Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

Now put a tiny paddle wheel into the flow, with its axle fixed in some direction, and watch it (Figure 2.2.7). If the water pushes harder on the blades on one side than on the other side, the wheel starts to turn. The curl of the field measures this tendency to rotate: its direction is the axle direction in which the wheel spins fastest (by the right-hand rule), and its size tells how fast. Surprisingly, a field whose arrows go round in circles need not have any curl, and a field with straight, parallel arrows can have one. Figure 2.2.7 shows both.

Paddle wheels (axle along z, out of the page) in three computed fields. (a) Rigid rotation, \vec A=-y\,\vec a_x+x\,\vec a_y: the arrows grow with distance from the centre, the outer blades are pushed harder, and the wheel turns counter-clockwise wherever it is placed. (b) Shear flow, \vec A=(y+1.1)\,\vec a_x: the lines are straight, but the flow is faster above the wheel than below it, so the wheel turns clockwise: curl -\vec a_z. (c) Vortex, \vec A=(-y\,\vec a_x+x\,\vec a_y)/(x^2+y^2): the lines are circles, but the speed falls off as 1/\rho just fast enough for the pushes on the inner and outer blades to balance, so the wheel is carried round the centre without spinning. The curl values follow from (2.2.8). — **Figure 2.2.7.** Paddle wheels (axle along $z$ , out of the page) in three computed fields. (a) Rigid rotation, $A = - y a_{x} + x a_{y}$ : the arrows grow with distance from the centre, the outer blades are pushed harder, and the wheel turns counter-clockwise wherever it is placed. (b) Shear flow, $A = (y + 1.1) a_{x}$ : the lines are straight, but the flow is faster above the wheel than below it, so the wheel turns clockwise: curl $- a_{z}$ . (c) Vortex, $A = (- y a_{x} + x a_{y}) / (x^{2} + y^{2})$ : the lines are circles, but the speed falls off as $1/ ρ$ just fast enough for the pushes on the inner and outer blades to balance, so the wheel is carried round the centre without spinning. The curl values follow from (2.2.8).

Circulation and the definition of curl

The circulation of $A$ around a closed path $C$ is the line integral $\oint_{C} A \cdot d ℓ$ : we walk once around $C$ and add up the components of $A$ along our direction of travel times the lengths of the steps (line integrals are treated in detail in Section 2.3.3). If $A$ is a force, the circulation is the work done around the loop. For the paddle wheel, the circulation around the rim measures the net push of the fluid along the rim, i.e. how strongly the wheel is driven round.

Definition · Curl

The curl of a vector field $A$ is a vector whose magnitude is the maximum net circulation of $A$ per unit area, as the area tends to zero, and whose direction is the normal direction of the area when the area is oriented so as to make the circulation maximum. Equivalently, its component along any unit vector $a_{n}$ is

(\nabla \times A) \cdot a_{n} = Δ s \to 0 lim \frac{\oint _{C} A \cdot d ℓ}{Δ s}, (2.2.7)

where $Δ s$ is a small plane area with normal $a_{n}$ bounded by the curve $C$ , and $C$ is traversed in the direction given by the right-hand rule: with the thumb along $a_{n}$ , the fingers curl in the direction of travel.

The two statements agree: by (2.1.7) the left-hand side of (2.2.7), $∣\nabla \times A ∣ cos θ$ , is largest when $a_{n}$ is parallel to $\nabla \times A$ , and its largest value is the magnitude of the curl.

From the definition to the formula: a small rectangle

To obtain the $z$ -component of the curl, take $a_{n} = a_{z}$ and a small rectangle in a plane $z = z_{0}$ , centred at $P (x_{0}, y_{0}, z_{0})$ , with sides $Δ x$ and $Δ y$ (Figure 2.2.8). By the right-hand rule it is traversed counter-clockwise when seen from the $+ z$ side. On each side we take the tangential component of the field at the midpoint:

side	direction of travel	contribution to $\oint A \cdot d ℓ$
1 (bottom, $y = y_{0} - Δ y /2$ )	$+ a_{x}$	$+ A_{x} (x_{0}, y_{0} - Δ y /2) Δ x$
2 (right, $x = x_{0} + Δ x /2$ )	$+ a_{y}$	$+ A_{y} (x_{0} + Δ x /2, y_{0}) Δ y$
3 (top, $y = y_{0} + Δ y /2$ )	$- a_{x}$	$- A_{x} (x_{0}, y_{0} + Δ y /2) Δ x$
4 (left, $x = x_{0} - Δ x /2$ )	$- a_{y}$	$- A_{y} (x_{0} - Δ x /2, y_{0}) Δ y$

Sides 2 and 4 together give, after a Taylor expansion about $P$ exactly as for the divergence,

[A_{y} (x_{0} + \frac{Δ x}{2}, y_{0}) - A_{y} (x_{0} - \frac{Δ x}{2}, y_{0})] Δ y \approx \frac{\partial A _{y}}{\partial x} Δ x Δ y,

and sides 1 and 3 give

- [A_{x} (x_{0}, y_{0} + \frac{Δ y}{2}) - A_{x} (x_{0}, y_{0} - \frac{Δ y}{2})] Δ x \approx - \frac{\partial A _{x}}{\partial y} Δ x Δ y .

Dividing the circulation by $Δ s = Δ x Δ y$ and letting the rectangle shrink,

(\nabla \times A)_{z} = \frac{\partial A _{y}}{\partial x} - \frac{\partial A _{x}}{\partial y},

which is the $a_{z}$ term of (2.2.8) below. Small rectangles in the planes $x = x_{0}$ and $y = y_{0}$ give the other two components (simply rotate the indices cyclically, $x \to y \to z \to x$ ). The three components fit exactly into the determinant form of the cross product (2.1.8), with $\nabla$ in the middle row (the derivatives act on the components in the bottom row):

Curl in Cartesian coordinates

\nabla \times A = a_{x} \frac{\partial}{\partial x} A_{x} a_{y} \frac{\partial}{\partial y} A_{y} a_{z} \frac{\partial}{\partial z} A_{z} = a_{x} (\frac{\partial A _{z}}{\partial y} - \frac{\partial A _{y}}{\partial z}) + a_{y} (\frac{\partial A _{x}}{\partial z} - \frac{\partial A _{z}}{\partial x}) + a_{z} (\frac{\partial A _{y}}{\partial x} - \frac{\partial A _{x}}{\partial y}) . (2.2.8)

The curl turns a vector field into another vector field.

The small rectangle used to derive (\nabla\times\vec A)_z. Its normal is \vec a_z (pointing out of the page), so it is traversed counter-clockwise. The labels give the tangential field component at the midpoint of each side. — **Figure 2.2.8.** The small rectangle used to derive $(\nabla \times A)_{z}$ . Its normal is $a_{z}$ (pointing out of the page), so it is traversed counter-clockwise. The labels give the tangential field component at the midpoint of each side.

Physical picture

Imagine a tiny paddle wheel placed in a flowing fluid with velocity field $A$ , with its axle along $a_{n}$ . It turns if the fluid pushes harder on the blades on one side than on the other, and its rate of rotation is proportional to $(\nabla \times A) \cdot a_{n}$ . The direction of $\nabla \times A$ is the axle direction in which the wheel spins fastest (Figure 2.2.7 and Figure 2.2.10, right). A non-zero curl at a point is sometimes called a vortex source of the field there. Curl does not mean that the field lines go round in circles:

The shear field $A = y a_{x}$ has straight field lines, but $\nabla \times A = - a_{z} \neq = 0$ , because the flow is faster above the wheel than below it.
The field $A = (- y a_{x} + x a_{y}) / (x^{2} + y^{2})$ , whose lines are circles around the $z$ -axis with strength decreasing as $1/ x^{2} + y^{2}$ , has $\nabla \times A = 0$ everywhere except on the $z$ -axis, where it is not defined. Around a small loop made of two arcs (radii $ρ_{1} < ρ_{2}$ , same opening angle) and two radial segments, the outer arc is longer by the factor $ρ_{2} / ρ_{1}$ by which the field there is weaker, so the two arc contributions cancel and the radial segments contribute nothing.
The rigid rotation $A = - y a_{x} + x a_{y}$ has $\nabla \times A = 2 a_{z}$ .

A field whose curl is zero everywhere is called irrotational.

Place the paddle wheel anywhere in the rigid rotation $- y a_{x} + x a_{y}$ below (the tool lists it as "vortex", and it is the field of Figure 2.2.7a): it turns at the same rate everywhere, because the curl is $2 a_{z}$ at every point, not only at the centre.

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In plain words

The curl answers "would a tiny paddle wheel spin here, about which axis, and how fast?". It is the circulation around a tiny loop, per unit area. Only the change of each component across its own direction matters: $\partial A_{y} / \partial x$ versus $\partial A_{x} / \partial y$ , and so on.

In electrostatics the electric field is irrotational, $\nabla \times E = 0$ , which is why it can be written as the gradient of a potential (Section 3.3).

Quick check

Without writing the full determinant, find the curl of $A = x a_{x}$ and of $A = y a_{x}$ .

Only $A_{x}$ is non-zero, so only the terms $\partial A_{x} / \partial z$ (in the $a_{y}$ bracket) and $- \partial A_{x} / \partial y$ (in the $a_{z}$ bracket) can survive. For $A_{x} = x$ both vanish: $\nabla \times A = 0$ (it grows along its direction, not across it). For $A_{x} = y$ : $\nabla \times A = - a_{z}$ , the shear flow of Figure 2.2.7b.

Now try the shear flow $y a_{x}$ itself: the streamlines are straight and parallel, yet the paddle wheel turns clockwise everywhere, because $(\nabla \times A)_{z} = - 1$ .

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Common mistake

Mixing up the order of the two terms in a bracket of (2.2.8), which flips the sign of that component. The pattern is cyclic: if $(i, j, k)$ is $(x, y, z)$ , $(y, z, x)$ or $(z, x, y)$ , the $a_{i}$ bracket is $\partial A_{k} / \partial j - \partial A_{j} / \partial k$ . Thus the first terms are $\partial A_{z} / \partial y$ , $\partial A_{x} / \partial z$ and $\partial A_{y} / \partial x$ . When in doubt, write out the determinant.

2.2.4 Curl

Circulation and the definition of curl

From the definition to the formula: a small rectangle