Part 2 · Light

2.1 Vector operators and Maxwell's equations

Source: Postorino, week 2 slides, pp. 12–14

Objectives

Vocabulary. Nabla, gradient, divergence, curl, Laplacian; charge and current density, polarisation, magnetisation, $D$ and $B$ .
Picture. Divergence is outflow from a point, curl is circulation around it.
Theorem. Maxwell’s equations in matter, in compact form, and in vacuum.
Key. In vacuum they are linear and symmetric in $E$ and $B$ : each field’s change makes the other.

Notation, as on the slides: vectors are bold; the cross product and the curl are written with $\land$ ; a product of two vectors written side by side, $kr$ , is their scalar product $k \cdot r$ .

Vector operators

Definition 2.1.1Nabla, gradient, divergence, curlslides w2, p. 12

Nabla is a vector operator, $\nabla = i \frac{\partial}{\partial x} + j \frac{\partial}{\partial y} + k \frac{\partial}{\partial z},$ with $i, j, k$ the unit vectors along $x, y, z$ . It gives three differential operators.

Gradient of a scalar field $f (x, y, z)$ : $\nabla f = i \partial_{x} f + j \partial_{y} f + k \partial_{z} f = grad f$ .
Divergence of a vector field $v = v_{1} i + v_{2} j + v_{3} k$ : $\nabla \cdot v = \partial_{x} v_{1} + \partial_{y} v_{2} + \partial_{z} v_{3} = div v$ .
Curl (rotore): $\nabla \land v = i \partial_{x} v_{1} j \partial_{y} v_{2} k \partial_{z} v_{3} = rot v .$

Expanded beyond the lectures: $\nabla \land v = (\partial_{y} v_{3} - \partial_{z} v_{2}) i + (\partial_{z} v_{1} - \partial_{x} v_{3}) j + (\partial_{x} v_{2} - \partial_{y} v_{1}) k .$

Key observation 2.1.2What they measurebeyond the lectures

The gradient points uphill, in the direction of fastest increase of $f$ ; its length is that rate of increase.
The divergence is the net outflow of $v$ per unit volume: positive at a source, negative at a sink.
The curl measures how much $v$ circulates around a point: a little paddle wheel placed in the flow spins about the curl’s direction, at a rate proportional to its size.

Loading the demo…

Figure 2.1.1 A vector field in the plane. Drag the

probe

: the small box shows what flows in and out of it (divergence), the paddle wheel how the field turns around it (curl, the

k

component). Try the presets: a source has divergence and no curl, a vortex the reverse, and a shear flow, made of parallel arrows, still has curl.

Definition 2.1.3Laplacianslides w2, p. 12

The Laplacian is nabla dotted with itself, $\nabla \cdot \nabla$ : $\nabla^{2} = \frac{\partial ^{2}}{\partial x ^{2}} + \frac{\partial ^{2}}{\partial y ^{2}} + \frac{\partial ^{2}}{\partial z ^{2}} .$ On a vector field it acts on each component.

$\nabla^{2} f$ compares $f$ at a point with its average around the point beyond the lectures. It is the 3D version of the $\partial^{2} / \partial x^{2}$ in the 1D wave equation.

Maxwell’s equations in matter

The electromagnetic field in matter is defined by four quantities:

the charge density $ρ$ , with $\int ρ d V = Q$ ( $d V$ a volume element);
the current density $J$ , current per unit area, with $\int J \cdot d A = I$ ; $d A$ is an area element, perpendicular to the surface;
the electric polarisation per unit volume $P$ , the density of electric dipoles;
the magnetic polarisation per unit volume $M$ (magnetisation).

N.B. All of these are macroscopic: averaged over small but macroscopic volumes, so that the discrete nature of the charges at the atomic level can be ignored.

Theorem 2.1.4Maxwell’s equations in matterslides w2, p. 13

Maxwell’s equations (1865) link these quantities to the macroscopic fields $E$ and $H$ : $\nabla \cdot E = - \frac{1}{ε _{0}} \nabla \cdot P + \frac{ρ}{ε _{0}}, \nabla \cdot H = - \nabla \cdot M,$ $\nabla \land E = - μ_{0} \frac{\partial H}{\partial t} - μ_{0} \frac{\partial M}{\partial t}, \nabla \land H = ε_{0} \frac{\partial E}{\partial t} + \frac{\partial P}{\partial t} + J .$ $μ_{0}$ is the magnetic permeability of vacuum, $ε_{0}$ the electric permittivity of vacuum.

The slide calls $ε_{0}$ “permeabilità elettrica”, a slip for permittività.

RemarkReading the matter termsbeyond the lectures

$- \nabla \cdot P$ is the bound charge that polarisation piles up where $P$ is not uniform: it adds to the free $ρ$ . $\partial P / \partial t$ is the current of bound charges moving as the dipoles change: it adds to the free $J$ . Both come back as source terms in §3.1 Dielectrics: the Lorentz model.

Compact form, and vacuum

Introduce two vectors:

the electric displacement (spostamento dielettrico) $D = ε_{0} E + P$ ;
the magnetic induction $B = μ_{0} (H + M)$ .

Theorem 2.1.5Maxwell’s equations, compact formslides w2, p. 14

$\nabla \cdot D = ρ, \nabla \cdot B = 0, \nabla \land E = - \frac{\partial B}{\partial t}, \nabla \land H = \frac{\partial D}{\partial t} + J .$

Proof

$D$ and $B$ absorb the matter terms: $\nabla \cdot D = ε_{0} \nabla \cdot E + \nabla \cdot P = ε_{0} (- \frac{1}{ε _{0}} \nabla \cdot P + \frac{ρ}{ε _{0}}) + \nabla \cdot P = ρ,$ $\nabla \cdot B = μ_{0} (\nabla \cdot H + \nabla \cdot M) = μ_{0} (- \nabla \cdot M + \nabla \cdot M) = 0.$ For the curls (not written out on the slide): in Faraday’s law the right side $- μ_{0} \partial_{t} (H + M)$ is $- \partial_{t} B$ ; in Maxwell’s law $ε_{0} \partial_{t} E + \partial_{t} P$ is $\partial_{t} D$ .

In vacuum there are no free charges or currents and no matter to polarise: $ρ = 0, J = 0, P = 0, M = 0, D = ε_{0} E, B = μ_{0} H .$

Theorem 2.1.6Maxwell’s equations in vacuumslides w2, p. 14

$\nabla \cdot E = 0, \nabla \cdot B = 0, \nabla \land E = - \frac{\partial B}{\partial t}, \nabla \land B = μ_{0} ε_{0} \frac{\partial E}{\partial t} .$ In order:

Gauss’s law;
Gauss’s law for magnetism: there are no magnetic monopoles;
Faraday’s law: a magnetic field that changes in time makes an electric field;
Maxwell’s law: an electric field that changes in time makes a magnetic field.

Key observation 2.1.7Symmetric and linearslides w2, p. 14

A system of linear differential equations, symmetric in the fields $E$ and $B$ . Each field’s change makes the other. That mutual feeding is what lets the fields travel on their own, as a wave (§2.2 Electromagnetic waves in vacuum).

Exercises

Exercise 2.1.1beyond the lectures

Show that $\nabla \cdot (\nabla \land v) = 0$ for any field. What does it imply for $B$ , given Faraday’s law?

Solution

$\partial_{x} (\partial_{y} v_{3} - \partial_{z} v_{2}) + \partial_{y} (\partial_{z} v_{1} - \partial_{x} v_{3}) + \partial_{z} (\partial_{x} v_{2} - \partial_{y} v_{1}) = 0$ , since mixed partial derivatives commute. Taking the divergence of Faraday’s law gives $\partial_{t} (\nabla \cdot B) = 0$ : if there are no monopoles at one time, there never are. The equations are consistent.

Exercise 2.1.2beyond the lectures

Compute the divergence and curl of $v = (x, y, 0)$ , of $v = (- y, x, 0)$ , and of $v = (y, 0, 0)$ .

Solution

$(x, y, 0)$ : div $= 2$ , curl $= 0$ : a source. $(- y, x, 0)$ : div $= 0$ , curl $= (0, 0, 2)$ : a rigid rotation. $(y, 0, 0)$ : div $= 0$ , curl $= (0, 0, - 1)$ : a shear flow circulates although its arrows are parallel.

Exercise 2.1.3beyond the lectures

Compute the gradient of $f = x^{2} + y^{2} + z^{2}$ and its Laplacian.

Solution

$\nabla f = (2 x, 2 y, 2 z) = 2 r$ : radially outward, steeper further out. $\nabla^{2} f = 2 + 2 + 2 = 6$ .

Exercise 2.1.4beyond the lectures

Take the divergence of Maxwell’s law in matter, $\nabla \land H = \partial_{t} D + J$ , and use $\nabla \cdot D = ρ$ . What do you get, and what does it mean?

Solution

$0 = \partial_{t} (\nabla \cdot D) + \nabla \cdot J = \partial_{t} ρ + \nabla \cdot J$ : the continuity equation. Charge is conserved: the charge in a volume decreases only by flowing out. The $\partial_{t} D$ term, Maxwell’s addition, is what makes this work.