"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, F_B the force it experiences and B the magnetic field causing the force.

• Note that this equation is the magnetic equivalent of the electric expression  F_E = 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 F_B 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 = 180^o) then sinφ = 0 and F_B = 0.  Therefore, charged particles moving along magnetic field lines experience no magnetic force.
   • When v and B are at 90^o F_B has its maximum value, F_B = qvB.


3. F_B 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 F_B  is reversed.
   


The work done by the magnetic force when a charged particle is displaced by ds is given by,




where θ is the angle between F_B  and ds.

The displacement ds and the velocity of the particle, v, are in the same direction, so θ is also the angle between F_B  and v.  But from the form of the magnetic force we know F_B  and v are perpendicular (θ = 90^o) 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