Lecture 2

Maxwell’s Equations in

Differential Operator Form

2.1 Gauss’s Divergence Theorem

The divergence theorem is one of the most important theorems in vector calculus [29, 31–33]

First, we will need to prove Gauss’s divergence theorem, namely, that:

∭_V dV ∇ · D = ∮_S D · dS (2.1.1)

In the above, ∇ · D is defined as

∇ · D = lim_{ΔV→0} (∮_{ΔS} D · dS) / ΔV (2.1.2)

and eventually, we will find an expression for it. We know that if ΔV ≈ 0 or small, then the above,

ΔV ∇ · D ≈ ∮_{ΔS} D · dS (2.1.3)

First, we assume that a volume V has been discretized1 into a sum of small cuboids, where the i-th cuboid has a volume of ΔVi as shown in Figure 2.1. Then

V ≈ Σ_{i=1}^N ΔVi (2.1.4)

1Other terms are “tesselated”, “meshed”, or “gridded”.

Then from (2.1.2),

ΔVi ∇ · Di ≈ ∮_{ΔSi} Di · dSi (2.1.5)

By summing the above over all the cuboids, or over i, one gets

Σ_i ΔVi ∇ · Di ≈ Σ_i ∮_{ΔSi} Di · dSi ≈ ∮_S D · dS (2.1.6)

It is easily seen the the fluxes out of the inner surfaces of the cuboids cancel each other, leaving only the fluxes flowing out of the cuboids at the edge of the volume V as explained in Figure 2.2. The right-hand side of the above equation (2.1.6) becomes a surface integral over the surface S except for the stair-casing approximation (see Figure 2.1). Moreover, this approximation becomes increasingly good as ΔVi → 0, or that the left-hand side becomes a volume integral, and we have

∭_V dV ∇ · D = ∮_S D · dS (2.1.7)

The above is Gauss’s divergence theorem.

Next, we will derive the details of the definition embodied in (2.1.2). To this end, we evaluate the numerator of the right-hand side carefully, in accordance to Figure 2.3.

Accounting for the fluxes going through all the six faces, assigning the appropriate signs in accordance with the fluxes leaving and entering the cuboid, one arrives at

∮_{ΔS} D · dS ≈ −Dx(x0, y0, z0)ΔyΔz + Dx(x0 + Δx, y0, z0)ΔyΔz

− Dy(x0, y0, z0)ΔxΔz + Dy(x0, y0 + Δy, z0)ΔxΔz

− Dz(x0, y0, z0)ΔxΔy + Dz(x0, y0, z0 + Δz)ΔxΔy (2.1.8)

Factoring out the volume of the cuboid ΔV = ΔxΔyΔz in the above, one gets

∮_{ΔS} D · dS ≈ ΔV {[Dx(x0 + Δx, . . .) − Dx(x0, . . .)] /Δx

+ [Dy(. . . , y0 + Δy, . . .) − Dy(. . . , y0, . . .)] /Δy

+ [Dz(. . . , z0 + Δz) − Dz(. . . , z0)] /Δz} (2.1.9)

Or that

∮ D · dS / ΔV ≈ ∂Dx/∂x + ∂Dy/∂y + ∂Dz/∂z (2.1.10)

In the limit when ΔV → 0, then

lim_{ΔV→0} (∮ D · dS) / ΔV = ∂Dx/∂x + ∂Dy/∂y + ∂Dz/∂z = ∇ · D (2.1.11)

where

∇ = x̂ ∂/∂x + ŷ ∂/∂y + ẑ ∂/∂z (2.1.12)

D = x̂Dx + ŷDy + ẑDz (2.1.13)

The divergence operator ∇· has its complicated representations in cylindrical and spherical coordinates, a subject that we would not delve into in this course. But they are best looked up at the back of some textbooks on electromagnetics.

Consequently, one gets Gauss’s divergence theorem given by

∭_V dV ∇ · D = ∮_S D · dS (2.1.14)

2.1.1 Gauss’s Law in Differential Operator Form

By further using Gauss’s or Coulomb’s law implies that

∮_S D · dS = Q = ∭ dV ρ (2.1.15)

which is equivalent to

∭_V dV ∇ · D = ∭_V dV ρ (2.1.16)

When V → 0, we arrive at the pointwise relationship, a relationship at a point in space:

∇ · D = ρ (2.1.17)

2.1.2 Physical Meaning of Divergence Operator

The physical meaning of divergence is that if ∇·D ≠ 0 at a point in space, it implies that there are fluxes oozing or exuding from that point in space [34]. On the other hand, if ∇ · D = 0, if implies no flux oozing out from that point in space. In other words, whatever flux that goes into the point must come out of it. The flux is termed divergence free. Thus, ∇ · D is a measure of how much sources or sinks exists for the flux at a point. The sum of these sources or sinks gives the amount of flux leaving or entering the surface that surrounds the sources or sinks.

Moreover, if one were to integrate a divergence-free flux over a volume V , and invoking Gauss’s divergence theorem, one gets

∮_S D · dS = 0 (2.1.18)

In such a scenerio, whatever flux that enters the surface S must leave it. In other words, what comes in must go out of the volume V , or that flux is conserved. This is true of incompressible fluid flow, electric flux flow in a source free region, as well as magnetic flux flow, where the flux is conserved.

Figure 2.4: In an incompressible flux flow, flux is conserved: whatever flux that enters a volume V must leave the volume V .

2.1.3 Example

If D = (2y^2 + z)x̂ + 4xy ŷ + xzẑ, find:

1. Volume charge density ρv at (−1, 0, 3).

2. Electric flux through the cube defined by

0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1.

3. Total charge enclosed by the cube.

2.2 Stokes’s Theorem

The mathematical description of fluid flow was well established before the establishment of electromagnetic theory [35]. Hence, much mathematical description of electromagnetic theory uses the language of fluid. In mathematical notations, Stokes’s theorem is

∮_C E · dl = ∬_S ∇ × E · dS (2.2.1)

In the above, the contour C is a closed contour, whereas the surface S is not closed.2

First, applying Stokes’s theorem to a small surface ΔS, we define a curl operator 3 ∇× at a point to be

∇ × E · n̂ = lim_{ΔS→0} (∮_{ΔC} E · dl) / ΔS (2.2.2)

Figure 2.5: In proving Stokes’s theorem, a closed contour C is assumed to enclose an open surface S. Then the surface S is tessellated into sum of small rects as shown. Stair-casing error vanishes in the limit when the rects are made vanishingly small.

First, the surface S enclosed by C is tessellated into sum of small rects (rectangles). Stokes’s theorem is then applied to one of these small rects to arrive at

∮_{ΔCi} Ei · dli = (∇ × Ei) · ΔSi (2.2.3)

2 In other words, C has no boundary whereas S has boundary. A closed surface S has no boundary like when we were proving Gauss’s divergence theorem previously.

3 Sometimes called a rotation operator.

Next, we sum the above equation over i or over all the small rects to arrive at

Σ_i ∮_{ΔCi} Ei · dli = Σ_i ∇ × Ei · ΔSi (2.2.4)

Again, on the left-hand side of the above, all the contour integrals over the small rects cancel each other internal to S save for those on the boundary. In the limit when ΔSi → 0, the left-hand side becomes a contour integral over the larger contour C, and the right-hand side becomes a surface integral over S. One arrives at Stokes’s theorem, which is

∮_C E · dl = ∬_S (∇ × E) · dS (2.2.5)

Figure 2.6: We approximate the integration over a small rect using this figure. There are four edges to this small rect.

Next, we need to prove the details of definition (2.2.2). Performing the integral over the small rect, one gets

∮_{ΔC} E · dl = Ex(x0, y0, z0)Δx + Ey(x0 + Δx, y0, z0)Δy

− Ex(x0, y0 + Δy, z0)Δx − Ey(x0, y0, z0)Δy

= ΔxΔy (Ex(x0, y0, z0)/Δy − Ex(x0, y0 + Δy, z0)/Δy

− Ey(x0, y0, z0)/Δx + Ey(x0, y0 + Δy, z0)/Δx) (2.2.6)

We have picked the normal to the incremental surface ΔS to be ẑ in the above example, and hence, the above gives rise to the identity that

lim_{ΔS→0} (∮_{ΔS} E · dl) / ΔS = ∂/∂x Ey − ∂/∂y Ex = ẑ · ∇ × E (2.2.7)

Picking different ΔS with different orientations and normals n̂, one gets

∂/∂y Ez − ∂/∂z Ey = x̂ · ∇ × E (2.2.8)

∂/∂z Ex − ∂/∂x Ez = ŷ · ∇ × E (2.2.9)

Consequently, one gets

∇ × E = x̂(∂/∂y Ez − ∂/∂z Ey) + ŷ(∂/∂z Ex − ∂/∂x Ez) + ẑ(∂/∂x Ey − ∂/∂y Ex) (2.2.10)

where

∇ = x̂ ∂/∂x + ŷ ∂/∂y + ẑ ∂/∂z (2.2.11)

2.2.1 Faraday’s Law in Differential Operator Form

Faraday’s law is experimentally motivated. Michael Faraday (1791-1867) was an extraordinary experimentalist who documented this law with meticulous care. It was only decades later that a mathematical description of this law was arrived at.

Faraday’s law in integral form is given by4

∮_C E · dl = − d/dt ∬_S B · dS (2.2.12)

Assuming that the surface S is not time varying, one can take the time derivative into the integrand and write the above as

∮_C E · dl = − ∬_S ∂/∂t B · dS (2.2.13)

One can replace the left-hand side with the use of Stokes’ theorem to arrive at

∬_S ∇ × E · dS = − ∬_S ∂/∂t B · dS (2.2.14)

4 Faraday’s law is experimentally motivated. Michael Faraday (1791-1867) was an extraordinary experimentalist who documented this law with meticulous care. It was only decades later that a mathematical description of this law was arrived at.

The normal of the surface element dS can be pointing in an arbitrary direction, and the surface S can be very small. Then the integral can be removed, and one has

∇ × E = − ∂B/∂t (2.2.15)

The above is Faraday’s law in differential operator form.

In the static limit is

∇ × E = 0 (2.2.16)

2.2.2 Physical Meaning of Curl Operator

The curl operator ∇× is a measure of the rotation or the circulation of a field at a point in space. On the other hand, ∮_{ΔC} E · dl is a measure of the circulation of the field E around the loop formed by C. Again, the curl operator has its complicated representations in other coordinate systems, a subject that will not be discussed in detail here.

It is to be noted that our proof of the Stokes’s theorem is for a flat open surface S, and not for a general curved open surface. Since all curved surfaces can be tessellated into a union of flat triangular surfaces according to the tiling theorem, the generalization of the above proof to curved surface is straightforward. An example of such a triangulation of a curved surface into a union of triangular surfaces is shown in Figure 2.7.

Figure 2.7: An arbitrary curved surface can be triangulated with flat triangular patches. The triangulation can be made arbitrarily accurate by making the patches arbitrarily small.

2.2.3 Example

Suppose E = x̂3y + ŷx, calculate ∮ E · dl along a straight line in the x-y plane joining (0,0) to (3,1).

2.3 Maxwell’s Equations in Differential Operator Form

With the use of Gauss’ divergence theorem and Stokes’ theorem, Maxwell’s equations can be written more elegantly in differential operator forms. They are:

∇ × E = − ∂B/∂t (2.3.1)

∇ × H = ∂D/∂t + J (2.3.2)

∇ · D = ρ (2.3.3)

∇ · B = 0 (2.3.4)

These equations are point-wise relations as they relate field values at a given point in space. Moreover, they are not independent of each other. For instance, one can take the divergence of the first equation (2.3.1), making use of the vector identity that ∇ · ∇ × E = 0, one gets

− ∂∇ · B/∂t = 0 → ∇ · B = constant (2.3.5)

This constant corresponds to magnetic charges, and since they have not been experimentally observed, one can set the constant to zero. Thus the fourth of Maxwell’s equations, (2.3.4), follows from the first (2.3.1).

Similarly, by taking the divergence of the second equation (2.3.2), and making use of the current continuity equation that

∇ · J + ∂ρ/∂t = 0 (2.3.6)

one can obtain the second last equation (2.3.3). Notice that in (2.3.3), the charge density ρ can be time-varying, whereas in the previous lecture, we have “derived” this equation from Coulomb’s law using electrostatic theory.

The above logic follows if ∂/∂t ≠ 0, and is not valid for static case. In other words, for statics, the third and the fourth equations are not derivable from the first two. Hence all four Maxwell’s equations are needed for static problems. For electrodynamic problems, only solving the first two suffices.

Something is amiss in the above. If J is known, then solving the first two equations implies solving for four vector unknowns, E, H, B, D, which has 12 scalar unknowns. But there are only two vector equations or 6 scalar equations in the first two equations. Thus we need more equations. These are provide by the constitutive relations that we shall discuss next.

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