(a) What value of magnetic field would make a beam of...

(a) What value of magnetic field would make a beam of electrons, traveling to the right at a speed of $4.8 \times 10^{6} \mathrm{~m} / \mathrm{s}$ go undeflected through a region where there is a uniform electric field of $8400 \mathrm{~V} / \mathrm{m}$ pointing vertically up? $(b)$ What is the direction of the magnetic field if it is known to be perpendicular to the electric field? $(c)$ What is the frequency of the circular orbit of the electrons if the electric field is turned off?

Key Concepts

Lorentz Force

The Lorentz force is the net force experienced by a charged particle moving in electric and magnetic fields. It is given by F = q(E + v × B), where q is the charge of the particle, E is the electric field, v is the velocity of the particle, and B is the magnetic field. This concept is crucial for understanding how charged particles interact with fields and for determining conditions for equilibrium or specific types of motion.

Velocity Selector

A velocity selector is a device that uses perpendicular electric and magnetic fields to filter charged particles by speed. Only particles with a specific velocity, where the electric force (qE) is exactly balanced by the magnetic force (qvB), pass undeflected. This equilibrium condition, E = vB, is fundamental for applications in mass spectrometers and particle accelerators.

Magnetic Field Direction Determination

Determining the direction of the magnetic field involves understanding the vector nature of the forces acting on moving charges and employing the right-hand rule. For a positive charge, the right-hand rule indicates that the direction of the magnetic force is perpendicular to both the velocity vector and the magnetic field. For electrons, which are negatively charged, the direction of the force is opposite. This method is key for predicting and controlling the trajectories of charged particles in electromagnetic fields.

Cyclotron Motion

Cyclotron motion describes the circular movement of a charged particle in a uniform magnetic field, where the magnetic force continuously acts as a centripetal force. The frequency of this circular orbit, known as the cyclotron frequency, is determined by the charge-to-mass ratio of the particle and the magnetic field strength. It is given by ? = qB/m, which is a central concept in the study of charged particle dynamics in magnetic fields and is applied in devices like cyclotrons.

Transcript

00:01 Hi there, so for this problem for part a, we are asked about what value of magnetic field will make a bin of electrons traveling to the right at a speed that is given and that speed is equal to 4 .8, this times 10 to the 6 meters per second to go undeflicted through a region where there is an uniform electric field.

00:28 And its magnitude is 8 ,400 balls per meter, pointing vertically up.

00:39 Okay? so for the beam of electrons to be undeflicted, the magnet of the magnetic force must equal the magnet of the electric force.

00:50 We assume that the magnetic field will be perpendicular to the velocity of the electron, of the electrons, so that the maximum magnetic force, is obtained.

01:00 So first of all, we know that the magnetic force is equal to the electric force.

01:05 So we know from this that this is the product between the charge, the speed, and the magnetic field.

01:11 And then this is equal to the part between the charge and the electric field.

01:15 Then solving in here for the magnetic field, we can cancel this with this, then we'll have that that is just simply the electric field divided by the speed.

01:24 So that will be 8 ,400 in units of 4.

01:28 Volts per meter.

01:30 This divided by the speed that we're given for this, that is 4 .8 times 10 to the 6 meters per second.

01:36 So from this we obtain a value of 1 .75 times 10 to the minus 3 tesla.

01:44 So that's a solution for part a of this problem.

01:49 Now for part b, we are asked about what is the disruption of the magnetic field if it is known to be perpendicular to the electric field? now, since the electric field is pointing up, the electric force is down.

02:05 Thus, the magnetic force must be up.

02:08 Using the right home rule with the negative electrons, the magnetit must be out of the plane.

02:14 Out of the plane.

02:20 So that's the solution for this.

02:22 It is out of the plane, formed by the electron velocity and the electric field...

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