Voltage Drop, Potential Difference and EMF

in SI units, where ρ is the electric charge density, c is the speed of light in the medium between r and r′ (assumed in this note to be vacuum), R = |r− r′|, and 0 is the permittivity of the vacuum. This convention follows Stratton [7], p. 352. A prominent application of the notion of voltage is in circuit analysis, where prescriptions exist for the “voltage drop” across various circuit elements (secs. 3.2.1-5), and where Kirchhoff’s (voltage) law states that the sum of the “voltage drops” around any loop is zero. However, Kirchhoff’s “law” is only approximately valid, and the “voltage drops” it considers are, in general, neither differences in the electric scalar potential V , nor the EMF = ∫ E · dl along some path between the ends of the circuit element. In “ordinary” circuit analysis, as defined below, the “voltage drops” are well approximated by both the difference in the electric scalar potential (in the Lorenz gauge) and by the EMF along a suitable path. But, great care is required when considering examples outside “ordinary” circuit analysis where these familiar approximations can be very poor (sec. 3.4). An “ordinary” circuit, operating at angular frequency ω, is a circuit for which: