Maxwell's equations in integral form

Consider a fixed volume $v$ bounded by the closed surface $S$, whose outward unit normal is $\vec{n}$ as shown in Fig. 1.1.

Figure 1.1: Geometry for Gauss theorem.

Integration of (1.3,1.4) and (1.5,1.6) over $v$, followed by the use of the divergence theorem, yields:

$$\oint_S \underline{\mathcal{ D}} \cdot \vec{n} \, dS = \int_v \rho_e \, dv = Q_e \space ,$$ (1.14)

$$\oint_S \underline{\mathcal{ B}} \cdot \vec{n} \, dS = \int_v \rho_m \, dv = Q_m \space ,$$ (1.15)

$$\oint_S \underline{\mathcal{ J}}_e \cdot \vec{n} \, dS = -\frac{dQ_e}{dt} \space ,$$ (1.16)

$$\oint_S \underline{\mathcal{ J}}_m \cdot \vec{n} \, dS = - \frac{dQ_m}{dt} \space ,$$ (1.17)

where $Q_e$ and $Q_m$ are the total electric charge (in C) and magnetic charge (in Wb) inside the volume $v$, respectively.

Equation (1.14) means that the total electric charge contained in $v$ equals the outgoing flux of $\underline{\mathcal{D}}$ through the surface $S$ of $v$. A similar interpretation applies to (1.15); since $Q_m$ is zero, the total outgoing flux of $\underline{\mathcal{B}}$ through any fixed closed surface is zero.

Equation (1.16) means that the rate of decrease of the total electric charge $Q_e$ inside $v$ equals the amount of electric charge which leaves $v$ in unit time by traveling outward through $S$; thus, (1.3) is obviously a statement of conservation of electric charge. A similar interpretation applies to (1.17); since $Q_m$ is zero, the outgoing flux of $J_m$ through $S$ is also zero.

As a particular application of (1.14), consider a point charge $Q_e$ located at the center of a sphere of radius $r$ in free space. Because of symmetry $\underline{\mathcal{D}}$ is radially directed and has the same magnitude at all points on $S$. Hence (1.14) gives:

$$\underline{\mathcal{ D}}=\frac{Q_e}{4 \pi r^2} \hat{r}$$ (1.18)

and, with the use of (1.7):

$$\underline{\mathcal{ E}}=\frac{Q_e}{4 \pi \varepsilon_0 r^2} \hat{r}$$ (1.19)

which is Coulomb's law of electrostatic.

Let us now consider a fixed open surface $S$ bounded by a closed curve $l$, as shown in Fig. 1.2. The unit normal $\hat{n}$ on $S$ and the unit tangent $\hat{l}$ along $l$ are chosen according to the right-handed corkscrew rule. Integration of (1.1-1.2) over $S$ and use of Stokes' theorem yields:

$$\oint_l \underline{\mathcal{ H}} \cdot \, d\underline{l} = \int_S \underline{\mathcal{ J}}_e \cdot \hat{n} \, dS + \frac{d}{dt} \int_S \underline{\mathcal{ D}} \cdot \hat{n} \, dS \space ,$$ (1.20)

$$\oint_l \underline{\mathcal{ E}} \cdot \, d\underline{l} =- \int_S \underline{\mathcal{ J}}_m \cdot \hat{n} \, dS - \frac{d}{dt} \int_S \underline{\mathcal{ B}} \cdot \hat{n} \, dS \space ,$$ (1.21)

Figure 1.2: Geometry for Stokes theorem.

For time-invariant, or stationary, fields, eq. (1.20) becomes:

$$\oint_l \underline{\mathcal{ H}} \cdot \, d\underline{l}=\int_S \underline{\mathcal{ J}}_e \cdot \hat{n} \, dS=I_e \space ,$$ (1.22)

where $I_e$ is the total electric current which flows through the closed loop $l$. Equation (1.22) is Ampère's law, whose generalization to time-varying fields requires the addition of the second term in the right-hand side of (1.20); this added term, whose existence was postulated by Maxwell in 1861, is called the displacement current.

For the case $\underline{\mathcal{ J}}_m \equiv 0$, equation (1.21) is Lenz's law which in turn represents a generalization of Kirchhoff's second law of circuit theory to the case of time-varying fields by the inclusion of an induction term.