Integral Definition of Curl

Integral Definition of Curl
(∇ × v) · ˆn = lim
dA→0
1
dA
∮
v · dl
The component of the curl of a vector field in some direction ˆn is equal to the
circulation per unit area of the field around a loop to which ˆn is the unit normal.
slide 1

Stokes’ Theorem
Divide closed loop C into 2 parts, C1 and C2.
C12
C21
C
C1
Consider the circulation of some vector field v around C
∮ ∫ ∫
v · dl = v · dl + v · dl
C
C1 C2
∫ ∫ ∫ ∫
= v · dl + v · dl + v · dl −
∮C1
C12∮
C2
= v · dl + v · dl
C1+C12
C2+C21
C2
C12
v · dl
In other words the contributions from the common curve in each pair of loops
cancel.
slide 2

Now consider an open surface S bounded by a curve C. Divide the surface in
to an infinite number of infinitesimal rectangles. Add up the circulation of v
around all these rectangles.The contribution from all the sides common
between two rectangles i.e. all the sides except those that lay on the curve C,
cancel. This leaves only the contribution from the sides on curve C.So
∮
v · dl = ∑ ∮
v · dl
C i loop i
But for an infinitesimal rectangular loop
∮
loop i
v · dl = (∇ × v) i · ˆn i dA i
ˆn i normal to surface S for loop i (righthand rule) and dA i is the area of the
loop.
slide 3

So
∮
C
v · dl = ∑ i
(∇ × v) i · ˆn i dA i
In the limit where the loops are infinitesimal, the sum becomes an integral so
∮ ∫
v · dl = (∇ × v) · ˆn dA
C
Replacing ˆndA with dA, we get
- Stokes’ theorem
Example
S
∮
C
v · dl =
∫
S
(∇ × v) · dA
slide 4

Maxwell’s Equations
Connect one scalar field and three vector fields:
1. ρ = density of electric charge (C/m 3 ) - scalar field
2. E = electric field strength - vector field
3. B = magnetic field strength - vector field
4. J = current density (A/m 2 ) - vector field
1 ∇ · E = ρ ǫ 0
2 ∇ · B = 0
3 ∇ × E = − ∂B
∂t
4 ∇ × B = µ 0 J + 1 c 2 ∂E
∂t
slide 5

Faraday’s Law
Electric field induced by a changing magnetic field of flux Φ of magnetic field
B:
∫
Φ = B · dA - definition
S
− ∂Φ ∮
= E · dl - Faraday’s law
∂t
− ∂ (∫ ) ∮
B · dA = E · dl
∂t S
∮ ∫
E · dl = (∇ × E) · dA
S
∫
× E) · dA = −
S(∇ ∂ (∫ ) ∫
∂B
B · dA = −
∂t S
S ∂t · dA
But this is true for any surface, so
∇ × E = − ∂B
∂t - Maxwell 3
slide 6

Ampère’s Law
Magnetic field B due to a current I
∮
B · dl = µ 0 I
C
where the path C encloses the current I = ∫ s J · dA.
Apply Stokes’ theorem
∫
∮
B · dl =
∫
C
S
∫
(∇ × B) · dA = µ 0
S
S
- must be true for any surface, so
∇ × B = µ 0 J
(∇ × B) · dA
J · dA
This isn’t Maxwell 4 - here the current is time independent.
slide 7

But ...
Can’t be true in general. To see why take the divergence of this equation -
∇ · (∇ × B) = 0 - for any B
so ∇ · J = 0
Not in general true. True only for time independent current.
Try
∇ × B = µ 0 J + ǫ 0 µ 0
∂E
∂t
- take the divergence
∇ · (∇ × B) = ∇ · (µ 0 J + ǫ 0 µ 0
∂E
∂t )
but since ∇ · E = ρ ǫ 0
0 = ∇ · J + ǫ 0
∂
∂t ∇ · E
∇ · J + ∂ρ
∂t
= 0 - charge continuity equation
slide 8

Maxwell’s Equations
1 ∇ · E = ρ ǫ 0
⇐⇒ Gauss’ law
2 ∇ · B = 0 ⇐⇒ No magnetic charges
3 ∇ × E = − ∂B ⇐⇒ Faraday’s Law
∂t
4 ∇ × B = µ 0 J + 1 ∂E
⇐⇒ Ampère’s Law
c 2 ∂t
slide 9

EM Waves
Can use Maxwell equations to derive wave equation for em waves.Assume free
space and no current.
∇ × E = − ∂B
∂t
∂E
∇ × B = ǫ 0 µ 0
∂t
Take curl
∂∇ × E
∇ × (∇ × B) = ǫ 0 µ 0
∂t
∇(∇ · B) − ∇ 2 ∂ 2 B
B = −ǫ 0 µ 0
∂t 2
∇ 2 B =
1 ∂ 2 B
c 2 ∂t 2
= −ǫ 0 µ 0
∂ 2 B
∂t 2
where c = 1/ √ ǫ 0 µ 0 - The wave equation for EM waves.
slide 10