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A singular integral equation for an electric dipole has been obtained, which makes it possible to take into account the finite conductivity of the metal from which it is made. The derivation of the singular integral equation is based on the application of the Greens function for free space, written in a cylindrical coordinate system, taking into account the absence of the dependence of the field on the azimuthal coordinate, on a point source located on the surface of an electric dipole. Methods for its solution are proposed. In contrast to the well-known mathematical models of an electric dipole, built in the approximation of an ideal conductor, the use of the singular integral equation obtained in this work makes it possible to take into account heat losses and calculate the efficiency.

In recent years, freshwater resource contamination by non-point source pollution has become particularly prominent in China. To control non-point source (NPS) pollution, it is important to estimate NPS pollution exports, identify sources of pollution, and analyze the pollution characteristics. As such, in this study, we established the modified export coefficient model based on rainfall and terrain to investigate the pollution sources and characteristics of non-point source total nitrogen (TN) and total phosphorus (TP) throughout the Huangqian Reservoir watershed—which serves as an important potable water source for the main tributary of the lower Yellow River. The results showed that: (1) In 2018, the non-point source total nitrogen (TN) and total phosphorus (TP) loads in the Huangqian Reservoir basin were 707.09 t and 114.42 t, respectively. The contribution ratios to TN export were, from high to low, rural life (33.58%), farmland (32.68%), other land use types (20.08%), and livestock and poultry breeding (13.67%). The contribution ratios to TP export were, from high to low, rural life (61.19%), livestock and poultry breeding (21.65%), farmland (12.79%), and other land use types (4.38%). The non-point source pollution primarily originated from the rural life of the water source protection zone. (2) Non-point source TN and TP pollution loads and load intensities showed significantly different spatial distribution patterns throughout the water source protection area. Specifically, their load intensities and loads were the largest in the second-class protected zone, which is the key source area of non-point source pollution. (3) When considering whether to invest in agricultural land fertilizer control or rural domestic sewage, waste, and livestock manure pollution control, the latter is demonstrably more effective. Thus, in addition to putting low-grade control on agricultural fertilizer loss, to rapidly and effectively improve potable water quality, non-point source pollution should, to a larger extent, also be controlled through measures such as establishing household biogas digesters, introducing village sewage treatment plants, and improving the recovery rate of rural domestic garbage. The research results discussed herein provide a theoretical basis for formulating a reasonable and effective protection plan for the Huangqian Reservoir water source and can potentially be used to do the same for other similar freshwater resources.