High-frequency electromagnetic structures such as waveguides must be described in terms of quantities that vary in space as well as in time; a complete description might thus involve specification of the electric and magnetic fields everywhere. There exist structures, however, called *lumped elements*, that are described quite accurately at low frequencies in terms of integrals over the spatial variables; these, as we will now see, are usually voltages [line integrals of the electric field, expressed in volts (V)] and currents [flow of charge per unit time through a given surface, expressed in coulombs (C) per second, or amperes (A)]. We will first examine the low-frequency behavior of isolated lumped elements and then proceed to *networks* or *circuits*, structures that are obtained by interconnecting lumped elements and that can also be described in terms of voltages and currents.

Let us start by recalling three of the four fundamental laws of electromagnetism, valid as shown in vacuum and in the materials we will consider for now (the fourth law states that there are no magnetic charges, but we will not need it). In integral form these laws are as follows.

- Gauss's law for a closed surface,
where

**E**is the electric field,*q*is the charge enclosed in the volume bounded by the surface, and ∊_{0}is the vacuum permittivity, given in terms of the speed ...

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