A Cycloidal Path. A particle with mass m and positive charge...

A Cycloidal Path. A particle with mass $m$ and positive charge $q$ starts from rest at the origin shown in Fig. $P 27.91$ . There is a uniform electric field $\vec { E }$ in the $+ y$ -direction and a uniform magnetic field $\vec { \boldsymbol { B } }$ directed out of the page. It is shown in more advanced books that the path is a cycloid whose radius of curvature at the top points is twice the $y$ -coordinate at that level. (a) Explain why the path has this general shape and why it is repetitive. (b) Prove that the speed at any point is equal to $\sqrt { 2 q E y / m }$ (Hint: Use energy conservation.) (c) Applying Newton's second law at the top point and taking as given that the radius of curvature here equals $2 y ,$ prove that the speed at this point is 2$E / B$ .

A Cycloidal Path. A particle with mass $m$ and positive charge $q$ starts from rest at the origin shown in Fig. $P 27.91$ .
There is a uniform electric field $\vec { E }$ in the $+ y$ -direction and a uniform magnetic field $\vec { \boldsymbol { B } }$ directed out of the page. It is shown in more
advanced books that the path is a cycloid whose radius of curvature at the top points is twice the $y$ -coordinate at that level. (a)
Explain why the path has this general shape and why it is repetitive. (b) Prove that the speed at any point is equal to $\sqrt { 2 q E y / m }$
(Hint: Use energy conservation.) (c) Applying Newton's second
law at the top point and taking as given that the radius of curvature
here equals $2 y ,$ prove that the speed at this point is 2$E / B$ .

Key Concepts

Lorentz Force

The Lorentz force is the combined electric and magnetic force acting on a charged particle: F = q(E + v × B). This force governs the motion of the particle in the presence of electromagnetic fields. In this problem, the electric field accelerates the particle in the y-direction, while the magnetic field exerts a perpendicular force which curves its path. This interplay is foundational for understanding the development of the cycloidal trajectory.

Cycloidal Motion

Cycloidal motion describes the path that a point on the rim of a circle will trace as it rolls along a straight line. In the context of charged particle dynamics, although the particle is not physically attached to a circle, the influence of constant electromagnetic forces produces a repeating path that mathematically resembles a cycloid. This is due to the periodic interplay between the linear acceleration by the electric field and the constant curvature imposed by the magnetic field, resulting in a repetitive curved path.

Energy Conservation in Electromagnetic Systems

Energy conservation, particularly the work-energy theorem, applies to charged particles in electric fields. As the particle accelerates due to the electric field, the work done on the particle converts entirely into kinetic energy, which explains why its speed at any point depends only on the y-coordinate. This relationship is crucial to deriving expressions like v = ?(2qEy/m), illustrating one way that potential energy (due to position in the field) transforms into kinetic energy.

Centripetal Force and Radius of Curvature

The concept of centripetal force is essential when analyzing the curvature of a particle's trajectory. At any point in circular motion, the centripetal force required to keep the particle moving in a curved path is provided by the magnetic component of the Lorentz force. By applying Newton’s second law at the top of the cycloidal path and linking the required centripetal acceleration (v²/?) to the magnetic force, one can derive relationships involving the radius of curvature, such as the given condition where ? = 2y at the top points, which further connects to the particle’s speed.

Transcript

00:01 So we need to show that the charge particle moving in the uniform electric field and uniform magnetic field showing this graph would follow a cycloader path.

00:12 So if we want to show this rigorously, we need to solve the equation of motion for both x and y and show that they satisfy the cyclotor equation.

00:24 But for this problem, we only need to do it qualitatively.

00:30 So initially, the particle is at rest, so the lecture field would give a change of velocity along the positive y direction.

00:43 And using the right -hand rule, we can show that initially the lorentz force points towards the positive x direction.

00:54 Since the lorentz force is always perpendicular to the direction of velocity, it will slowly rotate the direction of the direction of.

01:07 The velocity clockwise.

01:13 And as the particle moves upward along the wide direction, the velocity would increase is.

01:23 And as the velocity increases, the magnitude of the range force increases.

01:39 So the curvature of radius would slowly increase.

01:49 And when the direction of the velocity has rotated 90 degrees, the charged particle would add its highest point, and then it will slowly move downward.

02:16 And when it hits the x -axis, the velocity would become zero again, because the x -axis is the equal potential surface for the electric field, and the lorentz force doesn't do any work on the charged particle.

02:35 So when the particle hits the x -axis, the velocity, the velocity, it gets zero again, and the whole motion starts again, so it's a repetitive motion.

02:51 Right.

02:51 Now for part b, we need to prove that the speed at any point is equal to this form...

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