles, with the direction of the force determined by the right-hand rule and the sign of the charge.

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