Figure 14.12 in Balanis’s antenna book (3rd Ed) has a misprint. The “W” should be “L”. It is corrected when cited as Figure 1.2 in the book “Microstrip Antenna Design Handbook” (2001). For PEC, the surface charge is directly related to the normal electric field at the surface. This point can be understood by considering the integral form of Gauss's Law. On the cross section of a transmission line, the direction of electric field at the surface of a conductor must have the same sense, inward or outward, otherwise the voltage can not be uniquely defined. Therefore the charge distribution across the width of the patch antenna has the same sign, while it is not across the length. Voltage has different signs along the transmission line.
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