Electromagnetic Field Theory 1 - cvut.cz CTU-FEE in Prague, Department ofElectromagneticField GENERAL

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  • Lukas Jelinek

    lukas.jelinek@fel.cvut.cz

    Department of Electromagnetic Field

    Czech Technical University in Prague

    Czech Republic

    Ver. 2019/09/23

    Electromagnetic Field Theory 1 (fundamental relations and definitions)

  • CTU-FEE in Prague, Department of Electromagnetic Field

    Electromagnetic Field Theory 1GENERAL INTRO

    Fundamental Question of Classical Electrodynamics

    2 / XXX

    A specified distribution of elementary charges is in a state of arbitrary

    (but known) motion. At certain time we pick one of them and ask what

    is the force acting on it.

    Rather difficult question – will not be fully answered

  • CTU-FEE in Prague, Department of Electromagnetic Field

    Electromagnetic Field Theory 1GENERAL INTRO

    Elementary Charge

    3 / XXX

    Coulomb

    191.602176634 10 Ce −= ⋅

    smallest known

    amount of charge

    As far as we know, all charges in nature have values ,Ne N± ∈ ℤ

  • CTU-FEE in Prague, Department of Electromagnetic Field

    Electromagnetic Field Theory 1GENERAL INTRO

    Charge conservation

    4 / XXX

    Amount of charge is conserved in every frame (even non-inertial).

    Neutrality of atoms has been verified to 20 digits

  • CTU-FEE in Prague, Department of Electromagnetic Field

    Electromagnetic Field Theory 1GENERAL INTRO

    Continuous approximation of charge distribution

    5 / XXX

    ( ) ( ) ( )d d d V S l

    Q V S lρ σ τ= = =∫ ∫ ∫r r r

    Net charge

    Continuous approximation allows for using powerful mathematics

    Volumetric

    density of

    charge

    3C m− ⋅  

    C   

    Surface

    density of

    charge

    2C m− ⋅  

    Line

    density of

    charge

    1C m− ⋅  

  • CTU-FEE in Prague, Department of Electromagnetic Field

    Electromagnetic Field Theory 1ELECTROSTATICS

    Fundamental Question of Electrostatics

    There exist a specified distribution of static elementary charges. We

    pick one of them and ask what is the force acting on it.

    This will be answered in full details

    6 / XXX

  • CTU-FEE in Prague, Department of Electromagnetic Field

    Electromagnetic Field Theory 1ELECTROSTATICS

    Coulomb(’s) Law

    ( ) ( ) 3

    0 4

    qq

    πε

    ′ ′− =

    ′−

    r r F r

    r r

    Measuring

    charge

    C   

    Source

    charge

    C   

    Radius vector of

    the measuring

    charge

    0

    12 1

    0 2

    0 0 1

    6 1

    0

    1 8.8541878128 F m

    299792458 m s

    1.25663706212

    0

    10 H m

    1 c

    c

    ε µ

    µ

    − −

    −= = ⋅

    =

    = ⋅ ⋅

    Radius vector

    of the source

    charge

    Permittivity of

    vacuum

    Permeability

    of vacuum

    Speed of light

    Farad

    Henry

    m    m   

    Force on

    measuring

    charge

    N   

    7 / XXX

  • CTU-FEE in Prague, Department of Electromagnetic Field

    Electromagnetic Field Theory 1ELECTROSTATICS

    Coulomb(’s) Law + Superposition Principle

    ( ) ( ) 3

    0 4

    n n

    n n

    qq

    πε

    ′ ′− =

    ′− ∑

    r r F r

    r r

    Entire electrostatics can be deduced from this formula

    8 / XXX

  • CTU-FEE in Prague, Department of Electromagnetic Field

    Electromagnetic Field Theory 1ELECTROSTATICS

    Electric Field

    ( ) ( ) 3

    0

    1

    4

    n n

    n n

    q

    πε

    ′ ′− =

    ′− ∑

    r r E r

    r r

    Force is represented by field – entity generated by charges and permeating the space

    ( ) ( )q=F r E r

    9 / XXX

    Intensity of

    electric field

    1V m− ⋅  

  • CTU-FEE in Prague, Department of Electromagnetic Field

    Electromagnetic Field Theory 1ELECTROSTATICS

    Continuous Distribution of Charge

    ( ) ( )( ) 3

    0

    1 d

    4 V

    V ρ

    πε ′

    ′ ′− ′=

    ′− ∫

    r r r E r

    r r

    Continuous description of charge allows for using powerful mathematics

    ( ) ( ) 3

    0

    1

    4

    n n

    n n

    q

    πε

    ′ ′− =

    ′− ∑

    r r E r

    r r

    10 / XXX

  • CTU-FEE in Prague, Department of Electromagnetic Field

    Electromagnetic Field Theory 1ELECTROSTATICS

    Continuous Description of a Point Charge

    ( ) ( )n n n

    qρ δ= −∑r r r ( ) ( ) ( )dn n V

    Vδ= −∫F r F r r r

    11 / XXX

    Dirac’s delta

    “function” Defining property of

    Dirac’s delta “function”

  • CTU-FEE in Prague, Department of Electromagnetic Field

    Electromagnetic Field Theory 1ELECTROSTATICS

    Gauss(’) Law

    ( ) ( ) 0

    ρ

    ε ∇⋅ =

    r E r ( )

    0 0

    1 d d

    S V

    Q Vρ

    ε ε ⋅ = =∫ ∫E S r�

    Total charge

    enclosed by

    the surface

    C   

    Differential law

    (local)

    Integral law

    (global)

    Mind the

    orientation of

    the surface

    12 / XXX

  • CTU-FEE in Prague, Department of Electromagnetic Field

    Electromagnetic Field Theory 1ELECTROSTATICS

    Rotation of Electric Field

    d 0 l

    ⋅ =∫ E l�

    Differential law

    (local)

    Integral law

    (global)

    0∇× =E

    13 / XXX

  • CTU-FEE in Prague, Department of Electromagnetic Field

    Electromagnetic Field Theory 1ELECTROSTATICS

    Various Views on Electrostatics

    The physics content is the same, the formalism is different.

    Differential laws of

    electrostatics

    Integral laws of

    electrostatics

    0∇× =E

    ( ) ( ) 0

    ρ

    ε ∇⋅ =

    r E r

    d 0 l

    ⋅ =∫ E l�

    0

    d S

    Q

    ε ⋅ =∫ E S�

    ( ) ( )( ) 3

    0

    1 d

    4 V

    V ρ

    πε ′

    ′ ′− ′=

    ′− ∫

    r r r E r

    r r

    Coulomb’s law

    14 / XXX

  • CTU-FEE in Prague, Department of Electromagnetic Field

    Electromagnetic Field Theory 1ELECTROSTATICS

    Electric potential

    ( ) ( )ϕ= −∇E r r

    Scalar description of electrostatic field

    0∇× =E

    15 / XXX

    ( ) ( ) 0

    1 d

    4 V

    V K ρ

    ϕ πε ′

    ′ ′= +

    ′−∫ r

    r r r

    Electric potential

    Defined up to

    arbitrary constant

  • CTU-FEE in Prague, Department of Electromagnetic Field

    Electromagnetic Field Theory 1ELECTROSTATICS

    Voltage

    B

    A

    W d qU   F l

    Voltage represents connection of abstract field theory with experiments

        B

    A

    d B A U     E l

    16 / XXX

    Work necessary to

    take charge q from

    point A to point B

    Voltage

    V   

    Potential

    difference is a

    unique number

  • CTU-FEE in Prague, Department of Electromagnetic Field

    Electromagnetic Field Theory 1ELECTROSTATICS

    Electrostatic Energy

    Be careful with point charges (self-energy)

    Energy is

    carried by

    charges

    ( ) 2

    0

    1 d

    2 V

    W Vε= ∫ E r( ) ( ) ,0

    0

    1

    8

    1 d d

    8

    i j

    i j i j

    j i

    V V

    q q W

    W V V

    πε

    ρ ρ

    πε

    = −

    ′ ′=

    ′−

    ∫ ∫

    r r

    r r

    r r

    Energy is carried

    by charges and

    fields

    17 / XXX

    Energy is carried

    by fields

    ( ) ( )1 d 2 V

    W Vρ ϕ= ∫ r r

  • CTU-FEE in Prague, Department of Electromagnetic Field

    Electromagnetic Field Theory 1ELECTROSTATICS

    Electrostatic Energy vs Force

    Electrostatic forces are always acting so as to minimize energy of the system

    Energy of a

    system of point

    charges

    ,0

    1

    8

    i j

    i j i j

    j i

    q q W

    πε ≠

    = −

    ∑ r r

    18 / XXX

    ( ) ( )3 0

    4

    j

    j j

    jj

    q W q

    ξξ

    ξ ξ

    ξξ

    πε ≠

    − = −∇ =

    − ∑

    r r F r

    r r

    Coulomb’s law

  • CTU-FEE in Prague, Department of Electromagnetic Field

    Electromagnetic Field Theory 1ELECTROSTATICS

    Electric Stress Tensor

    All the information on the volumetric Coulomb‘s force is c