Assume the region to the right of a certain plane contains a...

Assume the region to the right of a certain plane contains a uniform magnetic field of magnitude 1.00 mT and the field is zero in the region to the left of the plane as shown in Figure P 29.22. An electron, originally traveling perpendicular to the boundary plane, passes into the region of the field. (a) Determine the time interval required for the electron to leave the “field-filled” region, noting that the electron’s path is a semicircle. (b) Assuming the maximum depth of penetration into the field is 2.00 cm, find the kinetic energy of the electron.

Key Concepts

Uniform Circular Motion in a Magnetic Field

A charged particle entering a uniform magnetic field at a right angle experiences a force that continuously redirects its motion, leading to uniform circular motion. In this context, the magnetic force acts as the centripetal force required to keep the particle moving in a circle, and the radius of this circular path is determined by the balance of the magnetic and centripetal forces.

Cyclotron Frequency and Period

The cyclotron frequency characterizes the rate at which a charged particle orbits in a uniform magnetic field and is calculated as ? = |q|B/m. The corresponding period, which is the time for a complete orbit, is given by T = 2?m/(|q|B). For motion along a semicircular path, the time taken is half of this period, linking the magnetic properties to the temporal aspects of the particle's trajectory.

Lorentz Force

The Lorentz force is the force exerted on a charged particle moving in a magnetic field. When the particle's velocity is perpendicular to the magnetic field, the force has a magnitude equal to the product of the charge, the speed, and the magnetic field strength, and it acts perpendicular both to the velocity and the field direction, resulting in a change in the direction of motion rather than its speed.

Kinetic Energy of a Charged Particle

The kinetic energy of a particle in motion is expressed as KE = 1/2 mv². In scenarios involving circular motion in a magnetic field, the velocity of the particle—and thus its kinetic energy—can be indirectly determined from the radius of the circular path via the relation established by equating the magnetic force and the required centripetal force.

Transcript

00:01 So we have an electron here that's traveling into magnetic field that has a magnitude of 1milatelsa.

00:08 And for part a, we need to find the time it takes for this electron to leave this magnetic field, assuming that its path is a semicircle.

00:22 So to start solving this problem, i'm going to write out the force of b, which is the force acting on the electron in the magnetic field, is equal to the force of e, which is the force acting on the electron due to its velocity.

00:43 So we're going to rewrite this as q times v times b is equal to mv squared divided by r.

01:01 And we can rewrite this as qb is equal to mv divided by r.

01:16 So now what we can do is we can say that qb is equal to m times r times omega divided by r.

01:35 And then we can solve this for the angular velocity omega, and that's going to be equal to q times b divided by m.

01:50 So now this angular velocity is equal to the charge of the electron times the given maximum.

02:06 Magnetic field and then divided by the mass of the electron.

02:26 And so this angular velocity is equal to 0 .1 -786 times 10 to the 9 radians per second.

02:44 So we're going to find the time it takes for the electron to follow the full circle path, and we're going to label that as t, capital letter t, is equal to 2 times pi, divided by the angular velocity.

03:06 And so this time is going to be equal to 2 pi divided by the angular velocity that we just calculated...

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