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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
- 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.

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

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