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Applied Magnetism. Wang C. Ng Sep. 21, 2009 (916)558-2638. References:. ; 9-13-2009. ; 9-13-2009. ; 9-13-2009.

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Applied Magnetism

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Applied magnetism
Applied Magnetism

Wang C. Ng

Sep. 21, 2009



  •; 9-13-2009.

  •; 9-13-2009.

  •; 9-13-2009.

  •; 9-14-2009.

  • J.J.Cathey & S.A.Nasar; Basic Electrical Engineering; McGraw-Hill, NY, 1984.

Magnetic properties of solids
Magnetic Properties of Solids

  • Materials may be classified by their response to externally applied magnetic fields as diamagnetic, paramagnetic, or ferromagnetic.

  • These magnetic responses differ greatly in strength.

  • Diamagnetism is a property of all materials and opposes applied magnetic fields, but is very weak.

Magnetic properties of solids1
Magnetic Properties of Solids

  • Paramagnetism is stronger than diamagnetism and produces magnetization in the direction of the applied field, and proportional to the applied field.

  • Ferromagnetic effects are very large, producing magnetizations sometimes orders of magnitude greater than the applied field and as such are much larger than either diamagnetic or paramagnetic effects.


  • The magnetization of a material is expressed in terms of density of net magnetic dipole momentsm in the material.

  • We define a vector quantity called the magnetization M = μtotal/Volume.

  • The total magnetic fieldB in the material is given byB = B0 + μ0M

    where μ0 is the magnetic permeability of space and B0 is the externally applied magnetic field.


  • Another way to view the magnetic fields which arise from magnetization of materials is to introduce a quantity called magnetic field strength H .

  • It can be defined by the relationship

    H = B0/μ0 = B/μ0 – M

  • H has the value of unambiguously designating the driving magnetic influence from external currents in a material, independent of the material's magnetic response.


  • H and M have the same units, amperes/meter.

  • The relationship for B and H above can be written in the equivalent form

    B = μ0 (H + M) = μr μ0 H, where μr = (1 + M/H)

  • The relative permeability mr can be viewed as the amplification factor for the internal field B due to an external field H.


  • The orbital motion of electrons creates tiny atomic current loops, which produce magnetic fields.

  • When an external magnetic field is applied to a material, these current loops will tend to align in such a way as to oppose the applied field.

  • This may be viewed as an atomic version of Lenz's law: induced magnetic fields tend to oppose the change which created them. Materials in which this effect is the only magnetic response are called diamagnetic.


  • All materials are inherently diamagnetic, but if the atoms have some net magnetic moment as in paramagnetic materials or in ferromagnetic materials, these stronger effects are always dominant.

  • Diamagnetism is the residual magnetic behavior when materials are neither paramagnetic nor ferromagnetic.


  • Any conductor will show a strong diamagnetic effect in the presence of changing magnetic fields because circulating currents will be generated in the conductor to oppose the magnetic field changes.

  • A superconductor will be a perfect diamagnet since there is no resistance to the forming of the current loops.


  • Some materials exhibit a magnetization which is proportional to the applied magnetic field in which the material is placed.

  • These materials are said to be paramagnetic and follow Curie's law:


  • All atoms have inherent sources of magnetism because electron spin contributes a magnetic moment and electron orbits act as current loops which produce a magnetic field.

  • In most materials the magnetic moments of the electrons cancel, but in materials which are classified as paramagnetic, the cancelation is incomplete.


  • Iron, nickel, cobalt and some of the rare earths (gadolinium, dysprosium) exhibit a unique magnetic behavior which is called ferromagnetism because iron (ferrum in Latin) is the most common and most dramatic example.

  • Samarium and neodymium in alloys with cobalt have been used to fabricate very strong rare-earth magnets.


  • Ferromagnetic materials exhibit a long-range ordering phenomenon at the atomic level which causes the unpaired electron spins to line up parallel with each other in a region called a domain.

  • Within the domain, the magnetic field is intense, but in a bulk sample the material will usually be unmagnetized because the many domains will themselves be randomly oriented with respect to one another.


  • Ferromagnetism manifests itself in the fact that a small externally imposed magnetic field, say from a solenoid, can cause the magnetic domains to line up with each other and the material is said to be magnetized.

  • The driving magnetic field will then be increased by a large factor which is usually expressed as a relative permeability for the material.


  • There are many applications of ferromagnetic materials, such as the electromagnet.

  • Ferromagnets will tend to stay magnetized to some extent after being subjected to an external magnetic field.

  • This tendency to "remember their magnetic history" is called hysteresis.

  • The fraction of the saturation magnetization which is retained when the driving field is removed is called the remanence of the material, and is an important factor in permanent magnets.




  • All ferromagnets have a maximum temperature where the ferromagnetic property disappears as a result of thermal agitation.

  • This temperature is called the Curie temperature.

  • Ferromagntic materials respond mechanically to an impressed magnetic field, changing length slightly in the direction of the applied field.

  • This property, called magnetostriction, leads to the familiar hum of transformers as they respond mechanically to 60 Hz AC voltages.

Inductance review1
Inductance (review)

  • Increasing current in a coil of wire will generate a counter emf which opposes the current.

  • Applying the voltage law allows us to see the effect of this emf on the circuit equation.

  • The fact that the emf always opposes the change in current is an example of Lenz's law.

  • The relation of this counter emf to the current is the origin of the concept of inductance.

  • The inductance of a coil follows from Faraday's law.

Inductance review2
Inductance (review)

  • Inductance of a coil: For a fixed area and changing current, Faraday's law becomes

  • Since the magnetic field of a solenoid is

    then for a long coil the emf is approximated by

Inductance review3
Inductance (review)

  • From the definition of inductance

    we obtain

Relative permeability
Relative Permeability

  • The magnetic constant μ0 = 4π x 10-7 T m/A is called the permeability of space.

  • The permeabilities of most materials are very close to μ0 since most materials will be classified as either paramagnetic or diamagnetic.

Relative permeability1
Relative Permeability

  • But in ferromagnetic materials the permeability may be very large

  • It is convenient to characterize the materials by a relative permeability.

Relative permeability2
Relative Permeability

  • When ferromagnetic materials are used in applications like an iron-core solenoid, the relative permeability gives you an idea of the kind of multiplication of the applied magnetic field that can be achieved by having the ferromagnetic core present.

  • For an ordinary iron core you might expect a magnification of about 200 compared to the magnetic field produced by the solenoid current with just an air core.

Relative permeability3
Relative Permeability

  • This statement has exceptions and limits, since you do reach a saturation magnetization of the iron core quickly, as illustrated in the discussion of hysteresis.

Eddy currents
Eddy Currents

  • Currents that are induced into a conducting core due to the changing magnetic field.

  • Eddy currents produce heat which results a loss of power

  • This effect can reduce the efficiency of an inductor or a transformer.

  • The Eddy current loss is proportional to f 2.


  • Compound composed of iron oxide, metallic oxide, and ceramic.

  • The metal oxides include zinc, nickel, cobalt or iron.

  • A powdered, compressed and sintered magnetic material having high resistivity.

  • The high resistance makes eddy current losses low at high frequencies.

Ferrites vs iron cores
Ferrites vs. iron cores

  • Iron cores are used for frequencies below about 100 kHz.

  • Ferrite cores are used for frequencies up to say, 10 MHz.

  • Above 100MHz the core is usually air and the coil is self supporting.

Ferrites vs iron cores1
Ferrites vs. iron cores

  • At low frequencies the inductor may have hundreds of turns, above 1 MHz only a few turns.

  • Most inductors have a low DC resistance since they are wound from copper wire.

Inductance of a solenoid
Inductance of a Solenoid

Inductance of a solenoid1
Inductance of a Solenoid

Solenoid length = 10 cm with N = 200 turns,

Coil radius r = 1 cm gives area A = 3.14159 cm2.

Relative permeability of the core k = 200,

Then the inductance of the solenoid is

L = 31.58 mH.

Inductance of a solenoid2
Inductance of a Solenoid

  • Small inductors for electronics use may be made with air cores.

  • For larger values of inductance and for transformers, iron is used as a core material.

  • The relative permeability of magnetic iron is around 200.

  • This calculation makes use of the long solenoid approximation.

Approximate inductance of a toroid
Approximate Inductance of a Toroid

  • Finding the magnetic field inside a toroid is a good example of the power of Ampere's law.

  • The current enclosed by the dashed line is just the number of loops times the current in each loop.

Approximate inductance of a toroid1
Approximate Inductance of a Toroid

  • Amperes law then gives the magnetic field at the centerline of the toroid as

Approximate inductance of a toroid2
Approximate Inductance of a Toroid

  • The inductance can be calculated in a manner similar to that for any coil of wire.

Approximate inductance of a toroid3
Approximate Inductance of a Toroid

Toroidal radius r = 5 cm with N = 200 turns,

Coil radius = 1 cm gives area A = 3.14159 cm2.

Relative permeability of the core k = 200,

Then the inductance of the toroid is approximately

L = 10.053 mH.

Approximate inductance of a toroid4
Approximate Inductance of a Toroid

  • Small inductors for electronics use may be made with air cores.

  • For larger values of inductance and for transformers, iron is used as a core material.

  • The relative permeability of magnetic iron is around 200.

Approximate inductance of a toroid5
Approximate Inductance of a Toroid

  • This calculation is approximate because the magnetic field changes with the radius from the centerline of the toroid.

  • Using the centerline value for magnetic field as an average introduces an error which is small if the toroid radius is much larger than the coil radius.


  • A transformer makes use of Faraday's law and the ferromagnetic properties of an iron core to efficiently raise or lower AC voltages.

  • A transformer cannot increase power so that if the voltage is raised, the current is proportionally lowered and vice versa.

Magnetic circuits
Magnetic Circuits

  • A magnetic circuit is a path for magnetic flux, just as an electric circuit provides a path for the flow of electric current.

  • Transformers, electric machines, and numerous other electromechanical devices utilize magnetic circuits.

  • If the magnetic field B is uniform over a surface A and is everywhere perpendicular to the surface the magnetic flux  = B A.

Magnetic circuits1
Magnetic Circuits

  • The effectiveness of electric current in producing magnetic flux is defined as the magnetomotive force = N I.

  • The reluctance of a magnetic circuit is defined as: = / 

  • Ohm’s law: R = V / I

    R = l /  A

    = = l / μ A

Magnetic circuits2
Magnetic Circuits

  • Differences between a resistive circuit and a magnetic circuit:

    • There is I 2 R loss in a resistive circuit but no 2 loss in a reluctance.

    • Magnetic flux can leak to the surrounding space.

    • A magnetic circuit can have air gaps.

    • In general, the permeability μ is not a constant.

Magnetic circuits3
Magnetic Circuits

  • Mutual inductance:

  • (coupling coefficient) is defined as the fraction of the flux produced by the qth coil that links the pth coil.

  •  1 if fringing is negligible.

Magnetic circuits4
Magnetic Circuits

  • Finally, the self and mutual inductances can be computed from the following formula:

Magnetic circuits5
Magnetic Circuits

Example: The core of a magnetic circuit is of mean length 40 cm and uniform cross-sectional area 4 cm2. The relative permeability of the core material is 1000. An air gap of 1 mm is cut in the core, and 1000 turns are wound on the core. Determine the inductance of the coil if fringing is negligible.

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