A charged particle enters a uniform magnetic field and...

A charged particle enters a uniform magnetic field and follows the circular path shown in the drawing. (a) Is the particle positively or negatively charged? Why? (b) The particle’s speed is 140 $\mathrm{m} / \mathrm{s}$ , the magnitude of the magnetic field is $0.48 \mathrm{T},$ and the radius of the path is 960 $\mathrm{m}$ . Determine the mass of the particle, given that its charge has a magnitude of $8.2 \times 10^{-4} \mathrm{C}$

Key Concepts

Centripetal Force

Centripetal force is the inward force necessary for an object to maintain its circular motion. It is described by the equation Fc = mv²/r, where m is the mass of the particle, v is its speed, and r is the radius of the circular path. In the context of charged particles in a magnetic field, the centripetal force is provided by the Lorentz force when the velocity is perpendicular to the magnetic field.

Lorentz Force

The Lorentz force is the force exerted on a charged particle moving in a magnetic field. It is given by the equation F = q(v × B), where q is the charge, v is the velocity, and B is the magnetic field. This force always acts perpendicular to the velocity of the particle, influencing its direction of motion without doing work on it.

Circular Motion in Magnetic Fields

When a charged particle enters a magnetic field at a velocity perpendicular to the field, the magnetic component of the Lorentz force acts as a centripetal force, bending the particle’s trajectory into a circular path. The relationship qvB = mv²/r is derived from equating the magnetic force to the required centripetal force for circular motion.

Right-Hand Rule

The right-hand rule is used to determine the direction of the force on a moving charged particle in a magnetic field. When the fingers of the right hand point in the direction of the particle’s velocity and curl toward the direction of the magnetic field, the thumb indicates the direction of the magnetic force on a positively charged particle. For negatively charged particles, the force direction is opposite to that indicated by the thumb.

Transcript

00:02 Hello, this problem is similar to other mass spectrometer problems because we're considering a charge particle moving in a circular path in a uniform magnetic field.

00:12 What we need to first determine is what is the charge of this particle.

00:16 So what we're going to do is look at the motion of the particle, compare it to the conventions of the right -hand rule for a positive charge, and see what that tells us about the actual charge of this particle.

00:26 So what do we have in this problem? what is the situation? we have a region of space that has a uniform.

00:31 Form magnetic field always heading completely out of the screen.

00:40 I'm only going to draw a few of these vectors, but we usually represent them as a dot in a circle or something like that heading out of the screen.

00:47 We also have a charged particle moving in a circular path, say maybe starting around here and heading around one quarter of the circle like this.

00:58 And of course it would continue its circular path.

01:00 We're only drawing one quarter of it.

01:01 So we have a charge q here.

01:04 Now let's consider the most of the most of it.

01:05 Of this charge to help answer part a, meaning whether or not this charge is positive or negatively charged.

01:13 So let's look at this charge's motion.

01:15 We see that it has, at the beginning, let's pick a particular point in its motion.

01:19 Let's talk about the beginning.

01:21 In the beginning, its velocity is directly upward.

01:24 And because its circular motion is heading like this, we know that the acceleration, because of uniform circular motion, is always heading into the circle.

01:34 So it's going to be heading directly to the left.

01:35 Now, because it's accelerating to the left, that means the net force is also to the left, but the only source of any force is the magnetic field.

01:43 So the total force is headed to the left, and therefore the magnetic force is headed to the left.

01:49 Now, we know all the vectors that make up the right -hand rule for the magnetic force.

01:54 So we can apply the right -hand rule here, see if it agrees with a positive charge convention, and if it does, then this charge is positive, and if it doesn't, this charge is negative.

02:04 So let's apply the right -hand rule...

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