An electron and a proton are each moving at 735 km/s in perpendicular paths as shown in Fig. E28

An electron and a proton are each moving at 735 km/s in...

Key Concepts

Magnetic Field Produced by Moving Charges

Moving electric charges generate magnetic fields, and the contribution of a point charge moving with constant velocity is described by the Biot–Savart law. This law relates the velocity of the charge, the relative position vector, and the resulting magnetic field, emphasizing that the field is oriented perpendicularly to the motion and the position vector.

Coulomb's Law

Coulomb's law describes the electric force between two point charges, stating that the force magnitude is proportional to the product of the charges and inversely proportional to the square of the distance between them. It provides the basis for calculating electric fields and forces in electrostatic interactions.

Superposition Principle in Electromagnetism

The superposition principle asserts that the net electric or magnetic field produced by multiple sources is the vector sum of the fields created by each source individually. This concept is crucial for analyzing systems where several charges or currents are present, as it allows for independent calculation and subsequent summing of their respective contributions.

Lorentz Force Law

The Lorentz force law determines the force exerted on a charge moving in electric and magnetic fields, given by F = q(E + v × B). It explains how a moving charge experiences a force due to both electric fields (Coulomb interactions) and magnetic fields (originating from other moving charges), and it is essential for calculating the trajectories of charges in electromagnetic environments.

Right-Hand Rule

The right-hand rule is a mnemonic for determining the direction of the magnetic field resulting from a moving charge or current, as well as the direction of the force experienced by a moving charge in a magnetic field. It provides an intuitive and practical method for resolving the vector directions in electromagnetic problems.

Transcript

00:01 Okay, so problem 28 .8, you have an electron of proton that are each moving at a certain speed in perpendicular paths that's shown.

00:08 And at the instant that they are at these positions, we're supposed to find the magnitude direction of the total magnetic field they produce at the origin to start.

00:16 Okay, so first off, we're going to start off with the electron.

00:22 We have the magnetic field, and that's going to be due to a moving point charge.

00:30 So that's going to be mu not over 4 pi.

00:37 And then we have q, the charge of the electron.

00:41 And i write down down as negative.

00:44 And then we have the distance squared.

00:46 So that's 5 nanometers squared.

00:52 And then we have the velocity.

00:54 And that's the cross product of the velocity with the r vector.

00:59 So that's going to be this r vector here.

01:06 It's from the point to the point where the charge is to the point at the origin, that straight line down.

01:12 So then we have the velocity, which just had the number, the magnitude, and we have the direction for the velocity is negative i -hat.

01:22 That's to the left, so negative i -hat, cross -product with the vertically down vector, so that's negative j -hat.

01:33 Okay, and so what we're left with is numbers up to the point where we have a cross -product.

01:37 Product and so that's going to come out to a negative 0 .5 milatessla and the i -hat negative i -hat cross negative j hat comes out to be k -hat so then we can rewrite this as 0 .5 milat tesla in negative k -hat direction so at the origin this produces the magnetic field vector into the page.

02:06 Okay now for the proton we do the same thing.

02:11 What's different is we have a different distance, and we have different cross product.

02:19 So i'll write down this qe this time for positive.

02:21 That's the charge of a proton over four nanometers squared.

02:30 The velocity is the same, 735.

02:33 And then we have that as the velocity vector.

02:38 The direction is down, so negative j hat.

02:40 And then the r vector this time is in towards the origin.

02:46 So it's going to cross product with negative i hat.

02:51 And this is going to be negative j hat across the negative i hat.

02:54 This is going to be negative k hat.

02:56 So again, we're going to have a positive number out front like we have here, but then negative k hat.

03:03 So that's going to be 0 .7.

03:08 I'm glitching here.

03:10 0 .736.

03:16 Again, i'm getting some glitch in here, 0 .7 .436 mil -tesla and a negative k -hat direction.

03:30 Okay, and then we can add these together.

03:32 So then in all, this would be 1 .236 miltessla in the negative k hat direction.

03:49 Okay, so b, the magfield of the electron, this is at the location of the proton.

03:55 So it's a little different.

03:57 We have to find the magnetic field that the election causes at an angle.

04:02 It's not straight perpendicular to its velocity.

04:07 So this time it's an angle back to the proton.

04:11 So that's going to change our cross part a little bit.

04:14 So what we need to do is we need to just start all by writing down with the magnet field of moving charges.

04:21 So that's going to be a magnet field.

04:24 Fuel is going to be mu not over 4 pi, same as before, electron charge.

04:34 And we have over the distance squared.

04:41 So what's this distance squared? so that's going to be, let's see, pythagorean theorem.

04:48 So we have 5 squared plus 4 squared equals 41 squared.

04:58 Square, 41.

05:00 So then we can write is we write 41.

05:05 And let's distinguish 41 nanometer squared between the other 41 nanometer square.

05:11 In this case, we only are squaring the units themselves...

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