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# Magnetic Forces on Charged Particles

"A fact is a simple statement that everyone believes. It is innocent, unless found guilty. A hypothesis is a novel suggestion that no one wants to believe. It is guilty, until found effective"
Edward Teller

- Proceeding under the assumption that magnetic fields exist - created by an as yet not described mechanism - the magnetic force on a positively charged particle is found experimentally to be given by

> where q is the charge (positive) of the particle, v its velocity, FB the force it experiences and B the magnetic field causing the force.

- Note that this equation is the magnetic equivalent of the electric expression FE = qE

- The expression for the magnetic force is written in terms of a vector (cross) product, which means, like it or not, you have to work in three dimensions. Several important facts emerge from this equation.
  1. If v = 0 there is no force. Electric charges at rest in a magnetic field do not feel a magnetic force.
  2. The magnitude of FB is given by
     where φ is the angle between v and B.
     - So that, when v is parallel to B (φ = 0) or antiparallel to B (φ = 180o) then sinφ = 0 and FB = 0. Therefore, charged particles moving along magnetic field lines experience no magnetic force.
     - When v and B are at 90o FB has its maximum value, FB = qvB.
  3. FB is at right angles to v and B. The "sense" is given by the usual rules for the vector (cross) product, sometimes called (the) "Right Hand Rule".
     This leads to circular (or spiral) motion around the B field lines.
     Don't forget that the direction of the magnetic force obtained above is for a positive charge. For a negative charge the direction of FB is reversed.
  4. The work done by the magnetic force when a charged particle is displaced by ds is given by,
  where θ is the angle between FB and ds.
  The displacement ds and the velocity of the particle, v, are in the same direction, so θ is also the angle between FB and v. But from the form of the magnetic force we know FB and v are perpendicular (θ = 90o) therefore dW is always zero. In other words the magnetic force does no work, which means, unlike the electric force, it cannot give a charged particle energy (increase or decrease its kinteic energy). The effect of a magnetic force is to change the direction not the kinetic energy.

- If a charged particle feels both a magnetic and electric field the resultant force is given by

> this is known as the Lorentz Force Law.

- UNITS

> From the form of the magnetic force we see that the units of B are N/(C.m/s) = N/(A.m). This combination of basic units is defined as the Tesla.

> The Tesla is a very large unit of magnetic field. For this reason you may occasionally come across the smaller unit of magnteic field, the Gauss; where 1 Tesla = 104 Gauss.

Q: What did one quantum physicist say when he wanted to fight another quantum physicist?
A: Let me atom.

Dr. C. L. Davis
Physics Department
University of Louisville
email: c.l.davis@louisville.edu