Electromagnetic Fields

Lecture 1

Vector Calculus

Electromagnetic Fields, Lecture 1, slide 2

What is field?

Mathematics: a map Rn →Rm which assigns each point a quantity (scalar or vector).

Physics:property of space – action (observed as a force acting on objects)

In this course:“Field” means property or space in which this property is observed or function which describes this property.

Electromagnetic Fields, Lecture 1, slide 3

Vector calculus

Vector calculus (or vector analysis) is a branch of mathematics concerned with differentiation and integration of vector fields, primarily in 3 dimensional Euclidean space R3

Euclidean vector is a geometric object that has both a magnitude (or length) and direction.

Vectors are fundamental in the physical sciences.

Electromagnetic Fields, Lecture 1, slide 4

Why we use vectors

Under changes of coordinate systems they behave the same way points do.

Thus, vector notation of most of the equations used to describe physical fields' properties is independent on the coordinate system being used.

VECTORS ARE CONVENIENT!

[Physics] General covariance: the essential idea is that coordinates do not exist a priori in nature, but are only artifices used in describing nature, and hence should play no role in the formulation of fundamental physical laws.

Electromagnetic Fields, Lecture 1, slide 5

Coordinate systemsA coordinate system is a system which uses a set of numbers, or coordinates, to uniquely determine the position of a point or other geometric element.Using CS we can transform problems of geometry into problems concerning numbers (and then solve these problems by calculation).

Vectors allow nice and general presentation of ideas and general behaviour of field, but we, engineers, do need

numbers to quantitatively calculate, predict and design...

y

x

2.65

3.33

P(3.33,2.65)

Electromagnetic Fields, Lecture 1, slide 6

Cartesian coordinate system

Three perpendicular planes are chosen and the three coordinates of a point are the signed distances to each of the planes

x – the signed distance from the yz planey – the signed distance from the xz planez – the signed distance from the xy plane

Electromagnetic Fields, Lecture 1, slide 7

Cylindrical coordinate system

r – the radius is the Euclidean distance from the origin

φ – the azimuth (or azimuthal angle) is the signed angle measured from the azimuth reference direction to the orthogonal projection of the line segment OP on the reference plane.

z – the height is the signed distance from the chosen plane

The origin, the plane and the reference direction on this plane are chosen.

Electromagnetic Fields, Lecture 1, slide 8

Spherical coordinate system

r – the radius is the Euclidean distance from the origin

θ – the inclination (or polar angle) is the angle between the zenith direction and the line segment OP.

φ – the azimuth (or azimuthal angle) is the signed angle measured from the azimuth reference direction to the orthogonal projection of the line segment OP on the reference plane.

The origin, the plane and the azimuth reference direction are chosen. The zenith direction direction is perpendicular to the plane.

Electromagnetic Fields, Lecture 1, slide 9

Coordinate transformationCartesian(x,y,z)

Cylindrical(ρ,φ,z)

Spherical(r,θ,φ)

Cartesian(x,y,z)

Cylindrical(ρ,φ,z)

Spherical(r,θ,φ)

= x2 y2

=arctan y / x z=z

r= x2 y2z2

=arccos z /r

=

x=cos

y=sin

z=z

=2z2

=arctan y / x =arctan / z

x=r sin cos

y=r sinsin

z=r cos

=r sin

=

z=r cos teta

These are not all elements we must change! More follows....

Electromagnetic Fields, Lecture 1, slide 10

Unit vectors transformationCartesian(x,y,z)

Cylindrical(ρ,φ,z)

Spherical(r,θ,φ)

Cartesian

Cylindrical

Spherical

1=x1x

y1 y

1=−y1x

z1 y

1z=1z

1r=x1x y 1 yz1 z

r

1=x z1x y z1 y−21 z

r

1=−y1xx1y

1x=cos1−sin1

1 y=sin 1cos1

1z=1z

Still, these are not all elements we must change! More follows....

1x ,1 y ,1z

1r ,1 ,1

1 ,1 ,1z

1x=sin cos1rcos cos1−sin 1

1 y=sin sin 1rcoscos1cos1

1z=cos1r−sin 1

1r=

r1

zr1z

1=zr1−

r1 z

1=1

1=sin 1rcos1

1=1

1z=cos1r−sin 1

Electromagnetic Fields, Lecture 1, slide 11

Representation of vectorsVectors are similar to points and thus their can be

represented the same way the points are.

y

x

2.65

3.33

P(3.33,2.65)

v=[3.33,2.65]v=[0,0,5.3]

A

Z

Electromagnetic Fields, Lecture 1, slide 12

Basic operations

Scalar multiplication multiplication of a scalar (field) and a vector (field),

yielding a vector field:

w = a v

Vector addition (substraction) addition of two vectors (vector fields),

yielding a vector (field):

w = v + u

v

w=v + u

v u

Electromagnetic Fields, Lecture 1, slide 13

Dot productMultiplication of two vectors (vector fields), yielding a scalar (field):

s = v ∙ u = |v||u| cos θ

θ

v

u

|v| cos θ

Properties: the dot product is commutative:

v ∙ u = u ∙ v

the dot product is distributive over vector addition:

w ∙ ( v + u ) = w ∙ u + w ∙ vUsing coordinates:

v = [ vi, v

j, v

k ], w = [ w

i, w

j, w

k ]

v ∙ u = v

iw

i+v

jw

j+v

kw

k

Electromagnetic Fields, Lecture 1, slide 14

Cross productMultiplication of two vectors (vector fields), yielding a vector (field):

w = v × u = |v||u| sin θ 1n

θv

u

v×u

|v×u|Properties: the dot product is anticommutative:

v × u = - u × v

the dot product is distributive over vector addition:

w × ( v + u ) = w u + w × v

Using coordinates:v = [ v

i, v

j, v

k ], w = [ w

i, w

j, w

k ]

v × u = det [

i j kvi v j vkwi w j wk

]

Electromagnetic Fields, Lecture 1, slide 15

Del – a special ,,vector”A convenient mathematical notation for three differential operators used in vector calculus:

∇=[ ∂∂ x,

∂∂ y,

∂∂ z ]

What is partial derivative?

f(x,y,z)=2xy+sin y + y e- z

∂ f∂ x

=2 y

∂ f∂ y

=cos ye−z

∂ f∂ z

=−y e−z

Del operator represented by nabla symbol

Electromagnetic Fields, Lecture 1, slide 16

GradientThe gradient of a scalar field f is a vector field pointing in the direction of fastest increase of f.

∇ f x , y , z=[ ∂ f∂ x,∂ f∂ y,∂ f∂ z ]

f x , y =x2 y2

∇ f x , y =[2 x ,2 y ]

Electromagnetic Fields, Lecture 1, slide 17

Gradient example

Field of a dipole (+q,-q): 3D surface, equilines and gradient (direction only) (Example from http://www.gnuplot.info/demo/vector.html)

Electromagnetic Fields, Lecture 1, slide 18

Line integral of a vector field

∫Lu r ⋅d r=∫a

bu r t ⋅r ' t dt

where is a bijective parametrization of Lr : [a ,b ] L

|dr| u

ur

u

∫Lu r ⋅d r≈∑i

Liur i1

23

i... ...

Electromagnetic Fields, Lecture 1, slide 19

Gradient theorem

∫L∇ f⋅d r=f e−f b

b

e

b

L

We use this theorem when trying to integratea vector field u along a line. If u is a gradientof a scalar field f, then the integral is path-independent.

Electromagnetic Fields, Lecture 1, slide 20

Flux of a vector field

=∬Su⋅ndS

u

n

uun

Electromagnetic Fields, Lecture 1, slide 21

DivergenceThe divergence of a vector field u is a scalar field that measures source or sink of u at a given point.

∇⋅u=limr0

∯S r u⋅ndS

∣V r ∣u= [ux x , y , z , uy x , y , z ,uz x , y , z ]

∇⋅u=∂ux x , y , z

∂ x

∂uy x , y , z

∂ y

∂uz x , y , z

∂ z

u=[ x3

3,y2

2 ]

∇⋅u=x2 y

Electromagnetic Fields, Lecture 1, slide 22

Divergence theoremAlso knows as Gauss' theorem, Ostrogradsky's theorem or Gauss-Ostrogradsky theorem.

∭V∇⋅udV=∯∂V

u⋅ndS

Volume integral of the divergence of a vector field u is equal to the outward flux of u through the volume boundary (surface).

V

S

n

n

n n

Electromagnetic Fields, Lecture 1, slide 23

CurlThe curl of a vector field u is a vector field that measures rotation of u at a given point.

∇×u=limr0

∯S rn×udS

∣V r ∣ u= [ux x , y , z , uy x , y , z ,uz x , y , z ]

∇×u=∣1x 1 y 1z∂

∂ x∂

∂ y∂

∂ zux uy uz

∣u=[ y

3

3,x2

2,0]

∇×u= [0,0, x− y2 ]

Electromagnetic Fields, Lecture 1, slide 24

Curl – interpretation

u=[ y3

3,x2

2,0]

∇×u= [0,0, x− y2 ]

Electromagnetic Fields, Lecture 1, slide 25

Stokes' theoremStrictly speaking this is Kelvin-Stokes theorem (a special case of more general Stokes' theorem).

∬S∇×u dS=∮

∂Sud r

Flux of the curl of a vector field u through a surface is equal to the line integral of u along the surface boundary (closed line).

uu

dS

∂S

S

Electromagnetic Fields, Lecture 1, slide 26

Green's theoremLet's start with curl of a 2D vector, written in 3D as u=[ L, M, 0 ]:

Calculating a flux of this curl through any 2D surface dS=[dx×dy]:

According to the Stokes' theorem

Green's theorem:

∇×u= ∂0∂ y−

∂M∂ z i ∂L∂ z −∂0

∂ x j ∂M∂ x −∂ L∂ y k

∬s∇×u d S=∬s

∇×u⋅k dS=∬S ∂M∂ x −∂ L∂ y dS

∬S∇×u dS=∮

∂Sud r

∬S ∂M∂ x −∂ L∂ y dS=∮∂ S

u⋅d r=∮∂ S

[ L,M ,0]⋅[dx ,dy , dz ]=∮∂ S

LdxM dy

∮∂SLdxM dy =∬S ∂M∂ x −

∂L∂ y dS

Electromagnetic Fields, Lecture 1, slide 27

Second order derivatives

The most important second order differential operator is Laplacian:

which may be defined for vector field as:

f=∇⋅∇ f=∇2 f

In math we have several possibilities:for scalar field:

and

for vector field:

and and

∇⋅∇ f ∇×∇ f≡0

∇ ∇⋅u ∇×∇×u ∇⋅∇×u≡0

∇2u=∇∇⋅u−∇×∇×u

Electromagnetic Fields, Lecture 1, slide 28

Identities

∇×∇ f≡0

Curl of the gradient of any scalar field is always zero vector:

This identity allows us express any curl free vector field by a scalar field.

Divergence of the curl of any vector field is always zero:

This identity allows us express any divergence free vector field by another vector field.

∇⋅∇×u≡0

Electromagnetic Fields, Lecture 1, slide 29

Conservative field

u=∇ f

A vector field u is said to be conservative if it is a gradient of scalar field:

As we have seen earlier, curl of a conservative field must be zero:

From Stokes' theorem it follows that:

∇×u≡0

∮∂Sud r=∬S

∇×udS=0

∮Cud r=∮C

∇ f d r=0

∫L∇ f⋅d r=f e−f b

∫L'∇ f⋅d r=f e−f b

e

b

L

L'

Electromagnetic Fields, Lecture 1, slide 30

Solenoidal field

∇⋅u=0

A vector field u is said to be solenoidal if its divergence is zero:

As we have seen earlier, a solenoidal field can be expressed as the curl of another vector field:

From divergence theorem it follows that flux of solenoidal field through anyclosed surface must be zero:

∇⋅∇×w≡0 u=∇×w

∯∂Vud S=∭V

∇⋅u dV=0

Electromagnetic Fields, Lecture 1, slide 31

PseudovectorIt was said earlier, that vectors transforms with coordinate system changes like points do. Strictly speaking we can observe two slightly different behavior of “vectors” under mirroring:

Electric field E of single positive charge q.E is a vector field.

Magnetic field B of a single wire .B is a pseudovector field.

Electromagnetic Fields, Lecture 1, slide 32

TensorTensors can bee seen as an extension of series: scalar, vector, … In general they can be considered as a multidimensional array of numbers or functions.

In field theory they have many uses, but here we shall need them to express material properties of some non-trivial media.

T=[ t xx t xyt yx t yy ]