A positron (q=+e) moves at 5.0 ×10^7 m / s in a magnetic field of magnitude 0.47 T. The magnet

A positron (q=+e) moves at 5.0 ×10^7 m / s in a magnetic...

A positron $(q=+e)$ moves at $5.0 \times 10^{7} \mathrm{~m} / \mathrm{s}$ in a magnetic field of magnitude $0.47 \mathrm{~T}$. The magnetic force on the positron has magnitude $2.3 \times 10^{-12} \mathrm{~N}$. (a) What is the component of the positron's velocity perpendicular to the magnetic field? (b) What is the component of the positron's velocity parallel to the magnetic field? (c) What is the angle between the velocity and the field?

Key Concepts

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving in a magnetic field. It is given by the equation F = qvB sin?, where q is the charge, v is the speed of the particle, B is the magnetic field strength, and ? is the angle between the velocity vector and the magnetic field. This concept is fundamental in understanding how magnetic fields affect the motion of charged particles.

Decomposition of Velocity

When a charged particle moves at an angle with respect to a magnetic field, its velocity can be broken down into two components: one that is perpendicular to the magnetic field and one that is parallel. The perpendicular component (v?) is responsible for the circular motion induced by the magnetic force, while the parallel component (v?) is unaffected by the magnetic field and leads to motion along the field lines.

Trigonometric Relationships in Vector Decomposition

The relationships between the total velocity and its perpendicular and parallel components are defined using sine and cosine functions. Specifically, the perpendicular component is v? = v sin? and the parallel component is v? = v cos?. These trigonometric relationships are essential for determining the angle between the velocity vector and the magnetic field when one or more components are known.

Transcript

00:01 For this question, we know that the positron has a charge just like the proton.

00:05 So that is positive 1 .6 times 10 to the negative 19th coulums.

00:10 And we know that the speed with which it moves is 5 .0 times 10 to the 7th meters per second.

00:17 And we also know that it is moving through a magnetic field of magnitude 0 .47 tesla.

00:24 And it experiences, therefore, a magnetic force that is 2 .3 times 10 to the negative 12 neutens.

00:34 And if you read the parts of the question, it suggests that there is actually an angle between these two vectors, because it is talking about, it is asking for the perpendicular component of velocity and the parallel component of velocity, which means in order for, in order for, us to solve part a, part b, and part c, we must first solve for the angle that exists between these two vectors.

01:03 And then we can solve for perpendicular component of velocity, and finally, the parallel component of velocity.

01:11 So in that regard, i'm going to solve for part c first, and then part a and then part b.

01:19 So according to the formula for finding me, magnetic force, it is equal to fb is equal to q times v crossed with b.

01:34 And because there is an angle between v and b, it expands to qvb times sign of that angle.

01:44 So we have these four values.

01:47 To solve for the angle, we must use inverse sign of an expression that involves these four letters.

01:56 So let's first isolate sign of theta, which comes out to be fb divided by q times v times b on the bottom...

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