A particle with charge 2.15μ C and mass 3.20 ×10 ^

step 1 First, we are going to find the radius R, which is given by mv over QV. And this equation comes

A particle with charge 2.15$\mu \mathrm { C }$ and mass $3.20 \times$ $10 ^ { - 11 } \mathrm { kg }$ is initially traveling in the $+ y$ -direction with a speed $v _ { 0 } = 1.45 \times 10 ^ { 5 } \mathrm { m } / \mathrm { s }$ . It then enters a region containing a uniform magnetic field that is directed into, and perpendicular to, the page in Fig. $\mathrm { P } 27.89 .$ The magnitude of the field is 0.420$\mathrm { T }$ . The region extends a distance of 25.0$\mathrm { cm }$ along the initial direction of travel; 75.0$\mathrm { cm }$ from the point of entry into the magnetic field region is a wall. The length of the field-free region is thus 50.0$\mathrm { cm }$ . When the charged particle enters the magnetic field, it follows a curved path whose radius of curvature is $R .$ It then leaves the magnetic field after a time $t _ { 1 }$ , having been deflected a distance $\Delta x _ { 1 }$ . The particle then travels in the field-free region and strikes the wall after undergoing a total deflection $\Delta x$ (a) Determine the radius $R$ of the curved part of the path. (b) Determine $t _ { 1 }$ , the time the partiole spends in the magnetic field. (c) Determine $\Delta x _ { 1 } ,$ the horizontal deflection at the point of exit from the field. (d) Determine $\Delta x ,$ the total horizontal deflection.

A particle with charge 2.15$\mu \mathrm { C }$ and mass $3.20 \times$ $10 ^ { - 11 } \mathrm { kg }$ is initially traveling in the $+ y$ -direction with a speed
$v _ { 0 } = 1.45 \times 10 ^ { 5 } \mathrm { m } / \mathrm { s }$ . It then enters a region containing a uniform magnetic field that is directed into, and perpendicular to, the
page in Fig. $\mathrm { P } 27.89 .$ The magnitude of the field is 0.420$\mathrm { T }$ . The region extends a distance of 25.0$\mathrm { cm }$ along the initial direction of
travel; 75.0$\mathrm { cm }$ from the point of entry into the magnetic field
region is a wall. The length of the field-free region is thus 50.0$\mathrm { cm }$ .
When the charged particle enters the magnetic field, it follows a curved path whose radius of curvature is $R .$ It then leaves the magnetic field after a time $t _ { 1 }$ , having been deflected a distance $\Delta x _ { 1 }$ .
The particle then travels in the field-free region and strikes the wall after undergoing a total deflection $\Delta x$ (a) Determine the radius $R$
of the curved part of the path. (b) Determine $t _ { 1 }$ , the time the partiole spends in the magnetic field. (c) Determine $\Delta x _ { 1 } ,$ the horizontal
deflection at the point of exit from the field. (d) Determine $\Delta x ,$ the
total horizontal deflection.

Key Concepts

Magnetic Lorentz Force

The magnetic Lorentz force is the force exerted on a charged particle when it moves through a magnetic field. It is given by F = q(v × B), where q is the charge, v is the velocity vector, and B is the magnetic field. This force acts perpendicular to both the velocity of the particle and the magnetic field, causing the particle to experience a change in direction rather than a change in speed.

Radius of Curvature in Uniform Magnetic Field

In a uniform magnetic field, a charged particle moving perpendicular to the field lines follows a circular path due to the constant magnetic Lorentz force acting as a centripetal force. By equating the magnetic force, qvB, to the centripetal force, mv²/R, one can derive the expression R = mv/(qB). This radius of curvature is a fundamental property that characterizes the bending of the particle's trajectory in the magnetic field.

Time of Flight Along a Circular Arc

Since a charged particle in a uniform magnetic field moves along a circular arc, the time it spends within the field can be determined by dividing the length of the arc by the constant speed of the particle. This involves calculating the arc length using the angle subtended at the center of the circle (?) and the radius R, with t = (R?)/v.

Horizontal Deflection and Geometric Analysis

The horizontal deflection of a particle emerging from a curved path in a magnetic field is deduced from the geometry of the circular trajectory. By applying trigonometric relationships, one can relate the angle of the arc to the displacement in the horizontal direction. Additionally, any subsequent displacement in a region where no magnetic field is present is determined using kinematic principles, considering the horizontal component of the particle's velocity after it exits the field.

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