The integral forms of Maxwell's equations are usually easier to recognize in terms of the experimental laws from which they have been obtained by a generalization process. Experiments must treat phacroscopic quantities, and their results therefore are expressed in terms of integral relationships. A differential equation always represents a theory.

$\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$ ...(1)

Integrating (1) over a surface and applying Stokes' theorem, we obtain Faraday's law,

$\oint \mathbf{E} \cdot d \mathbf{L}=-\int_{S} \frac{\partial \mathbf{B}}{\partial t} \cdot d \mathbf{S}$ ...(A)

and the same process applied to (2) yields Ampére's circuital law,

$\nabla \times \mathbf{H}=\mathbf{J}+\frac{\partial \mathbf{D}}{\partial t}$ ...(2)

$\oint \mathbf{H} \cdot d \mathbf{L}=I+\int_{S} \frac{\partial \mathbf{D}}{\partial t} \cdot d \mathbf{S}$ ...(B)

Gauss's laws for the electric and magnetic fields are obtained by integrating (3) and (4) throughout a volume and using the divergence theorem:

$\nabla \cdot \mathbf{D}=\rho_{v}$ ...(3)

$\nabla \cdot \mathbf{B}=0$ ...(4)

$\oint_{S} \mathbf{D} \cdot d \mathbf{S}=\int_{\mathrm{vol}} \rho_{v} d v$ ...(c)

$\oint_{S} \mathbf{B} \cdot d \mathbf{S}=0$ ...(D)

These four integral equations enable us to find the boundary conditions on B, D, $\mathbf{H},$ and $\mathbf{E}$ which are necessary to evaluate the constants obtained in solving Maxwell's equations in partial differential form. These boundary conditions are in general unchanged from their forms for static or steady fields, and the same methods may be used to obtain them. Between any two real physical medial (where $\mathbf{K}$ must be zero on the boundary surface), (A) enables us to relate the tangential E-field components,

$E_{t 1}=E_{t 2}$ ...(5)

and from (B)

$H_{t 1}=H_{l 2}$ ...(6)

The surface integrals produce the boundary conditions on the normal components,

$$ D_{N 1}-D_{N 2}=\rho_{S} $$ ...(7)

and

$B_{N 1}=B_{N 2}$ ...(8)

It is often desirable to idealize a physical problem by assuming a perfect conductor for which $\sigma$ is infinite but $\mathbf{J}$ is finite. From Ohm's law, then, in a perfect conductor,

$\mathbf{E}=0$

and it follows from the point form of Faraday's law that

$\mathbf{H}=0$

for time-varying fields. The point form of Ampére's circuital law then shows that the finite value of $\mathbf{J}$ is

$\mathbf{J}=0$

and current must be carried on the conductor surface as a surface current $\mathbf{K}$ . Thus, if region 2 is a perfect conductor, (5) to (8) become, respectively,

$\begin{aligned} E_{t 1} &=0 \\ H_{t 1} &=K \quad\left(\mathbf{H}_{t 1}=\mathbf{K} \times \mathbf{a}_{N}\right) \\ D_{N 1} &=\rho_{s} \\ B_{N 1} &=0 \end{aligned}$

where $\mathbf{a}_{N}$ is an outward normal at the conductor surface.

Note that surface charge density is considered a physical possibility for either dielectrics, perfect conductors, or imperfect conductors, but that surface current density is assumed only in conjunction with perfect conductors.