A proton, a deuteron (q=+e, m=2.0 u), and an alpha particle (q=+2 e, m=4.0 u) all having the sam

A proton, a deuteron (q=+e, m=2.0 u), and an alpha particle...

Key Concepts

Mass-to-Charge Ratio

The radius of curvature in a magnetic field depends on the square root of the particle's mass and inversely on its charge, as seen in the derived relation r ? ?m/q when kinetic energy is constant. This mass-to-charge ratio is a key factor that determines how sharply a charged particle's path will curve in a magnetic field.

Kinetic Energy and Velocity Relationship

The kinetic energy of a particle is given by K = 1/2 mv². For particles with the same kinetic energy, the velocity can be expressed as v = ?(2K/m). This indicates that particles with higher mass will have lower speeds when kinetic energy is constant, which in turn affects the radius of their circular path in a magnetic field.

Magnetic Lorentz Force

This is the force experienced by a charged particle moving in a magnetic field. The force is given by F = q(v × B), which has a magnitude qvB when the velocity is perpendicular to the magnetic field. It is responsible for bending the path of the particle into a circular motion by acting as the centripetal force.

Circular Motion in a Magnetic Field

When a charged particle enters a uniform magnetic field at a right angle, it undergoes circular motion. The radius of the circular path can be derived from equating the magnetic force to the centripetal force, leading to the equation r = mv/(qB). This relationship shows how mass, speed, and charge influence the curvature of the path.

Transcript

00:01 Okay, folks, so in this video we're going to be talking about this problem.

00:05 We have a proton, a deuteron, and an alpha particle, all having the same kinetic energy, enter a region of uniform and magnetic field b, moving perpendicular to b.

00:19 Okay, and what we're looking for is the ratio of the radius rd of the deuteron path to the radius rp of the proton path.

00:29 So that's part a.

00:30 And par b is the ratio our alpha of the alpha particle path to the proton particle path.

00:41 So how are we going to do this? well, first of all, you need to understand when you have a charged particle entering a region of uniform magnetic field, it's not going to have, you know, a straight path.

00:54 It's not going to, you know, for example, if it was moving in a straight path before, then as soon as it enters this region, it's not going to.

01:01 To keep having a straight path anymore because you know it's going to it's going to experience a force that's perpendicular to its to its velocity so it's going to whirl around it's going to in this case it's going to be traveling in a uniform you know in a circle okay so first of all let's let's recall from a newton second law that the f equals mass times x and for a particle in a circular path that acceleration can be written by v squared over radius okay and what is the force in this case well it is the magnetic force which is given by q vb i am ignoring the sign theta term in the cross product because it says in the problem that the velocity vector and the b field vector are perpendicular to each other so that sign that gives you a 1.

02:05 So we can ignore it.

02:06 Let's simplify this equation a little bit.

02:09 So we have a v here on the right side, and a v squared here on the left side.

02:14 So we're going to cross it out.

02:17 So we have radius equals mv over qv.

02:24 Ok.

02:25 So if you have two particles, the ratio of the path 1 and over the radius of the of the second path is going to give you m1 v1 q1b and 2 v2 q2 b okay and this is going to be simplified into m1 v1 over m2 v2 over q1.

03:06 So for two particles, we can calculate the ratio of the radius of their path by this expression.

03:16 And this expression is simpler than this expression, because we don't know the radiuses of the three particles in this problem.

03:27 But what we do know is the mass and the charge of the particles.

03:32 Okay.

03:34 And but we have another quantity, which is the velocity.

03:39 We don't know the velocity, but here's something that we do know.

03:42 We know that the three particles all have the same kinetic energy, right? they all have the same kinetic energy.

03:50 That means they all have the same one -half in v squared.

03:54 So we can calculate a ratio of the velocities by, by, okay, so let's just say, that this is k that stands for kinetic energy.

04:07 So v is going to be equal to 2k over m square root, right? so if you have two particles, the ratio of their velocities is going to be given by v1 over v2.

04:23 It's going to be square root of 2k over m1, and then m2 over 2k.

04:32 So this is going to be square root of m2 over m1.

04:38 And this makes sense because if two particles have the same kinetic energy and one particle is heavier than the other.

04:47 So let's say that the second particle is heavier than the first particle.

04:51 So this ratio is going to be, this ratio is going to be bigger than one.

04:54 That means this ratio is going to be smaller, i mean also bigger than one.

04:58 That means this first particle is faster than the second particle because it's a lighter.

05:04 Okay, anyway, so that was just a little bit of a justification for this equation.

05:13 And so we now have converted a ratio of velocities into a ratio of masses.

05:21 And that's going to make our problem a lot easier to solve because we're going to plug this back in here.

05:27 So r1 over r2 is going to be q2 over q1 m1 over m2 square root of m2 2 over m1.

05:40 Okay, let's simplify this a little bit more...

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