An electron follows a helical path in a uniform magnetic...

An electron follows a helical path in a uniform magnetic field given by $\vec{B}=(20 \hat{i}-50 \hat{j}-30 \hat{k}) \mathrm{m} \mathrm{T}$. At time $t=0$, the electron's velocity is given by $\vec{v}=(20 \hat{i}-30 \hat{j}+50 \hat{k}) \mathrm{m} / \mathrm{s}$. (a) What is the angle $\phi$ between $\vec{v}$ and $\vec{B}$ ? The electron's velocity changes with time. Do (b) its speed and (c) the angle $\phi$ change with time? (d) What is the radius of the helical path?

Key Concepts

Uniform Magnetic Field

A uniform magnetic field is one in which the magnetic field vector has the same magnitude and direction at every point in space. In problems involving charged particle dynamics, it provides a consistent influence on the particle’s motion, ensuring that the only forces acting on the particle are those that arise from its interaction with the steady field.

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving in an electric and magnetic field, given by F = q(v × B). In a magnetic field alone, this force is always perpendicular to the velocity of the particle, meaning that it does no work and therefore does not change the particle’s speed. It is responsible for the circular motion of the particle's velocity component that is perpendicular to the magnetic field.

Velocity Decomposition

In a uniform magnetic field, the velocity of a charged particle can be decomposed into two components: one parallel to the magnetic field and one perpendicular to it. The parallel component is unaffected by the magnetic field and remains constant, while the perpendicular component is responsible for the circular (cyclotron) motion. This decomposition is fundamental in understanding helical trajectories in magnetic fields.

Helical Motion

Helical motion occurs when a charged particle moves through a uniform magnetic field with a velocity that is not aligned with the field. The constant parallel component results in linear motion along the field, while the perpendicular component causes circular motion. The superposition of these two motions results in a helical path, with a constant pitch and radius.

Cyclotron Motion and Radius

Cyclotron motion refers to the circular motion of the perpendicular component of a particle's velocity in a magnetic field. The radius of this motion, often called the Larmor radius, is given by r = m*v_perp/(q*B), where m is the mass of the particle, v_perp is the speed perpendicular to the magnetic field, q is the charge of the particle, and B is the magnetic field strength. This radius determines the curvature of the helical path.

Pitch Angle

The pitch angle is defined as the angle between the particle’s velocity vector and the magnetic field direction. It is determined by the ratio of the perpendicular and parallel components of the velocity. In a uniform magnetic field, the pitch angle remains constant over time since the magnetic field does no work on the particle, preserving the magnitudes of these velocity components.

Transcript

00:01 So this figure shows the electron following a helical path from here the velocity of electron is along this way now from here the electron follows helical path like this right the magnetic field is along this way now the cause of helical path is there are two causes for this path to occur one is due to the vertical component of this velocity b sine theta so the magnetic field acts on the vertical component of velocity so the force due to magnetic field tends to move the particle tends to move the electron in a circular path but also the horizontal component of the velocity because five which is not affected by the force due to magnetic field causes the electron to move forward so thus creating this creating spiral motion helical motion now it is given the intensity of b and v are given this is in miltasla this is meter per second so we are asked to find in part a of the question to find the the angle between v b and b so we have two vectors so we can use the identity of a dot product or a cross product so dot product will be easier to do at the moment so let me do that so b vector dot b vector by the definition of dot vector this should be equal to magnitude of b into magnitude of v into cost theta now if you find the magnitude of b and v so cost theta should be equal to b dot v upon b now what is b dot v this is 40 plus 1500 minus 1 ,500 now what is magnitude of b b let me write it here magnitude of b equal to root under 20 squared is how much 400 50 square is 2500 and 30 squared is 900 so this will give me the magnitude of b that is just b is equal to 61 .64 minnesota and minedasla and magnitude of he is also the the components are same 61 .64 meter per second.

03:14 All right so just use this values in here this will be 61 .644 square this will give me of theta equal to 2 by 19 and this implies theta equals to inverse of this cos inverse of this 83 .95 degrees.

03:49 So this gives me the angle between v and b.

03:53 So let's move on to the other part of the equations.

03:59 So the question says that the electrons velocity changes with time true...

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