PhysicsLAB Resource Lesson
Introduction to Magnetism

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Magnetic fields can be defined as the regions surrounding a magnet where another magnet or a moving electric charge will feel a force of attraction or repulsion. They are created by moving charges. The strengths of magnetic fields are measured in a unit called Teslas, T. In formulas, the strength of a magnetic field is represented by the variable B. In a later lesson we will learn that a tesla is equivalent to
1 tesla  = 1 newton per meter per amp
This definition is based on the fact that if two 1-meter long current-carrying wire segments, each having 1 ampere of current, are either attracted to or repelled from each other by one newton of force, then the strength of the magnetic field produced by either current-carrying segment equals 1 tesla at the location of the other wire. 
As shown below, magnetic field lines emerge from the North pole of a magnet and enter the South pole.
When magnetic fields are perpendicular to the plane of the page, "x's" are used to represent those flux lines which point into the plane (-z) of the page and "'s" are represent those which point out of the plane (+z) of the page.
into the plane
out of the plane
These definitions can be remembered by imagining an arrow. If the front, or tip, of the arrow was coming "at you" then you would see a round point. If the tail feathers of an arrow were going away from you, then you would see the crossed-feathers.
Field lines can be visualized by sprinkling small iron filings over a magnet as shown in the following animation sponsored by the magnetic resonance lab at Florida State University. Remember that if arrows were to be present indicating direction, these lines would begin on the North pole and terminate on the South pole.
When a compass (or any freely floating bar magnet) points north, it is actually aligning its north pole to the Earth's magnetic south pole. Yes, the  Earth's geographic north pole is a magnetic south pole!
Magnetic field lines surrounding poles that have the same polarity repel - just like those between similarly charged particles - and are hyperbolic in shape. Conversely, field lines between poles having the opposite polarity attract and are elliptical in shape. This can be visualized with this second animation from the Molecular Expressions gallery at Florida State University.
Forces on Moving Charges
As mentioned earlier, magnetic fields can only affect moving charged particles. The formula used to calculate the magnetic force on a moving charged particle through a magnetic field is a vector cross-product, where F, v and B are all mutually perpendicular vectors.
In this diagram, if the magnetic field, B, were pointed in the positive x-direction (fingers), and a positively charged particle were traveling in the positive y-direction (thumb), then the magnetic force F (palm) would point in the negative z-direction, or into the plane of the page.
As indicated in the formula, magnetic forces are maximized when the angle between the charge’s velocity and the magnetic field through which it is moving is 90º. If B is not perpendicular to the particle's velocity, you may take components of either vector: B or v. Usually the component B sin(θ) is perpendicular to the particle's velocity. In both of the diagrams shown below, the magnetic force would equal zero, since B and v have no components which are perpendicular to each other.
parallel.gif (2013 bytes)
antiparallel.gif (2022 bytes) 
The right hand rule, RHR, for determining the direction of the force experienced by a moving positive charge in a magnetic field is:
  • thumb points in the direction of a moving positive charge's velocity, v
  • fingers point in the direction of magnetic field, B
  • palm faces in the direction of the magnetic force, F
This right hand rule only applies to positive charges. You would need to use an equivalent left hand rule for electrons. Or just remember that if the force would be "up" for a positive charge, then the force will be "down" for a negative charge. That is, the force on a negative charge will always act 180º in the opposite direction.
rightperpendicular.gif (1895 bytes)
leftperpendicular.gif (1797 bytes)
rightvelocity.gif (1829 bytes)

right palm faces OUT OF the plane of the page, or in the (+z) direction
leftvelocity.gif (1823 bytes)

right palm faces INTO the plane of the page, or in the (-z) direction
In most of our problems, this 90º angle is achieved by having the magnetic field (B) point along either the positive/negative z-axis while the charged particle moves in the xy-plane.

If the charge is not moving, F equals zero, . Magnetic fields can NOT produce linear acceleration. Magnetic forces only cause charged particles to change their direction of motion. Magnetic forces on moving charged particles are centripetal forces and result in the charged particles traveling in circular paths..

In addition, magnetic forces do NOT do any work on moving charges. That is, they do not cause a particle to gain kinetic energy. Remember that work = Fs cos(θ). For magnetic forces, FB and s (and the particle's tangential velocity, v) are always at right angles and cos 90º = 0.

The formula that allows you to calculate the radius of these circular paths is:

solving for r yields

Note that the radius is directly proportional to the particle's momentum and inversely proportional to the magnitude of the charge and the strength of the magnetic field.

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