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

The curl was defined in Eq. (2.3.55) as the circulation per unit area around an infinitesimal loop. Stokes' theorem is the corresponding statement for a surface of finite size.

Stokes' theorem

Let $S$ be an open surface bounded by the closed curve $C$ , and let $A$ have continuous first derivatives on $S$ . Then

\oint_{C} A \cdot d ℓ = \int_{S} (\nabla \times A) \cdot d S, (2.4.3)

where the direction of $d S$ and the sense of traversal of $C$ are related by the right-hand rule: when the fingers of the right hand curl in the direction of travel along $C$ , the thumb points along $d S$ .

In words: the circulation of a vector field around a closed curve equals the flux of its curl through any surface bounded by that curve.

$(a) Stokes' theorem relates an open surface S and the closed curve C (its rim) that bounds it. Any surface with the same rim C gives the same value of \int_S(\nabla\times\vec A)\cdot d\vec S. (b), (c) The right-hand rule: curl the fingers of the right hand along the direction of travel on C, and the thumb gives the direction of d\vec S. Reversing the direction of travel reverses d\vec S and changes the sign of both sides of Eq. (2.4.3).$

Figure 2.4.7. (a) Stokes' theorem relates an open surface

S

and the closed curve

C

(its rim) that bounds it. Any surface with the same rim

C

gives the same value of

\int_{S} (\nabla \times A) \cdot d S

. (b), (c) The right-hand rule: curl the fingers of the right hand along the direction of travel on

C

, and the thumb gives the direction of

d S

. Reversing the direction of travel reverses

d S

and changes the sign of both sides of Eq. (2.4.3).

Why it is true

Divide $S$ into many small patches $Δ S_{k}$ and traverse the boundary of each in the same (right-handed) sense (Figure 2.4.8). By the definition (2.3.55) the circulation around patch $k$ is $(\nabla \times A)_{k} \cdot a_{n, k} Δ S_{k}$ , to first order. When all the circulations are added, every interior edge is traversed twice, in opposite directions, by the two patches that share it, and these contributions cancel. Only the outer edges, which together form $C$ , survive:

\oint_{C} A \cdot d ℓ = k \sum \oint_{C_{k}} A \cdot d ℓ ≃ k \sum (\nabla \times A)_{k} \cdot Δ S_{k} Δ S_{k} \to 0 \int_{S} (\nabla \times A) \cdot d S .

Why Stokes' theorem holds. (a) The circulation around one small patch is (\nabla\times\vec A)_k\cdot\Delta\vec S_k. (b) Two neighbouring patches, both traversed counter-clockwise, travel their common edge in opposite directions, so its contributions cancel. (c) When the whole surface is divided into patches traversed in the same sense, all interior edges cancel and only the outer rim C survives in the sum of the circulations. — **Figure 2.4.8.** Why Stokes' theorem holds. (a) The circulation around one small patch is $(\nabla \times A)_{k} \cdot Δ S_{k}$ . (b) Two neighbouring patches, both traversed counter-clockwise, travel their common edge in opposite directions, so its contributions cancel. (c) When the whole surface is divided into patches traversed in the same sense, all interior edges cancel and only the outer rim $C$ survives in the sum of the circulations.

Try it yourself: move and resize the loop in the vortex field below. The circulation around the loop always equals the flux of the curl through the enclosed area, also for a loop that does not surround the centre.

InteractiveLoading simulation…

Meaning and consequences

The left-hand side of Eq. (2.4.3) depends only on the curve $C$ . Hence the flux of a curl is the same for every surface spanning the same rim, as noted in Figure 2.4.7: a flat disc, a hemisphere or a "butterfly net" bounded by $C$ all give the same answer. We shall use this freedom in Example 2.4.5.
If $\nabla \times A = 0$ everywhere in a region (and the region has no holes through which a loop could be threaded), the circulation around every closed curve in it vanishes. Conversely, a field with non-zero circulation must have a curl somewhere on every surface spanning the loop.
Combining the theorems gives two identities. For $A = \nabla f$ the gradient theorem gives $\oint_{C} \nabla f \cdot d ℓ = 0$ for every loop, so $\int_{S} (\nabla \times \nabla f) \cdot d S = 0$ for every surface, which is possible only if

$\nabla \times (\nabla f) = 0. (2.4.4)$

Likewise, a closed surface has no rim, so Stokes' theorem gives $\oint_{S} (\nabla \times A) \cdot d S = 0$ . By the divergence theorem, $\int_{V} \nabla \cdot (\nabla \times A) d V = 0$ for every volume, so

$\nabla \cdot (\nabla \times A) = 0. (2.4.5)$

Both identities can of course be checked directly from the Cartesian formulas of Section 2.2. The first one is the reason why the electrostatic field $E = - \nabla V$ is curl-free.

Common mistake

Choosing the direction of travel along $C$ and the direction of $d S$ independently. They are tied together by the right-hand rule, and choosing them inconsistently flips the sign of one side of the theorem.

Quick check

The field $a_{ϕ} / ρ$ has zero curl for $ρ > 0$ (Figure 2.3.32). Compute its circulation around the circle $ρ = a$ in the $x y$ -plane (counter-clockwise). Does the result contradict Stokes' theorem?

$\oint (a_{ϕ} / a) \cdot a_{ϕ} a d ϕ = \int_{0}^{2 π} d ϕ = 2 π \neq = 0$ . No contradiction: every surface spanning this circle is pierced by the $z$ axis, where the field is singular, so the hypothesis of the theorem fails, just as for $a_{r} / r^{2}$ and the divergence theorem. (This field has the shape of the magnetic field of a long straight wire, which you will meet later in the course.)

Example 2.4.4Stokes' theorem for a square

$The field of Example 2.4.4 on the square x=0 (seen from +x, with arrow length proportional to |\vec A|). Along the right side C_2 the field has a strong upward component that helps the travel, whereas along the left side C_4 it is weak and opposes it. Along the top side C_3 the field points against the direction of travel and along the bottom side C_1 with it. The four contributions do not cancel, and the net circulation is positive, i.e. counter-clockwise.$

Figure 2.4.9. The field of Example 2.4.4 on the square

x = 0

(seen from

+ x

, with arrow length proportional to

∣ A ∣

). Along the right side

C_{2}

the field has a strong upward component that helps the travel, whereas along the left side

C_{4}

it is weak and opposes it. Along the top side

C_{3}

the field points against the direction of travel and along the bottom side

C_{1}

with it. The four contributions do not cancel, and the net circulation is positive, i.e. counter-clockwise.

Example 2.4.5Stokes' theorem for a circle

(a) \vec A=-y\,\vec a_x+3x\,\vec a_y in the xy-plane — (a) $A = - y a_{x} + 3 x a_{y}$ in the $x y$ -plane

Illustrating Example 2.4.5. (a) The field (arrows proportional to \vec A) swirls counter-clockwise, along the direction of travel on C, so the circulation is positive, 36\pi. (b) The flat disc and the upper hemisphere are both bounded by C. With d\vec S chosen by the right-hand rule (upwards on the disc, outwards on the hemisphere), both carry the same flux of the curl, 36\pi. — **Figure 2.4.10.** Illustrating Example 2.4.5. (a) The field (arrows proportional to $A$ ) swirls counter-clockwise, along the direction of travel on $C$ , so the circulation is positive, $36 π$ . (b) The flat disc and the upper hemisphere are both bounded by $C$ . With $d S$ chosen by the right-hand rule (upwards on the disc, outwards on the hemisphere), both carry the same flux of the curl, $36 π$ .

2.4.3 Stokes' theorem

Why it is true

Meaning and consequences

In this part

Examples · 2