Finding Radial Electric Field

In summary, the conversation discusses the properties and behavior of a beam of particles with mass m and charge Q traveling parallel to the z axis. To maintain the focus of the beam, an external uniform electric field Bo is applied parallel to the z axis and the beam is rotated with a constant angular velocity w. The conversation also covers the use of Gauss' Law to find the radial electric field in the beam on a cylinder of radius r < R, as well as finding the azimuthal velocity of a particle and the total force required to keep a particle on a circular path with radius r < R.
  • #1
jughead4466
6
0
Particles having mass = m and charge = Q travel parallel to the z axis, forming a beam of radius = R and uniform charge density = p. To keep the beam focused, an external uniform electric field [tex]_{}Bo[/tex], parallel to the z axis is provided, and the beam is made to rotate with a constant, uniform angular velocity = w

A: Use Gauss' Law to find the radial electric field in the beam on a cylinder of radius = r<R

I figured out this one and got E = pr/2[tex]\epsilon[/tex]

Sorry about the coding, I'm very new to this. Well that is supposed to be epsilon, and it should not be a power.

I could not figure out B or C though.

B: Find the azimuthal (tangential) velocity of a particle in the beam at r<R

C: Find the total (electric and magnetic) force required on a particle at r<R

D: Set force in "C" equal to the centripetal force required to keep a particle on a circular path of radius = r<R
 
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  • #2
edit. double post
 
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  • #3


Thank you for your question. I would be happy to provide a response to this content.

Firstly, to find the radial electric field in the beam on a cylinder of radius r < R, we can use Gauss' Law which states that the electric flux through a closed surface is equal to the enclosed charge divided by the permittivity of free space. In this case, the closed surface is a cylinder of radius r and the enclosed charge is the charge density multiplied by the volume of the cylinder, which is given by pπr^2w. Therefore, the electric flux is given by:

Φ = (pπr^2w)/ε0

Since the electric field is uniform and parallel to the z axis, the electric flux can also be written as:

Φ = E(r) * 2πrL

Where L is the length of the cylinder. Equating the two equations and solving for E(r), we get:

E(r) = (pwr)/ (2ε0)

This is the equation for the radial electric field in the beam on a cylinder of radius r < R.

Moving on to part B, to find the azimuthal or tangential velocity of a particle in the beam at r < R, we can use the equation for the Lorentz force, which states that the force on a charged particle in an electric and magnetic field is given by:

F = q(E + v x B)

Where q is the charge of the particle, E is the electric field, v is the velocity of the particle and B is the magnetic field. In this case, the particle is moving parallel to the z axis, so the magnetic field does not affect its velocity. Therefore, the tangential velocity can be found by equating the force to the centripetal force required to keep the particle on a circular path of radius r < R. This gives us:

qvB = mv^2/r

Solving for v, we get:

v = qBr/m

This is the equation for the tangential velocity of a particle in the beam at r < R.

Moving on to part C, to find the total force required on a particle at r < R, we can use the same equation for the Lorentz force and substitute in the values for E and B. This gives us:

F = q(E + v x B)

= q(E + v x 0)

= qE

Sub
 

1. What is the definition of radial electric field?

The radial electric field is a vector field that represents the electric field strength and direction at a particular point in space, pointing radially away from a positive charge and towards a negative charge.

2. How is the radial electric field calculated?

The radial electric field can be calculated using the equation E = kQ/r^2, where E is the electric field strength, k is the Coulomb's constant, Q is the charge, and r is the distance from the charge to the point in space where the field is being measured.

3. What factors affect the magnitude of the radial electric field?

The magnitude of the radial electric field is affected by the charge of the source, the distance from the source, and the medium through which the field is passing. It is also affected by any other charges in the vicinity.

4. How does the radial electric field influence the motion of charged particles?

The radial electric field exerts a force on charged particles, causing them to accelerate or decelerate depending on the direction of the field. This force is known as the electric force and is responsible for the movement of charged particles in an electric field.

5. What are some real-world applications of understanding radial electric field?

Understanding radial electric field is crucial in many fields, including electrical engineering, physics, and chemistry. It is used in designing electrical circuits, calculating the behavior of particles in particle accelerators, and studying the properties of materials. It also plays a role in technologies such as televisions, computer screens, and medical imaging devices.

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