Refer to Check Your Understanding Question 10 before starting this problem. Suppose that the tar

Refer to Check Your Understanding Question 10 before...

Key Concepts

Magnetic Force on a Moving Charge

A charged particle moving in a magnetic field experiences a force known as the Lorentz force, which is perpendicular to both the velocity of the particle and the direction of the magnetic field. This force is given by the equation F = qvB sin?, where q is the charge, v is the speed of the particle, B is the magnetic field strength, and ? is the angle between the velocity and the field direction. When the velocity is perpendicular to the magnetic field, the sine term is maximized (equal to 1), making the force directly proportional to the speed.

Centripetal Force in Uniform Circular Motion

When a particle experiences a force that is always perpendicular to its velocity, it is forced into uniform circular motion. The magnetic force acting on the charged particle, when perpendicular to its velocity, provides the necessary centripetal force to keep the particle moving in a circular path. The centripetal force required for circular motion is given by F = 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.

Equating Magnetic and Centripetal Forces

By recognizing that the magnetic force acting on a charged particle in perpendicular motion can be equated to the centripetal force required for circular motion, one obtains the relation qvB = m v^2 / r. This relationship can be rearranged to derive the radius of the circular path, r = m v / (qB), and is often used to solve for unknown parameters such as the speed of the particle when other quantities are provided. This approach is fundamental in problems involving charged particle trajectories in magnetic fields.

Transcript

00:01 This problem deals with calculating the speed needed to have a charged particle hit a target as it follows circular motion in a uniform magnetic field.

00:09 So what we're given is a coordinate system, something like this, and everywhere in this space we have a magnetic field that is heading into the screen.

00:21 So i'll draw it like this, and it's everywhere, not just that location, so we'll just remember that.

00:25 And we have a charged particle.

00:27 I'll use a little purple thing here, a positively charged proton here, and it's sitting there with the charge plus e.

00:39 Now we are asked to figure out what's the speed that's needed for this particle to have, such that it's going to hit a target that is at negative 0 .1 meters in the x direction and negative 0 .10 meters in the y direction.

01:02 Now, because of the magnetic forces involved, the only way to get this proton here to hit this target is to have it shoot it either directly vertically, so along the positive y direction, or along the negative x direction.

01:23 So if we launch this particle initially vertically upward, right, we're going to have a, from our right -hand rule, we're going to have a magnetic force to the left, and our circular motion is going to look something like this.

01:34 So we're going to come around like this, come around and hit there like that.

01:42 And if our goal is to hit, the reason i do this circle like this is because our goal eventually in this problem is to hit this target.

01:47 So i needed to include that negative point one, negative point one location in the trajectory of my circular motion.

01:55 Now, my particle can also be launched this way and create a circular path that also includes that.

02:03 So it looks a little bit like an oval, meant to be a circle there.

02:08 A circular path that also can have our proton hit the target there.

02:12 So the question is, if i do want my proton to hit the target, what is the velocity i need? well, we know an equation for the circular motion of a charge particle in a magnetic field, right? it's m times v, so the momentum of our charged particle, divided by the absolute value of the charge value, multiplied by the the magnetic field that the charge is moving around in.

02:38 So you'll see from our diagram that if we want to hit the target at either of these, at that location of negative point one, negative point one, we'll have to map out a circular path that includes that target and we'll see that the radius of our circle, if we measure that out right, the radius of the circle is just equal to that point one meter.

02:59 Maybe i shouldn't have it equal to negative point one, right? but these radii here that we have for our circles, that is going to be 0 .1 meters.

03:09 So we actually know the size of the circle we need to have this electron, or sorry, this proton, hit the target...

Please give Ace some feedback