1. Electromagnetics II

2. Contents Magnetic field Curl Stokes’ theorem Vector potential
Lorentz force
Inductance
Transformer
Time-varying fields and Maxwell’s equations
Faraday’s law
Displacement current
Maxwell’s equation
Transmission lines
Telegrapher’s equation
Phasor representation
Reflection
Transient analysis
Uniform plane wave
Refraction
Dispersion
Guided waves and radiation
Waveguide
antenna principles

3. Chapter 7. Steady magnetic field

4. Generation of magnetic field
A charged particle in motion generates magnetic field nearby.
n : number of charges in unit volume
v : velocity of charges
A : cross-section area of the wire
e: amount of charges in an electron
In the same way, currents generate magnetic field nearby.

5. Prediction of magnetic field : Biot-Savart law
Current segment
Direction of H-field
The magnetic field can be predicted by Biot-Savart’s law with known current distribution.

6. Biot-Savart law : integral form
Line current
surface current
Volume current

7. Prediction of magnetic field : Differential equation
In practice, current distributions on objects are unknown and are apt to change with external magnetic field.
Differential eq. on magnetic field and boundary condition needed!

8. EM problems
Current/charge distributions are known only at the point of sources.
If surface currents are known everywhere, integral formulas such as Biot-Savart’s and Coulomb’s law are useful.

9. 8.3 Curl (Ampere’s law in a differential form)
From the integral form, we will derive the differential form of Ampere’s law.
Line integrals from these currents add up to zero.
Line integrals over a closed path is equal to the sum of line integrals over infinitesimally small loops.

10. Ampere’s law over an infinitesimally small closed path

11. ; Ampere’s law in a differential form
In the same way, Jx, Jy can be derived and are shown as follows.
Because the expression for line integrals are different for each coordinate systems, ∇x operator is used.

12. Curl operator in other orthogonal coordinates
Rectangular coordinate :
Circular cylindrical coordinate :
Spherical coordinate :

13. 8.4 Stokes’ theorem
Line integrals can be transformed to surface integrals by curl operations.

14. Example 8.3 (1) Explicit line integral
Evaluate the line integral over the specified path.
(1) Explicit line integral
Since r =4 on the contour, r component becomes zero.
Over the path 1 and 3, the integrand becomes zero in that dϕ = 0.

15. (2) Stokes’ theorem

16. Important vector identities1)
(ϕ : arbitrary scalar function) 2)
(A: arbitrary vector function)

17. 8.5 Magnetic flux and magnetic flux density

18. Magnetic field and magnetic flux density
With the same incident magnetic field, magnitudes of B changes with a magnetic permeability of surrounding medium.
Magnetic flux density
Permeability

19. DC solution of Maxwell’s equations
(1) Scalar potential
Because the curl of E is always equal to zero, E can be represented by a gradient of an arbitrary scalar function.

20. (2) Vector potential
Because the divergence of B is always equal to zero, B can be represented by a curl of an arbitrary vector function.

21. Potential equations Steady state Maxwell’s equations
A solution of Poisson equation for free space.

22. Laplacian operator in other orthogonal coordinates
The differential equation for source-free region becomes a Laplace equation.
(rectangular coordinate)
(cylindrical coordinate)
(spherical coordinate)

23. Solution of a Poisson equation for a point charge
If there is only one point charge in the space, the potential field will have spherically symmetric distribution.
Free space
G : Green’s function
(1) Homogeneous solution

24. If the position of the source is changed to r’, the expression becomes,

25. For a point charge with ‘q’ coulomb at origin,

26. Equation for the vector potential

27. For an infinitesimally small current segment with current I and length dL at origin,

28. Example
Calculate the magnetic field inside the coaxial cable sheath carrying current I.