Feb. 27  2007

Magnetism as a relativistic side effect

of Electrostatics.  (simpler as Purcell)


 by Hans de Vries 



     The Magnetic field is really an Electric field.


The notion of Magnetism as relativistic side effect of Electrostatics can in principle be derived from the work of Lienard & Wiechert around 1900, and the correct form of the Lorentz transformations established a few years later. However,  it was only in the nineteen sixties that it was generally recognized that a magnetic field can be always interpreted as an electric field caused by the effects of Special relativity.  


A moving charged which is attracted or repelled by a wire carrying a current is in fact seeing an electric force in its rest frame which attracts or repels it. The originally neutral wire gets a non-zero charge density in the charge's rest frame because of the relativistic effect. This then causes the electric field. 




Over 100 years after Lienard & Wiechert we are teaching this concept to undergraduate students with the help of a popular derivation following the 1952 Physics Nobel laureate Edward Mills Purcell. ( for NMR, Nuclear magnetic resonance). Purcell presented this derivation in his 1963 textbook and popularized the teaching to undergraduate students.


     A simpler derivation, valid in all cases. 


The derivation of Purcell however is somewhat dubious. Mainly so because the velocity of the test charge is unrealistically taken to be always the same as the charge velocity in the wire. So, when the test charge doubles it speed, the current I through the wire is also doubled and the magnetic force is quadrupled. This however, makes it impossible to determine if the contributions to the higher magnetic force come from either the higher current I, the higher speed v, or both.


The electrons in a real live wire drift with a wide range of different velocities which together produce the current I. We�ll discuss our derivation, which starts of with just the current I through the wire and the speed v of the test-charge. Surprisingly, this derivation turns out to be even simpler as Purcell�s. (for the case of the charge moving parallel to the wire).


We�ll also derive the case where the charge is moving perpendicular to the wire. The required charge density is derived for a current carrying wire in order to be neutral in the rest-frame. To be self consistent we will derive the relativistic EM Potential and the relativistic Electrostatic Field for a point particle from the classical EM wave equations in a way which is both short

and simple.


      The paper.


The paper itself can be accessed by clicking on the PDF logo. It requires only an elementary undergraduate physics level for a good understanding of the first section. The more involved following sections require a more advance undergraduate physics level. Only the laws of the Electro Magnetic field and the Lorentz Transformations (The general and those for the EM field) are assumed. Care has been taken to make the document as accessible as possible. 



     Other sources 


An online presentation of Purcell's derivation is available here: from Dan Schroeder (The one from Peskin & Schroeder)



Regards, Hans