EKT 241/4: ELECTROMAGNETIC THEORY

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EKT 241/4: ELECTROMAGNETIC THEORY. UNIVERSITI MALAYSIA PERLIS. CHAPTER 4 – MAGNETOSTATICS. PREPARED BY: NORDIANA MOHAMAD SAAID dianams@unimap.edu.my. Chapter Outline. Maxwell’s Equations Magnetic Forces and Torques The total electromagnetic force, known as Lorentz force Biot- Savart’s law

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EKT 241/4:ELECTROMAGNETIC THEORY

UNIVERSITI MALAYSIA PERLIS

CHAPTER 4 – MAGNETOSTATICS

dianams@unimap.edu.my

Chapter Outline
• Maxwell’s Equations
• Magnetic Forces and Torques
• The total electromagnetic force, known as Lorentz force
• Biot- Savart’s law
• Gauss’s law for magnetism
• Ampere’s law for magnetism
• Magnetic Field and Flux
• Vector magnetic potential
• Properties of 3 different types of material
• Boundary conditions between two different media
• Self inductance and mutual inductance
• Magnetic energy
Maxwell’s equations
• Maxwell’s equations for magnetostatics:
• Relationship between B and H:
• unit: Tesla or Weber/m2
• Where: μ = magnetic permeability
• Where;
• J = current density
• H = magnetic field intensity
• B = magnetic flux density
Magnetic Forces and Torques
• The electric force Fe per unit charge acting on a test charge placed at a point in space with electric field E.
• When a charged particle moving with a velocity u passing through that point in space, the magnetic forceFm is exerted on that charged particle.

where B = magnetic flux density (Cm/s or Tesla T)

Magnetic Forces and Torques
• If a charged particle is in the presence of both an electric field E and magnetic field B, the total electromagnetic force acting on it is:
Magnetic Force on a Current- Carrying Conductor
• For closed circuit of contour C carrying I , total magnetic force Fm is:
• In a uniform magnetic field, Fm is zero for a closed circuit.
Magnetic Force on a Current- Carrying Conductor
• On a line segment, Fm is proportional to the vector between the end points.
Example 1

The semicircular conductor shown 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.

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.
The Biot–Savart’s Law

Biot–Savart’s lawstates that:

where:

dH = differential magnetic field dI = differential current element

The Biot–Savart’s Law
• To determine the total H:
The Biot–Savart’s Law
• Biot–Savart’s law may be expressed in terms of distributed current sources.
Example 2

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.

Solution o Example 2
• For segment OA and OC, the magnetic field at O is zero since is parallel and anti-parallel to .
• For segment AC, dl is in φ direction,
• Using Biot- Savart’s law:
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)

Gauss’s Law for Magnetism
• Gauss’s law for magnetismstates that:
• Magnetic field lines always form continuous closed loops.
Ampere’s law for magnetism
• Ampere’s law states that:
• The directional path of current C follows the right-hand rule.

I

Magnetic Field of an infinite length of conductor
• Consider a conductor lying on the z axis, carrying current I in +az direction.
• Using Ampere’s law:
• The path to evaluate is along the aφ direction, hence use dLφ.
Magnetic Field of an infinite length of conductor

Integrating and then re-arrange the equation in terms of Hφ:

Hence, the magnetic field vector, H:

Note: this equation is true for an infinite length of conductor

Example 3
• A toroidal coil with N turns carrying a current I , determine the magnetic field H in each of the following three regions: r < a, a < r < b,and r > b, all in the azimuthal plane of the toroid.
Solution to Example 3
• 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.
• For a < r < b, we apply Ampere’s law:
• Hence, H = NI/(2πr) .
Magnetic Flux
• The amount of magnetic flux, φ in Webers from magnetic field passing through a surface is found in a manner analogous to finding electric flux:
Example 4

An infinite length coaxial cable with inner conductor radius of 0.01m and outer conductor radius of 0.05m carrying a current of 2.5A exists along the z axis in the +azdirection.

Find the flux passing through the region between two conductors with height of 2 m in free space.

Solution to Example 4

The relation between B and H is:

To find magnetic flux crossing the region, we use:

unit: Weber

where dS is in the aφ direction.

Vector Magnetic Potential
• For any vector of vector magnetic potentialA:
• We are able to derive: .
• Vector Poisson’s equationis given as:

where

Magnetic Properties of Materials
• Magnetic behavior of a material is due to the interaction of magnetic dipole moments of its atoms with an external magnetic field.
• This behavior is used as a basis for classifying magnetic materials.
• 3 types of magnetic materials: diamagnetic, paramagnetic, and ferromagnetic.
Magnetic Properties of Materials
• Magnetization in a material is associated with atomic current loops generated by two principal mechanisms:
• Orbital motions of the electrons around the nucleus, i.e orbital magnetic moment, mo
• Electron spin about its own axis, i.e spin magnetic moment, ms
Magnetic Permeability
• Magnetization vectorM is defined as

where = magnetic susceptibility (dimensionless)

• Magnetic permeability is defined as:

and relative permeability is defined as

Magnetic Materials
• Diamagnetic materials have negative susceptibilities.
• Paramagnetic materials have positive susceptibilities.
• However, the absolute susceptibilities value of both materials is in the order 10-5. Thus, can be ignored. Hence, we have
• Diamagnetic and paramagnetic materials include dielectric materials and most metals.
Magnetic Hysteresis of Ferromagnetic Materials
• Ferromagnetic materials is characterized by magnetized domain - a microscopic region within which the magnetic moments of all its atoms are aligned parallel to each other.
• Hysteresis – “to lag behind”. It determines how easy/hard for a magnetic material to be magnetized and demagnetized.
• Hard magnetic material- cannot be easily demagnetized by an external magnetic field.
• Soft magnetic material – easily magnetized & demagnetized.
Magnetic Hysteresis of Ferromagnetic Materials
• Properties of magnetic materials as follows:
Magnetic Hysteresis of Ferromagnetic Materials
• Comparison of hysteresis curves for (a) a hard and (b) a soft ferromagnetic material is shown.
Magnetic boundary conditions
• Boundary between medium 1 with μ1 and medium 2 with μ2
Magnetic boundary conditions
• Boundary condition related to normal components of the electric field;
• By analogy, application of Gauss’s law for magnetism, we get first boundary condition:
• i.e the normal component of B is continuous across the boundary between two adjacent media
Magnetic boundary conditions
• Since ,
• For linear, isotropic media, the first boundary condition which is related to H;
• Reversal concept: whereas the normal component of B is continuous across the boundary, the normal component of D (electric flux density) may not be continuous (unless ρs=0)
Magnetic boundary conditions
• A similar reversal concept applies to tangential components of the electric field E and magnetic field H.
• Reversal concept related to tangential components:
• Whereas the tangential component of E is continuous across the boundary, the tangential component of H may not be continuous (unless Js=0).
• By applying Ampere’s law and using the same method of derivation as for electric field E:
Magnetic boundary conditions
• The result is generalized to a vector form:
• Where
• However, surface currents can exist only on the surfaces of perfect conductors and perfect superconductors (infinite conductivities).
• Hence, at the interface between media with finite conductivities, Js=0. Thus:
Inductance
• An inductor is the magnetic analogue of an electrical capacitor.
• Capacitor can store electric energy in the electric field present in the medium between its conducting surfaces.
• Inductor can store magnetic energy in the volume comprising the inductors.
Inductance
• Example of an inductor is a solenoid - a coil consisting of multiple turns of wire wound in a helical geometry around a cylindrical core.
Magnetic Field in a Solenoid
• For one cross section of solenoid,
• When l >a, θ1≈−90° and θ2≈90°,

Where, N=nl

=total number of turns over the length l

Self Inductance
• Magnetic flux, linking a surface S is given by:
• In a solenoid with uniform magnetic field, the flux linking a single loop is:
Self Inductance
• Magnetic flux linkage, Λis the total magnetic flux linking a given conducting structure.
• Self-inductance of any conducting structure is the ratio of the magnetic flux linkage, Λ to the current I flowing through the structure.
Self Inductance
• For a solenoid:
• For two conductor configuration:
Mutual Inductance
• Mutual inductance – produced by magnetic coupling between two different conducting structures.
Mutual Inductance
• Magnetic field B1 generated by current I1 results in a flux Φ12 through loop 2:
• If loop 2 consists of N2 turns all coupled by B1 in exactly the same way, the total magnetic flux linkage through loop 2 due to B1 is:
Mutual Inductance
• Hence, the mutual inductance:
Magnetic Energy
• Consider an inductor with an inductance L connected to a current source.
• The current I flowing through the inductor is increased from zero to a final value I.
• The energy expended in building up the current in the inductor:
• i.e the magnetic energy stored in the inductor
Magnetic Energy
• Magnetic energy density (for solenoid):
• i.e magnetic energy per unit volume