Circular Motion in a Magnetic Field

Instructor: Matthew Bergstresser
Circular motion depends on an unbalanced force pointing towards the center of a circle. In this lesson, we explore the intricacies of the circular paths charged particles take in magnetic fields, like the mass of the particle and radius of the path.

Rubber Duckies and Charged Particles

Imagine a rubbery ducky moving towards the drain when bath water is draining. When it reaches the drain, it moves in circles around the drain. This is an analogy for what happens to charged particles that venture into a magnetic field. Sometimes the charged particle moves in a spiral-like helix and sometimes it doesn't. Let's focus on the phenomenon where a charged particle moves with constant circular motion when it enters a magnetic field.

Motion in the Field

Charged particles experience a magnetic force when they enter magnetic field. The only way they will feel a force due to the magnetic field is if they are not moving parallel to it. The way we know the direction the force acts on the charged particle is to use the right-hand rule. You arrange your right hand so that a backward ''L'' is made between your thumb and pointer finger. Then, stick out your middle finger so that it is perpendicular to both your pointer finger and thumb.


Right-hand rule
RHR


  • Pointer finger points in direction of charged particle's velocity.
  • Middle finger points in direction of magnetic field.
  • Thumb points in direction of force.

This rule applies only to positive charges. If the charge is negative, simply reverse the direction of the force.

Let's say a positive charge is moving downward, and a switch is flipped turning on a magnetic field that points out of the screen. Immediately the charge feels a force to its right, which changes its trajectory, and it curves to its right. The charge now feels a force perpendicular to its new velocity direction, and it deflects to its right again. This continues for the remaining time the magnetic field is on, and the particle moves in a circular path as shown in Diagram 1.


Diagram 1. Circular motion of positive charge in a magnetic field
circle


Diagram 1 shows the centripetal acceleration and related centripetal force as red arrows. Centripetal means ''center-seeking'', and the magnetic force the charge feels is not countered by another force. This means the magnetic force is the centripetal acceleration. The centripetal acceleration of the charge is given by Equation 1.


Equation 1
ac

  • ac is centripetal acceleration in m/s2.
  • v is velocity in m/s.
  • r is the radius of the circle in meters.


Accelerations don't happen without unbalanced forces, and since we are dealing with circular motion, we use the centripetal force equation shown in Equation 2.


Equation 2
Fc


  • Fc is centripetal force in newtons (N).
  • m is mass in kilograms (kg).

The magnitude of the magnetic force on a charged particle moving through a magnetic field is given in Equation 3.


Equation 3
F_mag


  • Fm is the magnetic force in newtons.
  • q is the value of the electrical charge in coulombs (C).
  • B is the magnitude of the magnetic field in tesla (T).

The value of θ is 90° because the velocity is perpendicular to the magnetic field. The sin of 90° is 1, leaving qvB, which we plug into the left side of Equation 2 giving us Equation 4.


Equation 4
qvb


Separating Nuclear Isotopes

Let's work on a case study involving the separation of nuclear isotopes.

Around 99% of uranium ore mined from the earth is uranium-238, and uranium-235 is less than 1% of the uranium ore.

U-235 is the isotope of uranium used in nuclear power plants and nuclear weaponry. Most nuclear reactors require 0.7% to 5% U-235, and nuclear weapons require at least 90% U-235. If a country has been forbidden to enrich uranium beyond 5% U-235, inspectors can use equipment that separates the isotopes of uranium by using a machine that pushes them through a magnetic field. The ratio of the two isotopes can then be determined. Let's calculate the radii of a +1 charged U-238 ion, and a +1 charged U-235 ion pushed into a constant magnetic field of 1.5 tesla. They will enter the magnetic field at 6.3 x 105 m/s.

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