2.18: Quasistatic Integral Laws

There are at least three reasons for desiring Maxwell's equations in integral form. First, th integral equations are convenient for establishing jump conditions implied by the differential equations. Second, they are the basis for defining lumped parameter variables such as the voltage, charge, current, and flux. Third, they are useful in understanding (as opposed to predicting) physical processes. Since Maxwell's equations have already been divided into the two quasistatic systems, i is now possible to proceed in a straightforward way to write the integral laws for contours, surface and volumes which are distorting, i.e., that are functions of time. The velocity of a surface \(S\) is \(\overrightarrow{v_s}\).

To obtain the integral laws implied by the laws of Eqs. 2.3.23-27, each equation is either (i) integrated over an open surface S with Stokes's theorem used where the integrand is a curl operator to convert to a line integration on \(C\) and Equation 2.6.4 used to bring the time derivative outside the integral, or (ii) integrated over a closed volume \(V\) with Gauss' theorem used to convert integrations of a divergence operator to integrals over closed surfaces S and Equation 2.6.5 used to bring the time derivative outside the integration:

\[ \begin{align} &\oint_{S} (\varepsilon_o \overrightarrow{E} + \overrightarrow{P}) \cdot \overrightarrow{n} da = \int_{V} \rho_f dV \quad \quad \quad & &\oint_{C} \overrightarrow{H} \cdot \overrightarrow{d} l = \int_{S} \overrightarrow{J_f} \cdot \overrightarrow{n} da \label{1} \\ &\oint \overrightarrow{E} \cdot \overrightarrow{d} l = 0 \quad \quad \quad & &\oint_{S} \mu_o (\overrightarrow{H} + \overrightarrow{M}) \cdot \overrightarrow{n} da = 0 \label{2} \\ &\oint_{S} \overrightarrow{J_f^{'}} \cdot \overrightarrow{n} da + \frac{d}{dt} \int_{V} \rho_f dV = 0 \quad \quad \quad & &\oint_{C} \overrightarrow{E^{'}} \cdot \overrightarrow{d}l = -\frac{d}{dt} \int_{S} \mu_o (\overrightarrow{H} + \overrightarrow{M}) \cdot \overrightarrow{n} da - \oint_{C} \mu_o \overrightarrow{M} \times (\overrightarrow{v} - \overrightarrow{v_s}) \cdot \overrightarrow{d}l \label{3} \\ &\oint_{C} \overrightarrow{H^{'}} \cdot \overrightarrow{d}l = \int_{S} \overrightarrow{J_f^{'}} \cdot \overrightarrow{n} da + \frac{d}{dt} \int_{S} (\varepsilon_o \overrightarrow{E} + \overrightarrow{P}) \cdot \overrightarrow{n} da + \oint_{C} \overrightarrow{P} \times (\overrightarrow{v} - \overrightarrow{v_s}). \overrightarrow{d}l \quad \quad \quad & &\oint_{S} \overrightarrow{J_f} \cdot \overrightarrow{n} da = 0 \label{4} \\ &\oint_{S} \mu_o (\overrightarrow{H} + \overrightarrow{M}) \cdot \overrightarrow{n}da = 0 \quad \quad \quad & &\oint_{S}(\varepsilon_o \overrightarrow{E} + \overrightarrow{P}) \cdot \overrightarrow{n} da = \oint_{V} \rho_f dV \label{5} \\ \\ &\text{where} \quad \quad \quad & &\text{where} \nonumber \\ \nonumber \\ &\overrightarrow{J_f^{'}} = \overrightarrow{J_f} - \overrightarrow{v_s} \rho_f \quad \quad \quad & &\overrightarrow{E^{'}} = \overrightarrow{E} + \overrightarrow{v_s} \times \mu_o \overrightarrow{H} \nonumber \\ &\overrightarrow{H^{'}} = \overrightarrow{H} - \overrightarrow{v_s} \times \varepsilon_o \overrightarrow{E} \quad \quad \quad \nonumber \end{align} \nonumber \]

The primed variables are simply summaries of the variables found in deducing these equations. However, these definitions are consistent with the transform relationships found in Sec. 2.5, and the velocity of these surfaces and contours, \(\overrightarrow{v_s}\), can be identified with the velocity of an inertial frame instantaneously attached to the surface or contour at the point in question. Approximations implicit to the original differential quasistatic laws are now implicit to these integral laws.