Electricity and Magnetism Unit E3: Magnetostatics
When we discussed electrostatics we dealt with a situation in which all the charges are stationary. When we discussed currents (?) we had moving charges and just assumed that the same electrostatic forces held. This is, in fact, true, but when charges move there is in addition a new phenomenon called magnetism. We recall that the CGS units used by Purcell for the most part are also used in the discussions here and in the quizzes. More information on the SI (MKS) units can be found in the asides in Purcell, while a conversion table can be found in Appendix E. Some of the more mathematically complex sections (for example the discussion of the magnetic vector potential) will be skipped.
Magnetic Fields and Forces on Currents and Moving Charges
E. M. Purcell, Electricity and Magnetism, Berkeley Physics Course, Vol. 2, 2nd Ed. , Chapt. 5 - The Fields of Moving Charges, Sec. 5.1-5.2, ;Chapt. 6 - The Magnetic Field, Sec. 6.1-6.2, 6.4 (begin with p. 225), 6.5; study Problem 6.22 and the solution to it below; read the last pargraph of p. 407 ending at the top of p. 408 and study Fig. 11.5.
We have no deeper explanation for any of these facts. Investigations aimed at the foundations of the subject have been conducted over the years and are still being conducted: for example, the equality of the magnitudes of the charges of the electron and proton have been tested with ever increasing accuracy. Experiments have been and are still being conducted at the world's largest accelerators to search for particles with non-integral charges. Current theories of particle physics suggest that there are particles call quarks with 1/3 or 2/3 the charge of the electron. The size of the electron charge and the relation of the electromagnetic force to the other forces in the universe are still among the greatest puzzles in physics.
Conservation of Charge:: Although charge can move and change, the total amount of charges remains constant. For example, if an electron (charge -e) interacts with a positron ( an anti-electron with charge +e)the two particles can annihilate, resulting in two neutral gamma rays. The algebraic sum of the total charge, however, is zero before and after the reaction. For situations where there is a continuous distribution of charge of density (charge per volume) ρ , there is an associated current density j = ρv, where v is the velocity with which the charge moves. Now consider a certain volume V surrounded by a closed surface S. Then the scalar product j⋅dS is the rate at which charge moves out of the element of surface dS , and the integral i = ∫S j⋅dS of this scalar product over the entire closed surface S, the net rate at which charge is leaving the volume V. (We assume that dS is in the direction of the outward normal to the surface.) The total amount of charge q in the volume, however, is just the integral q = ∫V ρdV over the whole volume V. Therefore i = -dq/dt. (The minus sign accounts for the fact that if charge is escaping through the surface (i positive) then the total amount of charge surrounded by the surface must be decreasing.) In terms of the surface and volume integrals this gives the "equation of continuity": The integral of j⋅dS over the closed surface S surrounding V must equal minus the volume integral of the time rate of change of ρ integrated over the whole volume V. This expresses the connection between current and charge density that must hold if charge is to be conserved.
Problems:: Purcell 6.1, 6.2, 6.4, 6.11, 6.24