An electron moves with a speed of 1.0 ×10^7 m / s in the...

An electron moves with a speed of $1.0 \times 10^{7} \mathrm{m} / \mathrm{s}$ in the directions shown in Figure $\mathrm{P} 24.23 .$ A $0.50 \mathrm{T}$ magnetic field points in the positive $x$ -direction. For each, what is the magnetic force on the electron? Give your answers as a magnitude and a direction. a. b.

Key Concepts

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving in electric and magnetic fields. In the context of a magnetic field only, the force is given by F = q(v × B), where q is the particle's charge, v is its velocity vector, and B is the magnetic field vector.

Cross Product

The cross product in the formula F = q(v × B) determines both the magnitude and the direction of the magnetic force. The magnitude of v × B is |v||B|sin(?), where ? is the angle between the velocity and magnetic field vectors, and its direction is perpendicular to the plane formed by these vectors.

Right-Hand Rule

The right-hand rule is a mnemonic for determining the direction of the cross product of two vectors. For a positive charge, you point your fingers in the direction of the first vector (v) and curl them toward the second vector (B); your thumb then indicates the force direction. For a negative charge, such as an electron, the force has the opposite direction.

Particle Charge and Sign

The sign of the charged particle, specifically that of an electron being negative, plays a crucial role in determining the actual direction of the force. Even though the magnitude is derived from the cross product, the negative sign reverses the direction of the force relative to what the right-hand rule would indicate for positive charges.

Transcript

00:01 In the given problem here, these are the x -axis, the z -axis, and the y -axis, x -axis towards right, z -axis in upward direction, and y -axis into the plane of paper.

00:27 So here this will be negative x -axis towards left, negative x -axis in downward direction, and negative y -axis perpendicularly coming out of the plane of paper.

00:48 Here this is negative y -axis.

00:51 The direction of magnetic field is given to be a long positive x -axis, in the direction of velocity of electron is given to be a long negative y -axis.

01:13 And the magnitude of this magnetic field is 0 .50 tesla, and the speed of this electron is 1 .50 tesla, and the speed of this electron is 1 .10.

01:25 0 .0 into 10 dash 7 meter per second.

01:31 Angle between the velocity vector and magnetic field vector means theta is 90 degree.

01:39 So using the expression for magnetic lawrence force experienced by the electron, which says f is equal to e into v cross b and as a as this is electron, so we can use a negative sign here.

02:04 So it's magnitude will come out to be e, v, b, sign theta, and the direction of this force using right hand screw rule.

02:24 The direction of b cross b is along positive z axis, but as it is having a negative sign also.

02:35 So the direction will become along negative xx.

02:43 Now, plugging in all known values for electronic charge, this is 1 .6 into 10 -p.

02:49 Minus 19.

02:51 Kulam, for v, this is 1 .0 into 10 -p.

02:54 7 meter per second.

02:57 For magnetic field, this is 0 .50 tesla...