A long straight wire carries a current of 3.2 A in the positive x -direction. An electron, trav

A long straight wire carries a current of 3.2 A in the...

A long straight wire carries a current of $3.2 \mathrm{~A}$ in the positive $x$ -direction. An electron, traveling at $6.8$ $\times 10^{6} \mathrm{~m} / \mathrm{s}$ in the positive $x$ -direction, is $4.6 \mathrm{~cm}$ from the wire. What force acts on the electron?

Key Concepts

Magnetic Field around a Long Straight Conductor

A long straight current-carrying wire creates a magnetic field that circulates around the wire. The magnitude of this field at any point is given by the equation B = ??I/(2?r), where ?? is the permeability of free space, I is the current, and r is the distance from the wire. This concept is fundamental in electromagnetism and is derived from the Biot–Savart law.

Lorentz Force on a Moving Charge

A charged particle moving in a magnetic field experiences a force known as the Lorentz force. This force is given by F = q(v × B), where q is the charge, v is the velocity vector of the particle, and B is the magnetic field vector. The cross product indicates that the force is perpendicular to both the velocity and the magnetic field, which is essential in understanding the motion of charged particles in magnetic fields.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the magnetic field created by a current-carrying conductor, as well as the direction of the force on a moving charge in a magnetic field. When applied to a current-carrying wire, point your thumb in the direction of the current; your curled fingers then show the direction of the magnetic field lines around the wire. For the Lorentz force, the rule helps in finding the direction of v × B.

Transcript

00:01 All right, so we have a wire with a current and an electron, not in the wire, moving in the positive x direction.

00:11 Also, the current is moving in the positive x direction.

00:14 All right, to calculate the value, we would like to know what the force of the magnetic field is on this electron.

00:23 Of course, we need to calculate what the magnetic field is where the electron exists to, get the force.

00:31 All right.

00:32 So we need, so the force is the size of the charge, multiplied by the velocity of the electron, multiplied by the magnetic field, which is mu zero, times the current, which is the wire, the current through the wires, what's causing the magnetic field divided by 2 pi a, which is the distance away from the current that we are looking for the magnetic field in the force, so the distance away the electron is.

01:10 All right.

01:11 So the charge of an electron is 1 .6 times 10 to the negative 19.

01:17 The velocity of this electron is 6 .8 times 10 to the 6 .8 times 10 to the 6 .7 .5 .5 .5.

01:26 6 meters per second.

01:29 Then mu zero is the constant 4 pi times 10 to the negative 7 and the current is 3 .2 divided by 2 pi and the distance away the electron is from the current and that is a 4 .6 centimeters which of course is 0 .0 .0 .0.

01:57 No, too many zeros.

02:01 Just kidding.

02:02 There we go.

02:04 Four, six.

02:07 Calculate that all out, and you should get a force of 1 .5 times 10 to the negative 17 newtons.

02:19 Now i would like direction.

02:21 All right, so the direction of the magnetic field, if we take the right hand rule, we would take the current going to the right and the distance away going down.

02:36 So fingers to the right, thumb down, takes the palm out towards me.

02:41 No, wait a second.

02:43 Hold on...

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