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Unformatted text preview: Magnetostatics Introduction We wish to introduce the magnetic field from a different perspective. Suppose the only knowledge we have are electrostatics and special relativity, and we never heard of magnetic force before. Using a simple argument, we will show that such thing as the magnetic field must exist. Consider a wire containing stationary positive charges and moving negative charge (for simplicity lets assume all the ve charge move with the same speed v ) Suppose the charge densities of the +ve and ve charges are not equal , and are happened to be related by = 2 + , where = 2 2 c v 1 1 (5.1) So the wire has a net charge of linear density ( +  )A , where A is the crosssectional area of the wire. The wire generates an electric field given by E = r 2 A r 2 ) ( + = , so that a stationary charge q will be attracted by the wire with a force F = r 2 A q ) ( + (5.2) Now, to an observer moving to the left with a speed v : What are the charge densities + ' and  ' in the wire? They are different from + and  ! The reason is as followed: Because of length contraction , a moving rod is shortened! But charge remains the same. The length contraction is described by 2 2 c v 1 l l = ' (5.3) (Dont worry if you havent learned relativity, just take this for granted) Charge density increases as a result of length contraction. i.e. 2 2 2 2 c v 1 c v 1 1 Al q Al q = = = + + ' ' (5.4) That take care of the +ve charge density. However, for the ve charge,  ' is really that corresponds to the stationary charge density. 2 2 c v 1 = ' (5.5) So that for the moving observer, the wire carries a net charge density of + '  ' 2 2 2 2 c v 1 c v 1 = + Using Eq.(5.1), + '  ' 2 2 2 2 2 2 c v 1 c v 1 c v 1  = + + = 0. So there is no net charge on the wire according to the moving observer. If only electrostatic force exists in nature, the moving observer would conclude that the charge q will stand still, while the stationary observer would conclude that the charge q will be attracted toward the wire. We have a contradiction. So we are forced to infer that there must exist something called the magnetic field that acts on a moving charge and causes it to move toward the wire. If we assume a form F = qvB for the force as seen by the moving observer, and the force as seen by the stationary and moving observers are related by F = F by relativity (again dont worry about how this formula came from), we have, using Eq.(5.1) and (5.2) qvB = r 2 A q 1 ) ( + = ) ( 1 r 2 qA 1 2 + = ) ( ' 1 r 2 qA 1 2 + using Eq.(5.4) Since 2 2 2 2 2 2 2 c v c v 1 1 1 1 1 = = = ) ( qvB = 2 2 c v r 2 qA + ' B = 2 rc 2 v A + ' Now, v A + ' = area current density = current 2 1 c = (youll learn this in PHY 4211) We obtain...
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This note was uploaded on 10/14/2010 for the course PHYS 2051 taught by Professor Drkwong during the Spring '10 term at CUHK.
 Spring '10
 DrKwong
 Electrostatics, Force, Special Relativity

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