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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

PREPARED BY: NORDIANA MOHAMAD SAAID

dianams@unimap.edu.my

chapter outline
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
  • 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
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 torques1
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
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 conductor1
Magnetic Force on a Current- Carrying Conductor
  • On a line segment, Fm is proportional to the vector between the end points.
example 1
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
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
The Biot–Savart’s Law

Biot–Savart’s lawstates that:

where:

dH = differential magnetic field dI = differential current element

the biot savart s law1
The Biot–Savart’s Law
  • To determine the total H:
the biot savart s law2
The Biot–Savart’s Law
  • Biot–Savart’s law may be expressed in terms of distributed current sources.
example 2
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
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
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 Magnetism
  • Gauss’s law for magnetismstates that:
  • Magnetic field lines always form continuous closed loops.
ampere s law for magnetism
Ampere’s law for magnetism
  • Ampere’s law states that:
  • The directional path of current C follows the right-hand rule.
magnetic field of an infinite length of conductor

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 conductor2
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
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
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
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
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
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
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 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 materials1
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
Magnetic Permeability
  • Magnetization vectorM is defined as

where = magnetic susceptibility (dimensionless)

  • Magnetic permeability is defined as:

and relative permeability is defined as

magnetic materials
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
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 materials1
Magnetic Hysteresis of Ferromagnetic Materials
  • Properties of magnetic materials as follows:
magnetic hysteresis of ferromagnetic materials2
Magnetic Hysteresis of Ferromagnetic Materials
  • Comparison of hysteresis curves for (a) a hard and (b) a soft ferromagnetic material is shown.
magnetic boundary conditions
Magnetic boundary conditions
  • Boundary between medium 1 with μ1 and medium 2 with μ2
magnetic boundary conditions1
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 conditions2
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 conditions3
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 conditions4
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
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.
inductance1
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
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
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 inductance1
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 inductance2
Self Inductance
  • For a solenoid:
  • For two conductor configuration:
mutual inductance
Mutual Inductance
  • Mutual inductance – produced by magnetic coupling between two different conducting structures.
mutual inductance1
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 inductance2
Mutual Inductance
  • Hence, the mutual inductance:
magnetic energy
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 energy1
Magnetic Energy
  • Magnetic energy density (for solenoid):
  • i.e magnetic energy per unit volume