In Fig. 28-58, an electron of mass m, charge -e, and low...

In Fig. $28-58$, an electron of mass $m$, charge $-e$, and low (negligi- ble) speed enters the region between two plates of potential difference $V$ and plate separation $d$, initially headed directly toward the top plate. A uniform magnetic field of magnitude $B$ is normal to the plane of the figure. Find the minimum value of $B$ such that the electron will not strike the top plate.

In Fig. $28-58$, an electron of mass $m$, charge $-e$, and low (negligi-
ble) speed enters the region between two plates of potential difference $V$ and plate separation $d$, initially headed directly toward the top plate. A uniform magnetic field of magnitude $B$ is normal to the plane
of the figure. Find the minimum value of $B$ such that the electron will not strike the top plate.

Key Concepts

Lorentz Force

The Lorentz force is the fundamental force experienced by a charged particle in the presence of electric and magnetic fields. It is given by F = q(E + v × B) and explains how both fields influence the trajectory of a particle, such as causing deflections or deviations from a straight-line path.

Electric Field Between Parallel Plates

Between two parallel plates with a potential difference, a uniform electric field is established. This field is typically calculated as E = V/d, where V is the potential difference and d is the separation of the plates. This concept is essential in understanding how the particle is accelerated or deflected by electric forces.

Magnetic Force on a Moving Charge

A moving charged particle in a magnetic field experiences a force perpendicular to both its velocity and the magnetic field direction, described by F = |q| v B. This force causes the particle to move in a curved or circular path, which is a key idea in manipulating and analyzing particle trajectories in various electromagnetic setups.

Motion of Charged Particles in Combined Electric and Magnetic Fields

When charged particles enter a region where both electric and magnetic fields are present, the resultant motion is determined by the interplay of these forces. Understanding this interaction, including how to balance the magnetic deflection against the electric acceleration, is crucial for solving problems that require precise control of particle trajectories, such as ensuring a particle avoids a particular region.

Transcript

00:01 So you can have situations where there are both electric forces and magnetic force working on a single charge.

00:10 Here we're going to take a look at an example where you have an electric force on a charge in the amount of here it's an electron.

00:22 So e times an electric field.

00:24 We are going to have that operating due to a potential difference v between two plates separated by distance d.

00:39 So electric force is e v over d.

00:47 At the same time, we are going to have this charge an electron in this case in a magnetic field that is crossed with respect to the electric field.

00:59 Field.

01:00 We'll talk a little bit about that, but the magnitude of the magnetic force is equal to e times speed times magnetic field.

01:12 Hopefully that looks different than the potential.

01:17 And the idea is we're going to introduce the electron in between the two plates, the negative plate, and it will accelerate towards the upwards, upwards due to the positive plate.

01:34 And we'll be going faster and faster and faster.

01:38 And we want to know what is the bare minimum situation we need to get the electron from hitting the top plate.

01:48 So here's the thought.

01:50 The speed of the electron is changing.

01:55 We would like it to curve due to the magnitude.

01:59 Field, so the magnetic force we're going to set equal to mass v squared over r.

02:09 And because the speed is changing, the radius of the motion will change.

02:16 And we don't know what kind of arc this will follow.

02:21 It will follow some sort of curved arc, especially if we have the magnetic field at a crossed angles.

02:31 So let's, oh, let's just make the magnetic fields into the page, for example.

02:39 Not necessary, but it'll follow some kind of arc to the upper plate.

02:47 And if we do not want the electron to hit the plate, well, we could turn it around.

02:55 So let's take a look at a couple paths, path one versus path two.

03:07 So that radius is kind of a complicated function of both the v and the b.

03:13 But certainly path one we're going to hit and path two we're going to miss.

03:25 What we're after is a condition where it just barely misses.

03:37 And so i can draw what that path looks like.

03:40 So path two misses by a good mississippi mile, i suppose.

03:46 Say that, but to just barely miss what we want it to do is be moving parallel to the plate when it gets to the top...

Please give Ace some feedback