Ssm The electrons in the beam of a television tube have a...

ssm The electrons in the beam of a television tube have a kinetic energy of $2.40 \times 10^{-15} \mathrm{J}$ . Initially, the clectrons move horizontally from west to east. The vertical component of the earth's magnetic field points down, toward the surface of the earth, and has a magnitude of $2.00 \times 10^{-5} \mathrm{T} .$ (a) In what direction are the electrons deflected by this field component? (b) What is the acceleration of an electron in part (a)?

Key Concepts

Acceleration of Charged Particles

Acceleration in this context results from the magnetic force acting on the moving charged particle. According to Newton's second law (F = ma), the acceleration can be calculated by dividing the magnitude of the magnetic force by the particle's mass. This concept is essential in understanding how charged particles change their motion when influenced by magnetic fields.

Kinetic Energy and Velocity Relationship

A particle's kinetic energy is related to its velocity through the equation KE = ½mv². This relationship allows one to calculate the speed of a charged particle from its kinetic energy, which is critical for subsequently determining the magnitude of the magnetic force acting on it.

Right-Hand Rule

The right?hand rule is a useful mnemonic for determining the direction of the force exerted on a positive charge moving through a magnetic field. When dealing with negatively charged particles like electrons, the force direction is opposite to that determined by the rule, which is key in predicting deflection directions in magnetic fields.

Lorentz Force

The Lorentz force is the fundamental force acting on a charged particle moving in a magnetic field, described by the equation F = q(v × B). This force is always perpendicular to the particle’s velocity and the magnetic field, causing the particle to change direction. It is the basis for understanding the deflection of charged particles in various applications, from cathode ray tubes to cyclotrons.

Transcript

00:01 Hello everyone.

00:03 First, let's get a visualization for what is happening in this problem.

00:08 So we have a beam of electrons.

00:11 We'll say that they're heading to the right, but our problem says that that means they're heading horizontally from west to east.

00:20 So we'll say that this is our westward direction and this is our eastward direction, and we have a beam of electrons heading to the right.

00:33 So we have a beam full of electrons all headed in a particular direction.

00:39 Now, it also tells us that we have a particular component of the magnetic field that could affect these electrons and that it's heading directly downward.

00:49 So we have a component, the vertical component, of the earth's magnetic field, heading this way, and i'll represent it as a downward vector here.

00:57 So if we go and adjust our quarter, system up here with west is left and east to the right.

01:02 That means up and down are literally that.

01:05 Up is towards the sky.

01:07 Down here is down towards the ground.

01:10 And then into the page would be headed directly north and out of the page is headed directly south.

01:20 So to answer the first question here, a, what is the direction of the magnetic force acting on these electrons? well, we can just use our right -hand rule to figure this out.

01:32 So for part a, let's put our pointer finger in the direction of the velocity of the electrons, and i said that they are heading to the right.

01:39 So a vector here representing the direction of the velocity.

01:42 So we'll have our pointer finger going this way, headed to the right, and then we want to put our middle finger headed down.

01:50 That's going to be something like this right here.

01:53 So velocity to the right, magnetic field heading down.

01:57 And then we do that with our hand, we see that our thumb is actually heading into the screen.

02:03 Now, remember, so normally we'd say that the magnetic force would then be into the screen, but these are not positive charges that we're thinking about.

02:10 These are negative charges.

02:11 So we actually flip that result.

02:13 So the magnetic force actually isn't heading into the screen.

02:16 We flip that to the opposite.

02:20 So we see that the way in which the electrons will be deflected because the magnetic force is actually out of the screen, and then in this case would be heading due south.

02:34 So that is part a.

02:36 We figured out which way the electrons are going to be deflected...

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