At the equator, near the surface of the Earth, the magnetic...

At the equator, near the surface of the Earth, the magnetic field is approximately $50.0 \mu \mathrm{T}$ northward, and the electric field is about $100 \mathrm{~N} / \mathrm{C}$ downward in fair weather. Find the gravitational, electric, and magnetic forces on an electron in this environment, assuming that the electron has an instantaneous velocity of $6.00 \times 10^{6} \mathrm{~m} / \mathrm{s}$ directed to the east.

Key Concepts

Lorentz Force

This represents the total force experienced by a charged particle when both electric and magnetic fields are present. It is the vector sum of the electric and magnetic forces, expressed as F = q(E + v × B), and is a fundamental concept in electromagnetism for analyzing the motion of charged particles.

Magnetic Force

Magnetic force on a moving charged particle is given by F = q(v × B), where v is the particle’s velocity, B is the magnetic field, and × denotes the cross product. This force acts perpendicular to both the velocity of the particle and the magnetic field, with its direction determined by the right-hand rule (or the left-hand rule for negative charges).

Gravitational Force

This force is due to the attractive interaction between masses. For objects near Earth’s surface, it is commonly approximated as F = mg, where m is the mass and g is the acceleration due to gravity. It acts uniformly downward toward the center of the Earth.

Electric Force

This force acts on a charged particle placed in an electric field and is given by F = qE, where q is the electric charge and E is the electric field vector. The direction of the force depends on the sign of the charge, with positive charges being pushed in the direction of the field and negative charges in the opposite direction.

Transcript

00:04 For problem 29 -8, we're asked to find the forces experienced by an electron due to the force of gravity, the electric field, and the magnetic field produced by the earth as that electron moves near the earth's equator.

00:25 We are given the magnetic field has a strength of 50 .0 microtestlas, and it is pointed northward.

00:35 We're given that the electric field has a strength of 100 newtons per kulam, and it is pointed downward toward the center of the earth.

00:48 We know from classical mechanics that the acceleration due to gravity of any mass acting at the earth's surface, acting on any mass at the earth's surface, is 9 .81 meters per second, and that acceleration is our energy.

01:07 Always downward toward the center of the earth.

01:11 We know that the mass of the electron, and i did have to look this up, but that's a constant in it.

01:17 The electron has a massive 9 .11 times 10 to the negative 31 kilograms.

01:25 And the electron charge, i'm denoting that here as q sub e, has a value of negative 1 .60 times 10 to the 19th kulams.

01:39 In the problem, crucially, we're given that the electron is going to move eastward at a velocity of 6 .00 times 10 to the 6th meters per second.

01:52 So for me, in order to keep my directions straight in this scenario, the earth's surface near the equator, i prefer to assign conventional cartesian coordinate system.

02:09 To the geographical coordinates or directions of north, south, east, and west.

02:15 Here i've done it as follows.

02:17 I assign north to the positive going unit vector k direction or the z axis, the vertical axis.

02:24 I've assigned west to the y axis or the unit vector j direction.

02:32 And i have assigned upward from the earth's surface to the sky to the, the univector i direction or the x direction i've gone went ahead and denoted the field orientation of the magnetic field described here as well as the electric field that goes into the plane of the screen a perpendicular to the plane of the screen same with the acceleration due to gravity vector g that is also perpendicular to the plane of the screen and going away from the viewer into the screen.

03:14 And finally, the electron goes in a eastward direction, which i have incorrectly denoted here.

03:35 I have it going westward.

03:38 Well, no, no, i have it going eastward.

03:40 It looks as if i have my axis, my direction's incorrect.

03:44 Okay, so let's fix that.

03:58 Okay.

04:01 Let's put west to its proper place here.

04:05 East, oops, that's my issue.

04:07 Let's move that temporarily.

04:10 Let's put east to its proper place here...