An electron is moving vertically downwards at any place. The direction of magnetic force acting

An electron is moving vertically downwards at any place. The...

Key Concepts

Lorentz Force Law

This fundamental law in electromagnetism describes the force on a charged particle moving in a magnetic field. The force is determined by the cross product of the particle’s velocity vector and the magnetic field vector, making it perpendicular to both. This concept is crucial in understanding how moving charges interact with magnetic fields.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the force experienced by a positive charge moving in a magnetic field. By pointing the fingers in the direction of the velocity and curling them toward the magnetic field, the thumb points in the direction of the force. This rule helps visualize the cross product operation in the Lorentz force law.

Effect of Charge Sign on Force Direction

The sign of the charge affects the direction of the magnetic force relative to that predicted by the right-hand rule. For negatively charged particles, like electrons, the force direction is opposite to that given by the rule, which is an essential modification when determining the actual force direction on such particles.

Earth's Magnetic Field

Earth's magnetic field, which has both horizontal and vertical components, influences charged particles in its vicinity. The horizontal component plays a significant role in determining the direction of the magnetic force on moving charges. Understanding the orientation of this component is important when applying the Lorentz force law to scenarios involving Earth’s magnetic environment.

Transcript

00:04 For problem 29 -2, we are asked to consider an electron near the earth's equator, and we're asked to find which direction it would tend to deflect, given a series of velocities in various directions, downward toward the ground.

00:29 For part b, we're asked what direction the electron would deflect, for a northward velocity, the electron with a northward velocity.

00:41 For part c, with electron having a westward velocity, and for part d with the electron having a southeastward velocity.

00:52 To set up, we'll note that the electron, in the case of our lawrence force equation, electrons have a negative value.

01:07 They're negatively charged.

01:09 So for a q, which i've represented here is this green dot, we're going to give that the value negative e.

01:16 E standing for the amount of electron charge in coulons.

01:21 Secondly, we know about the earth that its geologic magnetic field points upward from the south pole to the earth.

01:36 The north pole, which i've indicated as such on my diagram, the earth being the blue, big blue circle, and its natural magnetic field, for lack of a better term, is pointing from south to north.

01:53 I've drawn the geographical directions for earth as such, but for me, it's easier for me to assign those geographical directions, typical cartesian coordinates of the x, y, and z axes.

02:17 I've oriented them here relative to the geographical axes as follows.

02:24 The positive z direction is parallel to our north direction.

02:28 Negative z goes southward.

02:32 Our y -axis, positive going, is eastward, and our x -axis is going out of the plane of the screen toward me.

02:46 That's upward from the ground.

02:49 The negative z -axis is going downward to the ground.

02:55 And that's the coordinate axes, cartesian three -dimensional coordinate axes we're going to use for this problem.

03:04 So for the case of earth's geological magnetic field, we're going to represent that as its magnitude b times its direction northward, which for us is the z unit vector in the positive direction.

03:24 So for the first case of this problem, we're asked what direction an electron would deflect in the presence of a northward magnetic field from the earth if that electron is moving at a velocity downward that is toward the ground, in our case would be in the negative x direction for the coordinate system that i've just described and set up here.

03:52 So our first couple factors of the lawrence -force equation are going to be scalar q times vector e equals electron charge e times v -a -b in the direction.

04:06 Of positive x now because the two negatives, one for the kulum, i mean one for the electron charge and one for the direction of the velocity and the negative x -word direction, those two negatives make a positive.

04:23 So next we apply the earth's magnetic field via cross product, the scalar portion i've written as e times v -a -b times field strength b.

04:41 And for the cross -product, i put that all over on as far right as possible in this factor of positive x direction, cross -positive z direction...

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