A positively charged particle of mass 7.2 ×10^-8 kg is traveling due east with a speed of 85 m /

A positively charged particle of mass 7.2 ×10^-8 kg is...

A positively charged particle of mass $7.2 \times 10^{-8} \mathrm{kg}$ is traveling due east with a speed of 85 $\mathrm{m} / \mathrm{s}$ and enters a $0.31-\mathrm{T}$ uniform magnetic field. The particle moves through one-quarter of a circle in a time of $2.2 \times 10^{-3} \mathrm{s}$ , at which time it leaves the field heading due south. All during the motion the particle moves perpendicular to the magnetic field. (a) What is the magnitude of the magnetic force acting on the particle? (b) Determine the magnitude of its charge.

Key Concepts

Period and Angular Displacement in Circular Motion

The time taken by a particle to travel along a segment of its circular path is related to the total period of the motion. Specifically, if a particle traverses a known fraction of the full circle, this information can be used to determine the angular displacement and further relate the particle’s speed and the radius of the circle, which is crucial for solving for unknown quantities like charge.

Centripetal Force

In the context of circular motion, the centripetal force is the net force required to keep an object moving in a circle. For a charged particle in a magnetic field, the Lorentz force serves as the centripetal force, and its relationship can be expressed as the particle’s mass times the square of its speed divided by the radius of the circular path.

Lorentz Force

This is the force exerted on a charged particle moving in a magnetic field. The magnitude of the Lorentz force is given by the product of the particle's charge, its speed, and the magnetic field strength when the motion is perpendicular to the field. It acts perpendicular to both the particle's velocity and the magnetic field direction.

Uniform Circular Motion

When a charged particle enters a magnetic field perpendicularly, the Lorentz force deflects it into a circular path, causing it to undergo uniform circular motion. In such motion, the particle’s speed remains constant while its direction continuously changes, and its path is determined by the balance between the magnetic force and the required centripetal force.

Transcript

00:01 In this problem, we have a particle entering a region of space with a uniform magnetic field, and it's traveling perpendicular that field.

00:07 When that happens, of course, the particle will travel along a circular path.

00:11 In this case, it only travels one quarter of a circle, heading east, then turning around and heading due south.

00:18 So let's just get a quick visualization of what that looks like.

00:20 So we would have a region of space here with a magnetic field, in this case, pointing outward, and then we'd have a particle heading with velocity v, say to the right, in this case, this would be due east, and then down here would be due south.

00:40 Now, when the particle enters this region of space with this uniform magnetic field, what happens is it follows a circular path.

00:49 In this case, we'll only go a quarter of the path, and then head to turn to color of black, right, for a particle, and then it's going to head due south this way.

01:00 So, we're asked several questions about the situation, and we're given several pieces of information.

01:06 We know the mass of our particle, we know the speed of our particle, we know how long it spends in this region.

01:12 We know the delta t here, and we also know the magnitude of the magnetic field through which it's traveling.

01:18 Now, this is one of those problems where it's actually easier to solve for part b first and then solve for part a.

01:24 So that's what we're going to do first.

01:26 Part b here is asking us to find out the magnitude of the charge, of this particle.

01:32 Now remember we know how to relate the motion of our particle to characteristics of the particle through our equation for the radius of the path as a particle moves through a uniform, moves perpendicularly through a uniform magnetic field.

01:47 That means if we want to find the charge of that particle we just solve for that by rearranging our equation a little bit.

01:54 Oops, a little bit.

01:56 That is not going to be our radius down here.

01:59 So we have this expression here.

02:03 Now, what do we not know? well, we don't know the radius here.

02:12 So we don't know the radius of this path, but we know some information about the velocity and we know some information about the time.

02:22 So we can figure out how to relate those things together.

02:27 So let's think about the motion that this particle is undergoing in this problem.

02:32 So it's going to do travel a square, quarter circle of a path here, and we'll say that that distance that it travels is a distance d.

02:42 So we know from physics one how all these things relate to each other.

02:46 We know that the velocity of the particle is going to be the distance the particle travels divided by how long that it took to travel.

02:55 But we know that this particle, the distance it's traveling, is only one quarter of the circumference of the circle...

Please give Ace some feedback