An alpha particle (charge +2 e ) and an electron move in...

An alpha particle (charge $+2 e$ ) and an electron move in opposite directions from the same point, each with the speed of $2.50 \times 10^{5} \mathrm{~m} / \mathrm{s}$ (Fig. E28.4). Find the magnitude and direction of the total magnetic field these charges produce at point $P,$ which is $8.65 \mathrm{nm}$ from each charge.

Key Concepts

Magnetic Field of a Moving Charge

A moving charge produces a magnetic field in the surrounding space, which can be mathematically described using the Biot–Savart law. The magnetic field depends on the magnitude of the charge, its speed, and the geometry of the situation (i.e. the relative position of the observation point to the track of the moving charge). This concept is central when calculating the magnetic effects of individual charged particles in motion.

Right-Hand Rule

The right?hand rule is a mnemonic that helps determine the direction of the magnetic field induced by a moving positive charge. By pointing the thumb in the direction of the velocity, the curl of the fingers indicates the circular direction of the magnetic field lines. This principle is important for establishing the orientation of the field produced by each charge, taking into account the signs of the charges when determining the actual direction of the magnetic field.

Superposition Principle

The superposition principle states that the total magnetic field at any point is the vector sum of the magnetic fields produced by all individual sources. When dealing with multiple moving charges, this principle allows us to compute the overall magnetic effect by first calculating each contribution separately and then adding them together while considering both magnitude and direction.

Transcript

00:01 Problem 28 .4, we have an awful particle that's this here, an electron that move in opposite directions from the same point.

00:09 And we have each with the same speed, and we're supposed to find a magnitude direction of the total magnetic field of these charges that they produce at point b, which is a certain distance way 8 .65 nanometers from each charge.

00:20 Okay, so we need to solve this using the equation for the b field of a moving point charge.

00:29 So that equation is a is 5x0 equal to mu not over 4 pi.

00:38 And we have q for velocity, velocity vector, cross r hat over r squared.

00:50 Ok, so q, that's just the charge of each of them, v.

00:54 That's the vector for each of them in its different directions.

01:00 And r is just this distance given.

01:02 R hat is going to be the vector extending from there.

01:06 Order point out to the point p.

01:08 So we need to write down what this is going to be for each of them.

01:13 So i'm going to start off with the equation for the alpha particle.

01:18 So that's going to be mu not over 4 pi.

01:24 And it's going to be q is 2e.