UNIVERSITÀ DI PISA

Electromagnetic RadiationsElectromagnetic Radiationsd Bi l i l I id Bi l i l I i

“Laurea Magistrale” in “Laurea Magistrale” in BiomedicalBiomedical EngineeringEngineering

and Biological Interactionsand Biological Interactions

First First semestersemester (6 (6 creditscredits), ), academicacademic yearyear 2011/122011/12

Prof. Paolo Prof. Paolo NepaNepa

Fields and Differential Operators

p.nepa@iet.unipi.itp.nepa@iet.unipi.it

Fields and Differential Operators

1

Edited by Dr. Edited by Dr. AndaAnda GuraliucGuraliuc28/09/2011

Lecture ContentLecture Content

Scalar and vector fields Static and dynamic fields 

Frequency and time domainPhasors

228/09/2011

Scalar fieldsScalar fieldsA scalar field U is defined in a region of space where it can be defined a scalar function:

Examples include the temperature distribution throughout space, the pressure distribution in afluid or in the atmosphere A scalar field is represented by surfaces (or lines) where U(P)=const

( ) ( )U U P U r= =

fluid or in the atmosphere. A scalar field is represented by surfaces (or lines), where U(P)=const.

Temperature mapTopographical map

328/09/2011

Vector fieldsVector fieldsA vector field U is defined in a region of space where it can be defined a vector function:

( ) ( ) ( ) ( ) ( )x y zx y zU U P U r U r i U r i U r i= = = + +

Vector fields are often used to model, for example, the strength and direction of some force,such as the magnetic or gravitational force, the speed and direction of a moving fluidthroughout space. This physical intuition leads to notions such as the divergence (whichrepresents the rate of change of volume of a flow) and curl (which represents the rotation of aflow).flow).

+ ‐

Field lines of two point charges  

Q+ Q−

An irrotational vector field

4

A vector field is described by a set of direction lines, also known as stream lines or flux lines. The direction line is a curve constructed so that the field is tangential to the curve in all points of the curve.

f p gf

28/09/2011

Static and dynamic fieldsStatic and dynamic fields

The temperature in a room is a field of temperature, composed of the values of temperature in a

If the physical quantity associated with a scalar or vector field is:time independent we are talking about static fields

The temperature in a room is a field of temperature, composed of the values of temperature in anumber of points of the room. We do not see the field, but it exists, and we can for instance visualizeconstant‐temperature or isothermal surfaces.

‐ time independent we are talking about static fields.‐ time dependent we are talking about dynamic fields. Considering, for instance, the

temperature field, the room may be heated or cooled, which makes the temperature field time dependent.

Generally, fields depend on time: or( )U r t ( , )U r tGenerally, fields depend on time:             or 

5

( , )U r t ( , )U r t

28/09/2011

Gradient of a scalar fieldGradient of a scalar fieldThe gradient of a scalar field is a vector field that points in the direction of the greatest rate of increaseof the scalar field, and whose magnitude is the rate of change., g g

For a scalar field                   , the variation df for (x+dx, y+dy, z+dz):  ( , , )f x y zf f fdf dx dy dzx y z∂ ∂ ∂

= + +∂ ∂ ∂ x y z

f f fdf i i i dl⎛ ⎞∂ ∂ ∂

= + + ⋅⎜ ⎟∂ ∂ ∂⎝ ⎠x y zd l dxi dyi dzi= + +

yx y z⎜ ⎟∂ ∂ ∂⎝ ⎠

gradient of f:

f f f⎛ ⎞∂ ∂ ∂ ⎛ ⎞∂ ∂ ∂x y z

f f fgradf f i i ix y z

⎛ ⎞∂ ∂ ∂= ∇ = + +⎜ ⎟∂ ∂ ∂⎝ ⎠ x y zi i i

x y z⎛ ⎞∂ ∂ ∂

∇ = + +⎜ ⎟∂ ∂ ∂⎝ ⎠For a length element dl, the max(df ) is obtained when      is oriented in the same direction of the gradient:dl

( ) ( h h hcosdf f dl f dl α=∇ ⋅ = ∇ ( ),f dlα = ∇ 0α = max( )df df=

/ 2α = π 0df =

(The gradient has the directionof the maximum f variation.)

(The gradient is perpendicular tosurfaces with f=const. and is directedtoward increasing level curves)

∇f∇f

6

d l∇f

d l

28/09/2011

Gradient Gradient ‐‐ examplesexamples

Examples:

( )1. Consider a room in which the temperature is given by a scalar field, T, so at each point (x,y,z) thetemperature is T(x,y,z). (We will assume that the temperature does not change over time.) At each pointin the room, the gradient of T at that point will show the direction the temperature rises most quickly.The magnitude of the gradient will determine how fast the temperature rises in that direction.

2. Consider a surface whose height above sea level at a point (x,y) is H(x,y). The gradient of H at a pointis a vector pointing in the direction of the steepest slope or grade at that point. The steepness of theslope at that point is given by the magnitude of the gradient vector.

728/09/2011

Gradient of a scalar field Gradient of a scalar field –– properties properties 

( )1 2 1 2c u c v c u c v∇ + = ∇ + ∇1. Linearity

( )uv u v v u∇ = ∇ + ∇

( )/ v u u vu v ∇ − ∇∇ =

2. Product rule

3. Division rule ( )2

/u vv

∇ =3. Division rule

( ) ( )b b

a a

f dl df f b f a∇ ⋅ = = −∫ ∫ ‐ It doesn’t depends on the path  from a to b.

dl

dl

a

b

a

828/09/2011

Reference systems Reference systems ‐‐ correspondencecorrespondence

f f ff i i i⎛ ⎞∂ ∂ ∂

∇ ⎜ ⎟

Gradient in Cartesian coordinate system:

x y z

f f ff i i ix y z

⎛ ⎞∂ ∂ ∂∇ = + +⎜ ⎟∂ ∂ ∂⎝ ⎠

Gradient in Cylindrical coordinate system:Gradient in Cylindrical coordinate system:

1z

f f ff i i izρ φρ ρ φ

⎛ ⎞∂ ∂ ∂∇ = + +⎜ ⎟∂ ∂ ∂⎝ ⎠

Gradient in Spherical coordinate system:

⎛ ⎞1 1sinr

f f ff i i ir r rθ φθ θ φ

⎛ ⎞∂ ∂ ∂∇ = + +⎜ ⎟∂ ∂ ∂⎝ ⎠

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Curl of a vector fieldCurl of a vector field

The curl (or rotor) is a vector operator that describes the infinitesimal rotation of a vector field.At every point in the field, the curl is represented by a vector. The attributes of this vector(length and direction) characterize the rotation at that point. The direction of the curl is the axisof rotation, as determined by the right‐hand rule, and the magnitude of the curl is themagnitude of rotation

For a vector field ( ) x y zx y zA P A i A i A i= + + y yx xz zx y z

A AA AA Arot A i i iy z z x x y

∂ ∂⎛ ⎞ ⎛ ⎞∂ ∂∂ ∂⎛ ⎞= − + − + −⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠ ⎝ ⎠

magnitude of rotation.

x y zi i i

rot A Ax y z∂ ∂ ∂

= ∇× =∂ ∂ ∂

Considering the “nabla” operator:

x y zA A A

A dl⋅∫Lni

0lim L

n dSi A

dS→⋅∇×

∫dS

li

10

l

28/09/2011

Curl of a vector field Curl of a vector field ‐‐ propertiesproperties

( ) ( )1 0f∇ × ∇ =

( )2 0a∇× = a u= ∇ (    ‐ irrotational field )a

( ) ( )3 a b a b∇× + = ∇× +∇×

( )

( ) ( )4 ( ) ( )f a f a f a∇ × = ∇ × + ∇ ×

( f‐ scalar field;    ‐ vector field)a

1128/09/2011

Curl Curl ‐‐ exampleexample

Example: consider a liquid that flows into a pipe . 

+ +v

z

+ +

x

y

z y0 av

yConsider a wheel inside the liquid, which is small enough to neglect its perturbation to themotion of the liquid. Along the symmetry of the pipe axis the wheel doesn’t move (curl is zero).The wheel rotation velocity depends on the x‐component of the curl where the wheel isinserted (curl sign indicates the rotation – clockwise/counterclockwise).

1228/09/2011

Curl in Cartesian coordinate system:

Reference systems Reference systems ‐‐ correspondencecorrespondence

Curl in Cartesian coordinate system:

y yx xz zx y z

A AA AA AA i i iy z z x x y

∂ ∂⎛ ⎞ ⎛ ⎞∂ ∂∂ ∂⎛ ⎞∇× = − + − + −⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠ ⎝ ⎠

Curl in Cylindrical coordinate system:

⎛ ⎞ ⎛ ⎞ ⎛ ⎞( )1 1z zz

A A AA AA i i A iz zφ ρ ρ

ρ φ φρρ φ ρ ρ ρ φ

∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂∇× = − + − + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠

Curl in Spherical coordinate system:

( ) ( ) ( )1 1 1 1sin r rA A AA A i rA i rA iθθ⎛ ∂ ⎞ ⎛ ⎞∂ ∂∂ ∂ ∂⎛ ⎞∇× = − + − + −⎜ ⎟⎜ ⎟ ⎜ ⎟( ) ( ) ( )sinsin sinrA A i rA i rA i

r r r r rθ φφ φ θθθ θ φ θ φ θ

∇× = − + − + −⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠ ⎝ ⎠

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Stokes’s theorem (curl theorem)Stokes’s theorem (curl theorem)

Curl theoremCurl theorem

( ) nA dl A i dS⋅ = ∇× ⋅∫ ∫∫S

niStokes’s theorem (     surface S whose ∀( )

L S∫ ∫∫

Lli

boundary coincides with L )∀

(    and L path are related by the right‐hand rule)i( p y g )ni

L dLProof: 

( )i

n idL

A dl A i dS⋅ = ∇× ⋅∫idSidL

Stokes’s theoremi

( )1 1

i

N N

n ii idL

A dl A i dS= =

⋅ = ∇× ⋅∑ ∑∫S

ni

Stokes s theorem 

14

S

28/09/2011

DivergenceDivergenceThe divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given

i t i t f i d l M t h i ll th di t th l d it f thpoint, in terms of a signed scalar. More technically, the divergence represents the volume density of theoutward flux of a vector field from an infinitesimal volume around a given point. For example, consider air asit is heated or cooled. The relevant vector field for this example is the velocity of the moving air at a point. Ifair is heated in a region it will expand in all directions such that the velocity field points outward from that

i Th f th di f th l it fi ld i th t i ld h iti l thregion. Therefore the divergence of the velocity field in that region would have a positive value, as theregion is a source. If the air cools and contracts, the divergence is negative and the region is called a sink.

∫∫ni

SdS

0lim

nS

V

A i dSA

V∆ →

⋅∇ ⋅

∆

∫∫Divergence

S

V∆Proof: 

V

For a vector field ( ) x y zx y zA P A i A i A i= + +yx z

AA AdivAx y z

∂∂ ∂+ +

∂ ∂ ∂⎛ ⎞∂ ∂ ∂

V∆

15

Considering the “nabla” operator: x y zdivA A i i i Ax y z

⎛ ⎞∂ ∂ ∂= ∇⋅ = + + ⋅⎜ ⎟∂ ∂ ∂⎝ ⎠

ni dS

28/09/2011

( ) ( )

Divergence Divergence ‐‐ propertiesproperties

( ) ( )1 0a∇ ⋅ ∇× =

( )2 0b∇⋅ = b a= ∇× (    ‐ soleinodal field )b

( ) ( ) ( ) ( )3 a b b a a b∇⋅ × = ⋅ ∇× − ⋅ ∇×( ) ( ) ( ) ( )3 a b b a a b∇ × = ∇× ∇×

( ) ( ) ( ) ( )4 ua u a u a∇⋅ = ∇ ⋅ + ∇ ⋅

( ) ( )5 a b a b∇⋅ + = ∇ ⋅ +∇ ⋅

Helmholtz’s theorem:  i sa a a= + with 0ia∇× =0ia∇⋅ = 0ia∇

irrotational field  solenoidal field 

1628/09/2011

Divergence Divergence ‐‐exampleexample

Example: the electrical charge generates an electric field: 

If at a point P there is no charge:

( )∃ charge inside of the volume V

0E∇⋅ =If: 0n

S

E i dS⋅ ≠∫∫

nS

E i dS Q⋅ =∫∫1S Sni

1S

2

0nS

E i dS⋅ =∫∫+

ni 2S

ni0n

S

E i dS⋅ =∫∫

n

1728/09/2011

Divergence in Cartesian coordinate system:

Reference systems Reference systems ‐‐ correspondencecorrespondence

yx zAA AA

x y z∂∂ ∂

∇ ⋅ = + +∂ ∂ ∂

Divergence in Cartesian coordinate system:

Divergence in Cylindrical coordinate system:

( )1 1 zA AA Az

φρρ

ρ ρ ρ φ∂ ∂∂

∇ ⋅ = + +∂ ∂ ∂

Divergence in Spherical coordinate system:

( ) ( )22

1 1 1sinr

AA r A A φ

θ θ∂∂ ∂

∇ ⋅ = + +( ) ( )2 sin sinrr r r rθθ θ θ φ∂ ∂ ∂

1828/09/2011

Divergence theoremDivergence theoremGauss’s theorem (divergence theorem)Gauss’s theorem (divergence theorem)

∫∫ ∫∫∫

ni

SdS

( )nS V

A i dS A dV⋅ = ∇ ⋅∫∫ ∫∫∫ Gauss’s theorem 

V

P f

ni ‐is pointing in the direction outward the volume

( )n iA i dS A V⋅ = ∇ ⋅ ∆∫∫

Proof: 

iS∆

( )1 1

i

N N

n ii iS

A i dS A V= =∆

⋅ = ∇ ⋅ ∆∑ ∑∫∫V S

N Gauss’s theorem 

19

iV S

28/09/2011

Laplace operatorLaplace operatorThe Laplacian operator is a differential operator given by the divergence of the gradient ofa function on Euclidean spacea function on Euclidean space.

For an f scalar function: ( ) 2f f∇⋅ ∇ =∇

LaplacianLaplacian

In Cartesian coordinates:2 2 2

2

2 2 2

f f ffx y z

∂ ∂ ∂∇ = + +

∂ ∂ ∂

If  2 0f∇ = f ‐ harmonic function

Vector Laplacian: ( )2 a a a∇ =∇ ⋅ ∇ ⋅ −∇×∇×p ( )

In Cartesian coordinates: ( ) ( ) ( )2 2 2 2

x y zx y za a i a i a i∇ = ∇ ⋅ + ∇ ⋅ + ∇ ⋅

2028/09/2011

Laplace operator Laplace operator ‐‐ examplesexamples1. Laplace's and Poisson's Equations

A useful approach to the calculation of electric potentials is to relate that potential to the chargedensity which gives rise to it. The electric field is related to the charge density by the divergencerelationship:

E ρ∇ ⋅ =

Eρ‐electric fieldcharge density

0

E∇⋅ =ε

and the electric field is related to the electric potential by a gradient relationship:

ρ0ε‐charge density‐permittivity

E = −∇Φ

The potential is related to the charge density by Poisson's equation: 2

0

ρ∇ ⋅∇Φ =∇ Φ = −

ε

In a charge‐free region of space, this becomes Laplace's equation: 2 0∇ Φ =

2 W E ti21 ∂ φ2. Wave Equation 2

2 2

1 0c t∂ φ

∇ φ+ =∂

2128/09/2011

Reference systems Reference systems ‐‐ correspondencecorrespondence

Laplacian in Cartesian coordinate system:2 2 2

2 f f ffx y z

∂ ∂ ∂∇ = + +

∂ ∂ ∂

Laplacian in Cartesian coordinate system:

Laplacian in Cylindrical coordinate system:

2 22

2 2 2

1 1f f ffz

ρρ ρ ρ ρ φ

⎛ ⎞∂ ∂ ∂ ∂∇ = + +⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

Laplacian in Spherical coordinate system:2

2 22 2 2 2 2

1 1 1sinf f ff r θ∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞∇ = + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠2 2 2 2 2sin sin

fr r r r rθ θ θ θ φ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

2228/09/2011

Frequency & time domainFrequency & time domain

F d i d i ti t l f l h i ti ti h tiFrequency‐domain descriptions are extremely useful when investigating monochromaticor narrow‐band phenomena, when the materials and systems of interest are submitted to asource of sinusoidal fields. The quantities they evaluate are complex numbers or vectors.They are not real, physically measurable quantities (because a sinusoidal source is noty , p y y q (physical), although they may have a real physical content. the general description in thefrequency domain implies complex quantities, with a real and an imaginary part,respectively, which are not physical either.

Time‐domain descriptions evaluate the effect of physical sources, where the phenomenaare described as a function of time and hence they are real and physically measurable.They must be used when investigating wide‐band phenomena.

The interaction of RF/microwave fields with biological tissues is investigated mostly in the/ g g yfrequency domain, with sources considered as sinusoidal.Today numerical signals, such as for telephony, television, and frequency modulated (FM)

radio, may, however, necessitate time domain analyses and measurements.

2328/09/2011

Frequency & time domain Frequency & time domain ‐‐ examplesexamples

Time domain  Frequency domain 

( ) ( )( ) [ ( )]

x t X fX f F x t

⇔= (Fourier Transform)

1

( ) [ ( )]( ) [ ( )]

X f F x tx t F x f−

==

(Fourier Transform)

(Fourier Inverse Transform)

2428/09/2011

Phasors (I)Phasors (I)Phasor is a representation of a sine wave whose amplitude (A), phase (φ), and angular frequency (ω) aretime invariant Phasors reduce the dependencies on these parameters to three independent factorstime‐invariant. Phasors reduce the dependencies on these parameters to three independent factors,thereby simplifying certain kinds of calculations. The frequency factor, which also includes the time‐dependence of the sine wave, is often common to all the components of a linear combination of sinewaves. Using phasors, it can be factored out, leaving just the static amplitude and phase information tobe combined algebraically (rather than trigonometrically)be combined algebraically (rather than trigonometrically).

An example of series RLC circuit andI

RLCV

RV

LV

CV

A system can be outlined: v(t)h(t) i(t)

An example of series RLC circuit andrespective phasor diagram for a specific ω.

RLCV

RV

CVI

LV

If the input to a linear circuit is a sinusoid, then the output from theh(t)circuit will be a sinusoid. Specifically, if we have a voltage sinusoid:

( ) cos( )Mv t A tω ϕ= +Then the current through the linear circuit will also be a sinusoid, although its magnitude and phase may be differentquantities: ( ) cos( )

Ni t A tω ϕ= +

An impulse response of a system can be used to determine the output of the system: ( ) ( ) ( )y t h t x t= ∗x(t)‐ inputh(t)‐ impulse responsey(t)‐ output

In frequency domain:

( ) [ ( )]X f F t( ) ( ) ( )Y f H f X f= In time domain: 1( ) [ ( )]y t F Y f−=

25( ) [ ( )]X f F x t=

Fourier Transform

Inverse Fourier Transform

28/09/2011

Phasors (II)Phasors (II)Phasors are used for a sinusoidal signal representation.  

( ) cos( )a t A tω ϕ+ h jA A e ϕ { } { }( )( ) R Rj t j tA Aω ϕ ω+

A phasor associated with a vector field:$ $( ) cos( ) cos( ) cos( )

M M Mx x y y z ze t E t x E t y E t zω ϕ ω ϕ ω ϕ= + + + + + $

$ $ $

( ) cos( )M

a t A tω ϕ= + phasor: j

MA A e ϕ= { } { }( )( ) Re Rej t j t

Ma t A e Aeω ϕ ω+= =

yx z

M M M

jj j

x y zE E e x E e y E e zϕϕ ϕ= + +

$ $ $ $[ cos cos cos ] [ sin sin sin ]M M M M M Mx x y y z z x x y y z z

E E x E y E z j E x E y E z A jBϕ ϕ ϕ ϕ ϕ ϕ= + + + + + = +$ $

{ }0

4

t E ATt E B

= ⇒ =

= ⇒ = −

Temporal variation:

the vector draws an ellipse:

In time domain: { }( ) Re cos( ) sin( )j te t Ee A t B tω ω ω= = −

ImA

B

e234

Tt E A

Tt E B

= ⇒ = −

= ⇒ =

the     vector draws an ellipse:

A B h lli b li A B⊥ the ellipse becomes a circumference

Re

A- B−

e

A B the ellipse becomes a line

A

B

A B⊥ the ellipse becomes a circumference

A

Bω

26

A A

! A sinusoidal field generates a vector field oscillating along a line, circumference or ellipse.28/09/2011

ReferencesReferences

1. G. Manara, A. Monorchio, P.Nepa, “Appunti di Campi Elettromagnetici”, , p , pp p g2. J. Slater, N. Frank, “Electromagnetism”3. M. Schwartz, “Principle of Electrodynamics”4. http://www.wikipedia.org/

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