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Electromagnetic Field Theory - · PDF file This energy must come from a non-conservative field. The source of non-conservative field may be electric batteries (conservation of chemical

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  • Electromagnetic Field Theory

    2nd Year EE Students

    Prof. Dr. Magdi El-Saadawi www.saadawi1.net

    [email protected]

    2016/2017 11/8/2016 1Prof. Dr. Magdi El-Saadawi

    http://www.saadawi1.net/ http://www.saadawi1.net/ http://www.saadawi1.net/ mailto:[email protected] mailto:[email protected] mailto:[email protected]

  • Chapter 4

    Stationary Current Fields

    11/8/2016 2Prof. Dr. Magdi El-Saadawi

  • Chapter 4

    Stationary Current Fields 4.1. Introduction

    4.2. Conduction Current

    4.3. Cause of Field through the Impressed Field Intensity

    4.4. Boundary Conditions for Steady Electric Currents

    4.5. Energy Dissipation and Joule’s Law

    4.6. Field Equations for the Electric Field of Stationary Currents

    4.7. Electrostatic Simulation

    4.8. Equation of Continuity and Kirchhoff’s Current Law

    4.9. Resistance Calculations 11/8/2016 3Prof. Dr. Magdi El-Saadawi

  • Stationary Current Fields

    4.1. Introduction

     In Chapter 3 we dealt with electrostatic problems, field

    problems associated with electric charges at rest.

     We now consider the charges in motion that

    constitute current flow.

    11/8/2016 4Prof. Dr. Magdi El-Saadawi

  •  Stationary charges produce electric fields that are

    constant in time; (electrostatics fields).

    Charges move with constant velocity in solids,

    liquids, gasses or in vacuum constitutes a

    stationary current flow or stationary current field.

    By steady current we mean a flow of charge

    which has been going on forever, never

    increasing, never decreasing.

     Steady currents produce magnetic fields that are

    constant in time (magnetostatics fields).

    In Summary p. 137

    11/8/2016 5Prof. Dr. Magdi El-Saadawi

  • Two types of electric current are caused by the

    motion of electric charges:

    (1) Convection current (2) Conduction current

    Convection currents: resulting from the motion

    of the electron, the ions, or the other charged

    particles in vacuum, a liquid or a gas, and these

    currents are not governed by Ohms law:

     Electron beams in a cathode-ray tube

     Violent motions of charged particles in a thunderstorm.

    للجسيماث المشحونت في عاصفت رعديت( غير طبيعيت)حركت عنيفت

    4.1. Introduction

    11/8/2016 6Prof. Dr. Magdi El-Saadawi

  • Conduction currents: resulting from the motion

    of charges in metallic conductor under the action

    of an electric field. These currents are caused by

    drift motion of conduction electrons or holes and

    they obey the Ohm’s law.

    Charges move in free space has nothing to impede

    but in conductor there is a special vibrating lattice

    structure which collide with them

    4.1. Introduction

    11/8/2016 7Prof. Dr. Magdi El-Saadawi

    we will concentrate on conduction currents

  • When an external electric field is applied on a

    conductor, an organized motion of conduction

    electrons, which may wander from one atom to

    another in a random manner, is produced.

    The conduction electrons collide with the atoms

    in the course of their motion, dissipating part of

    their kinetic energy as heat (thermal radiation).

    This phenomenon manifests itself تظهر نفسها as a

    damping force قوة تخميد or resistance, to current

    flow.

    4.2. Conduction Current

    11/8/2016 8Prof. Dr. Magdi El-Saadawi

  • 4.2. Conduction Current

    11/8/2016 9Prof. Dr. Magdi El-Saadawi

  • 4.2. Conduction Current

    10

  • 11/8/2016 Prof. Dr. Magdi El-Saadawi 11

  • 4.2. Conduction Current

    Table 4.1 shows the conductivities of several media in

    S/m

    11/8/2016 12Prof. Dr. Magdi El-Saadawi

  • 11/8/2016 Prof. Dr. Magdi El-Saadawi 13

  •  A stationary field is a field which reaches to a state of

    independence on time (constant state) and is coupled

    with an energy transformation.

     This energy must come from a non-conservative field.

    The source of non-conservative field may be electric

    batteries (conservation of chemical energy to electric

    energy) or electric generator (conservation of mechanical

    energy to electric energy) or other devices.

     These electrical energy sources, when connected in an

    electric circuit, provide a driving force for the charge

    carries, and manifests تتجلى فى itself as equivalent impressed

    field intensity E′

    4.3. Cause of Field through the Impressed Field Intensity (electromotive force)

    11/8/2016 14Prof. Dr. Magdi El-Saadawi

  • 4.4. Boundary Conditions for Steady Electric Currents

    11/8/2016 15Prof. Dr. Magdi El-Saadawi

  • 4.4. Boundary Conditions for Steady Electric Currents

    11/8/2016 16Prof. Dr. Magdi El-Saadawi

  •  In a conducting medium, the collision of free electrons

    with the atomic lattice will generate thermal energy, and

    this is an irreversible energy conversion process.

     The impressed source has to compensate the energy

    dissipation in order to maintain the steady electric

    current.

    4.5. Energy Dissipation in Steady Electric Current Fields

    11/8/2016 17Prof. Dr. Magdi El-Saadawi

  •  In a steady electric current field, we construct a small

    cylinder of length and end face area , and assume

    the two end faces of the cylinder are equipotential

    surfaces.

     Under the influence of the electric

    field, electric charge dq is moved

    to the right end face from the left

    end face in dt, with the

    Corresponding work done by the electric force as

    4.5. Energy Dissipation in Steady Electric Current Fields

    11/8/2016 18Prof. Dr. Magdi El-Saadawi

  • 11/8/2016 19Prof. Dr. Magdi El-Saadawi

  • A parallel plate capacitor consists of two imperfect dielectrics in series. Their

    permittivities are 1 and 2 , the conductivities are 1 and 2 , and the

    thickness are d1 and d2, respectively. If the impressed voltage is U, find the

    electric field intensities, the electric energies per unit volume, and the power

    dissipations per unit volume in the two dielectrics.

    Solution: Since no current exists outside the

    capacitor, the electric current lines in the capacitor

    can be considered to be perpendicular to the

    boundaries. Then we have

    J1n= J2n

    2211  EE 

    Solving the two equations we get:

    U dd

    E 1221

    2 1

    

      U

    dd E

    1221

    1 2

    

     

     1 1

     2 2

    d1

    d2

    U

    Example pp. 150

    UdEdE  2211

    11/8/2016 20Prof. Dr. Magdi El-Saadawi

  • The electric energies per unit volume in two dielectrics, respectively, are

    2

    222e

    2

    111e 2

    1 ,

    2

    1 EwEw  

    The power dissipations per unit volume in two dielectrics, respectively, are

    2

    222

    2

    111 , EpEp ll  

    Two special cases are worth noting:

    If , then , , , .02  01 E 0e1 w 01 lp 2

    2 d

    U E 

    If , then , , , .01  1

    1 d

    U E  02 E 02e w 02 lp

    d1

    d2

     1= 0

    E 2= 0

    U E 1= 0

     2= 0

    U

    11/8/2016 21Prof. Dr. Magdi El-Saadawi

  • 4.6. Equations for the Electric Field of Stationary Currents

    11/8/2016 22Prof. Dr. Magdi El-Saadawi

  •  The analogy between the electric current field and the

    electrostatic field is explained by Table 4.2.

    4.7. Electrostatic Simulation pp.153

    The electric current density J corresponds to the electric field

    intensity E, and the electric current lines to the electric field lines.

    11/8/2016 23Prof. Dr. Magdi El-Saadawi

  • Based on this similarity, the solution of the steady

    electric current field can be found directly from

    the results of the electrostatic field.

     In some cases, since the steady electric current

    field is easy to be constructed and measured, the

    electrostatic field can be investigated based on the

    steady electric current field with the same

    boundary conditions, and this method is called

    electrostatic simulation.

    4.7. Electrostatic Simulation p. 153

    11/8/2016 24Prof. Dr. Magdi El-Saadawi

  • The electrostatic field and the steady electric current field

    between two electrodes as follows:

    P N

    Steady electric current field

    P N

    Electrostatic f

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