An electron moves at 0.100c as shown in Fig. E28.3. Find the magnitude and direction of the magn

An electron moves at 0.100c as shown in Fig. E28.3. Find the...

Key Concepts

Right-Hand Rule and Charge Sign Considerations

The right-hand rule is employed to determine the direction of the magnetic field generated by a moving charge. When the charge is positive, the thumb points in the direction of the velocity and the curled fingers indicate the field direction. However, for a negative charge such as an electron, the observed field directions are reversed compared to those predicted by the conventional right-hand rule. This nuance is crucial for correctly identifying the orientation of the magnetic field at various observation points.

Biot-Savart Law

This fundamental principle in electromagnetism quantifies the magnetic field generated by a moving charge or a current element. It expresses the magnetic field as proportional to the charge, its velocity, and the sine of the angle between the velocity and the position vector, while inversely proportional to the square of the distance. The law is central to calculating the magnetic fields at various points around a moving charge.

Magnetic Field from a Moving Charge

This concept explains that a charged particle in motion produces a magnetic field around it, analogous to the field produced by a current-carrying wire. The magnitude and orientation of the field depend on the speed of the charge and its distance from the observation point. It is instrumental in understanding how different geometric positions relative to the moving charge experience different field directions and magnitudes.

Vector Cross Product

In the context of magnetic field calculations, the vector cross product is used to determine the direction of the field produced by a moving charge. It involves taking the cross product of the velocity vector of the charge and the relative position vector from the charge to the point of interest. This operation results in a vector that is perpendicular to the plane formed by these two vectors, thus setting the direction of the magnetic field.

Transcript

00:01 Hi there.

00:03 So for this problem, we need to calculate the magnetic end direction of the magnetic field that this electron produces at some given points.

00:18 Each of them are out of the distance are of two micrometers, and the electron has a velocity of 0 .1 times the speed of light.

00:33 The first question is to calculate the magnetic field at a point a and b.

00:43 First, we can see from the figure the point b is at some given angle from the electron.

00:55 To obtain the magnetic field for all of these points, we are going to use the magnetic field produced by a moving charge.

01:06 And that formula or equation is the following, where the angle theta is the angle between the vector of the velocity with the vector of the distance to the point where we want to measure the electric field.

01:51 So the magnitude can be obtained from this expression and the direction can be obtained from the right -hand route.

02:08 So first we are going to attain the magnitude.

02:12 In this case for a, we're going to call this for a, is the permeability of free space.

02:23 Over 4, the charge in this case is the same for everyone and we are trading with electron.

02:34 And we can see that the vector are from the electron to the point form an angle of 30 degrees.

02:55 Because if this angle it sits degrees and we know that the angle that form the two axis is 90 degrees, which is subtract from 90 degrees, 60 degrees, and that gives us the third degrees that we want.

03:21 And everything else is just what we are given for these problems.

03:29 So we are going to substitute all of those values.

03:32 We know that the permeability of free space, it's 4 pi times 10 to the minus 7 newton per second second square over column square, all of these over 4 pi.

04:09 And the charge is it corresponds to the charge of an electron which is 1 .6 times 10 to the minus 19 colon and the velocity of this electron is given a 0 .1 the speed of light we also know that the speed of light is three times 10 to the 8 meters per second so we put that in here 0, 10 to the 8 meters per second.

04:58 And the angle, as we said, is 30 degrees.

05:05 And the distance to that point is 2 micrometers, which is 2 times 10 to the minus 6 meters.

05:18 With all this, we obtain a value of, oh, remember that distance is to the square, and this gives us 6 times 10 to the minus 8 tesla.

05:38 That's the units of the electric fuel.

05:43 So that's for this point, and the direction could be obtained by the right -hand side.

05:52 And that is given by the product between the vector, the velocity vector, and the r -better.

06:08 So the right -hand sign is that you put your right -hand hand in the direction of the b -bet -bet -w.

06:24 And you go from there to just our bed door and you will see that your thumb is pointing inward.

06:41 So it is inward.

06:44 If we put like another ad sets, it is going to point in this direction inward.

06:55 And if we set that as, if we set this as positive and this as negative, this positive, this negative, then it will be negative and direction inward or negative if you set it in that way.

07:24 Now for part being, oh well, for this same part, but to obtain the, um, magnitude of the electric field for the point b, we will have that the angle between the vector b and the angle between the distance, the position vector of the point b, it's, we need to sum all of this, we need to some 90 degrees, i'll put in another color, you need to some, this 90 degrees, degrees and also this 60 degrees.

08:12 So we will have 90 plus 60 degrees, that's 150 degrees...

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