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25 Definitions in Electrostatic Units

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25 Definitions in Electrostatic Units 25 Definitions in Electrostatic Units As explained in Section 17.6, the observations made in Oersted's experiment gives rise to either define electrical quantities first, or magnetic quantities first, which are named ESU and EMU. The flowchart of definitions in ESU will be presented in this chapter, and the definitions in EMU will be presented in the following chapter. The definitions will be revised in the future chapters, without the use of magnetic charges. Figure 25.1: The definitions flowchart in electrostatic units. Each of the definitions will be summarized in this chapter. Note that there are no circular definitions of the same quantity, and therefore, the electrical quantities are well defined. It will become clear, why some of the equations presented in the previous chapters are valid only in ESU, or EMU, or common to both ESU and EMU. It will become clear that some of the equations, although the units of the variables are different between ESU and EMU, are common to both the systems of units. 25.1 Electric Charge q The electric charge is defined first in ESU using Coulomb's law, q q F~ = 1 2 r;^ (25.1, ESU) r2 172 where q1 and q2 are two points charges, r is the distance between them, andr ^ is the unit vector indicating the direction of the force. The unit electric charge is defined as two equal unit point charges, separated by 1 cm, exerting a force of 1 dyne on each other. 25.2 Electric Field E~ Once the electric charge has been defined, it follows that the electric field E~ can be defined from the force F~ exerted on charge q, F~ E~ = : (25.2) q 25.3 Voltage V The voltage V between two points A and B is the path integral of the electric field E~ , Z B V = E~ · dl:~ (25.3) A It was proven that the electric field is a conservative field, and the above value is independent of the path from A to B. 25.4 Electric Current i Since electric current is the flow of charges, using Equation 18.1, ∆q i = ; (25.4) ∆t current i is defined as the flow of charge ∆q in time ∆t. 25.5 Magnetic Field H~ Biot-Savart's law relates magnetic field H~ and electric current i, by the path integral along a current-carrying wire `, Z ~ i ~ H = 2 dl × r^ ; (25.5) ` r where r is the distance from dl~ to the point where the magnetic field is calculated, andr ^ is the unit vector in that direction. The relation between magnetic field H~ and i is the same in ESU and EMU. In ESU, Biot-Savart's law is used to define magnetic field from the definition of current. In EMU, however, it will be shown in the following chapter, current is defined from the definition of magnetic field. 173 The magnetic field around a long straight wire, carrying current i, was derived in Equation 23.19, 2i H~ = ';^ (25.6) a where' ^ is in the azimuth direction, and set by the first right-hand rule. Using the above equation, the magnetic field can be defined as the field of 2 units, generated at a unit distance from a long straight wire, which carries a unit current. The units of the electrical quantities will be summarized towards the end of the chapter. 25.6 Magnetic Potential Vm The magnetic potential Vm, between two points A and B, is the path integral of the magnetic field H~ , Z B ~ ~ Vm = H · dl: (25.7) A Assuming symmetry to electric charges, magnetic field due to magnetic charges are also conser- vative, and any path may be chosen from A to B, to calculate the magnetic potential. 25.7 Magnetic Charge m The magnetic charge m, can be defined from the force F~ exerted on the charge in a magnetic field H~ , F~ = mH:~ (25.8) Note that the above definition of m has no relation to the Coulomb's law of magnetic charges in Equation 13.1. The force between two magnetic charges m1 and m2 separated by distance r, can only be written as m m F~ / 1 2 r;^ (25.9) r2 wherer ^ is the unit vector in the direction of the force. The proportionality constant in ESU is to be determined. However, since the definitions will be revised in the future chapters without the need of magnetic charges, determining the proportionality constant in the above equation will not be pursued. 25.8 Units in ESU Table 25.1 summarizes the units in ESU. It is left as an exercise for the reader to derive them from dimensional analysis, similar to the examples in the earlier chapters. 174 Electrical Quantity (Symbol) Unit (ESU) Unit (ESU) Charge (q) statcoulomb gm1=2cm3=2s−1 Electric field (E~ ) { gm1=2cm−1=2s−1 Voltage (V ) statvolt gm1=2cm1=2s−1 Electric current (i) statampere gm1=2cm3=2s−2 Magnetic field (H~ ) { gm1=2cm1=2s−2 1=2 3=2 −2 Magnetic potential (Vm) { gm cm s Magnetic charge (m) { gm1=2cm1=2 Table 25.1: A summary of the units in ESU. 175.
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