1. Electromagnetic Theory
EKT 241
Electromagnetic Theory
Magnetostatic
Anayet Karim

2. Chapter Objectives Magnetic force
The total electromagnetic force, known as Lorentz force
Biot–Savart law
Gauss’s law for magnetism
Ampere’s law
Vector magnetic potential
3 different types of material
Boundary between two different media
Magnetic energy density

3. Chapter Outline 4-1) Magnetic Forces and Torques The Biot–Savart Law
Magnetic Force between Two Parallel Conductors
Maxwell’s Magnetostatic Equations
Vector Magnetic Potential
Magnetic Properties of Materials
Magnetic Boundary Conditions
Inductance
Magnetic Energy
4-2)
4-3)
4-4)
4-5)
4-6)
4-7)
4-8)
4-9)

4. Today We Will Focus the Following Topics
Magnetic Forces
Apart between Fe & Fm
Magnetic Torques
Few Examples

5. 4-1 Magnetic Forces and Torques
When charged particle moving with a velocity u, magnetic force Fm is produced.
where B = magnetic flux density (C.m/s or Tesla T)
When charged particle has E (electric field) and B (Magnetic field), total electromagnetic force is

6. Apart Fe & Fm Fe always in the direction of electric field
Fm always perpendicular to magnetic field
Fe acts on a charged particles, which is static
Fm acts on a charged particles, which is moving or in motion
E & D (Electric field & Density)
H & B (magnetic field & Density)
Fe=qE for “Force on charge q)
Fm = quXB “Force on charge q)
Stationary charges (Electrostatics) & Steady currents (Magnetostatics)

7. 4-1.1 Magnetic Force on a Current-Carrying Conductor
For closed circuit of contour C carrying I , total magnetic force Fm is
Fm is zero for a closed circuit. On a line segment, it is proportional to the vector between the end point.

8. 4-1.1 Magnetic Force on a Current-Carrying Conductor
The total magnetic force Fm on any closed current loop in a uniform magnetic field is zero

9. Example 4.1 Force on a Semicircular Conductor
The semicircular conductor shown lies in the x–y plane and carries a current I . The closed circuit is exposed to a uniform magnetic field Determine (a) the magnetic force F1 on the straight section of the wire and (b) the force F2 on the curved section. a) b)
Solution

10. 4-1.2 Magnetic Torque on a Current-Carrying Loop
Applied force vector F and distance vector d are used to generate a torque T
T = d× F (N·m)
Rotation direction is governed by right-hand rule.
Magnetic Field in the Plane of the Loop
F1 and F3 generates a torque in clockwise direction.

11. 4-1.2 Magnetic Torque on a Current-Carrying Loop
Magnetic Field Perpendicular to the Axis of a Loop
When loop consists of N turns, the total torque is
The vector m with normal vector is expressed as
T can be written as

12. What we learn today Electric force Fe Magnetic force Fm
Electromagnetic force F
Lorentz force
Apart between Fe & Fm (5 types)
Examples are very Important

13. Test Yourself What is Force Apart between Force & Torque
What is 2D axis
What is 3D axis

14. Q & A Feel Free to ASK ME Next Class would be various laws-
Do some study in advance

15. THANK YOU
TERIMA KASIH

16. 4-2 The Biot–Savart Law Biot–Savart law states that
where dH = differential magnetic field dl = differential length
To determine the total H we have

17. 4-2.1 Magnetic Field due to Surface and Volume Current Distributions
Biot–Savart law may be expressed in terms of distributed current sources.

18. Example 4.3 Magnetic Field of a Pie-Shaped Loop
Determine the magnetic field at the apex O of the pie-shaped loop as shown. Ignore the contributions to the field due to the current in the small arcs near O.
For segment AC,
Consequently,
Solution

19. 4-2.2 Magnetic Field of a Magnetic Dipole
To find H in a spherical coordinate system, we have
where R’ >> a

20. 4-3 Magnetic Force between Two Parallel Conductors
Force per unit length on parallel current-carrying conductors is
where F’1 = -F’2 (attract each other with equal force)

21. 4-4 Maxwell’s Magnetostatic Equation
There are 2 important properties: Gauss’s and Ampere’s Law.
Gauss’s law for magnetism states that
Net electric magnetic flux through a closed surface is zero.
Gauss’s Law for Magnetism

22. 4-4.2 Ampere’s Law Ampere’s law states that
The directional path of current C follows the right-hand rule.

23. Example 4.6 Magnetic Field inside a Toroidal Coil
A toroidal coil (also called a torus or toroid) is a doughnut-shaped structure (called its core) with closely
spaced turns of wire wrapped around it as shown. For a toroid with N turns carrying a current I , determine the magnetic field H in each of the following three regions: r < a, a < r < b, andr > b, all in the azimuthal plane of the toroid.

24. Solution 4.6 Magnetic Field inside a Toroidal Coil
H = 0 for r < a as no current is flowing through the surface of the contour
H = 0 for r > b, as equal number of current coils cross the surface in both directions.
Application of Ampere’s law then gives
Hence, H = NI/(2πr) and

25. 4-5 Vector Magnetic Potential
For any vector of vector magnetic potential A,
We are able to derive
Vector Poisson’s equation is given as
where

26. 4-6 Magnetic Properties of Materials
Magnetic behavior is due to the interaction of dipole and field.
3 types of magnetic materials: diamagnetic, paramagnetic, and ferromagnetic.
Electron generates around the nucleus and spins about its own axis.
Orbital and Spin Magnetic Moments

27. 4-6.2 Magnetic Permeability
Magnetization vector M is defined as
where = magnetic susceptibility (dimensionless)
Magnetic permeability is defined as
and relative permeability is defined as

28. 4-6.3 Magnetic Hysteresis of Ferromagnetic Materials
Ferromagnetic materials is described by magnetized domains.
Properties of magnetic materials are shown below.

29. 4-6.3 Magnetic Hysteresis of Ferromagnetic Materials
Comparison of hysteresis curves for (a) a hard and (b) a soft ferromagnetic material is shown.

30. 4-7 Magnetic Boundary Conditions
For 2 different media when applying Gauss’s law, we have
Boundary condition for H is
Vector defined by the right-hand rule is
At interface between media with finite conductivities, Js = 0 and

31. 4-8 Inductance An inductor is a magnetic capacitor.
An example is a solenoid as shown below.

32. 4-8.1 Magnetic Field in a Solenoid
For a cross section of solenoid,
When l >a, θ1≈−90° and θ≈90°,

33. 4-8.2 Self-Inductance Magnetic flux is given by
To compute the inductance we need area S.

34. Self-Inductance
The self-inductance L of conducting structure is defined as
where = total magnetic flux (magnetic flux linkage)
For a solenoid,
For two-conductor configurations,

35. Mutual Inductance
Magnetic field lines are generated by I1 and S2 of loop 2.
The mutual inductance is
Transformer uses torodial coil with 2 windings.

36. 4-9 Magnetic Energy
Total energy (Joules) expended in building up the current in inductor is
Some of the energy is stored in the inductor, called magnetic energy, Wm.
The magnetic energy density wm is defined as

37. Example 4.9 Magnetic Energy in a Coaxial Cable
Derive an expression for the magnetic energy stored in a coaxial cable of length l and inner and outer radii a
and b. The insulation material has permeability µ.
The magnitude of the magnetic field is
Magnetic energy stored in the coaxial cable is given by
Solution