Electromagnetic Theory

Electromagnetic theory for first year Engineering students by Dr Md Kaleem

1. ELECTROMAGNETIC THEORY DR MD KALEEM 12/31/2016 1 DR MD KALEEM/ ASSISTANT PROFESSOR

2. INTRODUCTION In year 1984 James Clerk Maxwell brought together and extended four basic laws in electromagnetism such as , Gauss’s Law in electrostatics, Gauss’s Law in magnetism, Ampere's Law and Faraday’s Law. A complete set of relations giving the connection between the charges at rest (Electrostatics) and charges in motion( current Electricity), electric fields and magnetic fields (electromagnetism) were divided theoretically and summarized in four equations by Maxwell, called Maxwell’s Equations James Clerk Maxwell 1831 - 1879 12/31/2016 2 DR MD KALEEM/ ASSISTANT PROFESSOR

3. Current Density Current density (J) at a point, within a conductor, is the vector quantity whose magnitude is the current through unit area of the conductor, around that point, provided the area is perpendicular to the direction of flow of the current at that point. J = I/A d I = J. d S The total current density through the surface S I = ∫ s J .d S Thus the current I is defined as the flux of the current density vector J through the given area 12/31/2016 3 DR MD KALEEM/ ASSISTANT PROFESSOR

4. Concept of Conduction, Convection and Radiation 12/31/2016 4 DR MD KALEEM/ ASSISTANT PROFESSOR

5. Conduction Current Density Conduction Current Density refers to the amount of current (charges) flowing on the surface of a conductor (conduction band) in a time t. This surface is always parallel to the current flow. It obeys Ohm’s Law 12/31/2016 5 DR MD KALEEM/ ASSISTANT PROFESSOR

6. Convection Current Density Convection current , as distinct from conduction current ,does not involve conductors and consequently does not satisfy Ohm's law. Electrons in a metal are subject to frequent collisions with atoms. Electrons accelerated by an electric field lose their energy through collisions which appears as heat (Joule or Ohmic dissipation). 12/31/2016 6 DR MD KALEEM/ ASSISTANT PROFESSOR

7. The equation of motion for an electron in the presence of collisions may be written as m( dv / dt ) = - eE – mν c v Where ν is the collision frequency. In steady state, we have - eE = mν c v Multiplying by –en with n the conduction electron density, we obtain 12/31/2016 7 DR MD KALEEM/ ASSISTANT PROFESSOR

8. Random and Drift Velocities Random Velocity : In absence of electric field the electrons moves randomly with zero net velocity. It is about 10 6 m/s . Drift Velocity : In presence of electric field electrons moves randomly with net motion in the direction opposite net electric field. It is about 10 -5 – 10 -4 m/s 12/31/2016 8 DR MD KALEEM/ ASSISTANT PROFESSOR

9. Introduction of EM Field When an event in one place has an effect on something at a different location, we talk about the events as being connected by a “field”. A field is a spatial distribution of a quantity; in general, it can be either scalar or vector in nature. 12/31/2016 DR MD KALEEM/ ASSISTANT PROFESSOR 9

10. Fundamental Vector Field Quantities In Electromagnetics Electric field intensity ( E) SI unit = volts per meter (V/m = kg m/A/s 3 ) Electric flux density (electric displacement ) ( D ) SI unit = coulombs per square meter (C/m 2 = A s /m 2 ) Magnetic field intensity ( H ) SI unit = amps per meter (A/m ) Magnetic flux density ( B ) SI units = teslas = webers per square meter (T = Wb / m 2 = kg/A/s 3 ) 12/31/2016 DR MD KALEEM/ ASSISTANT PROFESSOR 10

11. Gauss’s Law of Electrostatics Gauss ' Law is the first of Maxwell's Equations which dictates how the Electric Field behaves around electric charges . Gauss' Law states that electric charge acts as sources or sinks for Electric Fields. Gauss' Law can be written in terms of the Electric Flux Density and the Electric Charge Density  as 12/31/2016 DR MD KALEEM/ ASSISTANT PROFESSOR 11

12. Implication of Gauss’ Law D  and  E  field lines diverge away from positive charges D  and  E  field lines diverge towards negative charges D  and  E  field lines start and stop on Electric Charges Opposite charges attract and negative charges repel The divergence of the  D  field over any region (volume) of space is exactly equal to the net amount of charge in that region. 12/31/2016 DR MD KALEEM/ ASSISTANT PROFESSOR 12

13. Gauss Law for Magnetism Gauss' Magnetism law states that the divergence of the Magnetic Flux Density ( B ) is zero . 12/31/2016 DR MD KALEEM/ ASSISTANT PROFESSOR 13

14. Implication of Gauss Law for Magnetism Magnetic Monopoles Do Not Exist The Divergence of the  B  or  H  Fields is Always Zero Through Any Volume Away from Magnetic Dipoles, Magnetic Fields flow in a closed loop. This is true even for plane waves, which just so happen to have an infinite radius loop. 12/31/2016 DR MD KALEEM/ ASSISTANT PROFESSOR 14

15. Faraday’s Law of Electromagnetic Induction   The instantaneous emf induced in circuit is directly proportional to the time rate of change magnetic flux through it. Faraday's law shows that a changing magnetic field within a loop gives rise to an induced current, which is due to a force or voltage within that circuit. 12/31/2016 DR MD KALEEM/ ASSISTANT PROFESSOR 15

16. Implication of Faraday’s Law of Electromagnetic Induction Electric Current gives rise to magnetic fields. Magnetic Fields around a circuit gives rise to electric current. A Magnetic Field Changing in Time gives rise to an E-field circulating around it. A circulating E-field in time gives rise to a Magnetic Field Changing in time. 12/31/2016 DR MD KALEEM/ ASSISTANT PROFESSOR 16

17. Ampere’s Circuital Law  the curl of the magnetic field is equal to the Electric Current Density A flowing electric current ( J ) gives rise to a Magnetic Field that circles the current A time-changing Electric Flux Density ( D ) gives rise to a Magnetic Field that circles the  D  field 12/31/2016 DR MD KALEEM/ ASSISTANT PROFESSOR 17

18. Maxwell’s Modification to Ampere’s Circuital Law For time-varying currents, Ampère’s law is not true. Maxwell fixed this problem by making a postulate that is, in a way, the complement of Faraday’s postulate that a changing electric field produces a magnetic field; Maxwell proposed that a changing electric field induces a magnetic field. In particular, he proposed that 12/31/2016 DR MD KALEEM/ ASSISTANT PROFESSOR 18

19. Maxwell’s Equation in Differential Form 12/31/2016 DR MD KALEEM/ ASSISTANT PROFESSOR 19

20. Maxwell’s Equation in Integral Form 12/31/2016 DR MD KALEEM/ ASSISTANT PROFESSOR 20