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PPT No. 26
Uniformly Magnetized Sphere in
the External Magnetic Field
Electromagnets
Uniformly magnetized sphere in external magnetic field
The Topic
“Uniformly magnetized sphere in external magnetic field”,
is divided into three sub-topics as follows
A) A Uniformly Magnetized Sphere
B) A Magnetized Sphere in a Uniform Magnetic Field
C) A Soft Iron Sphere in a Uniform Magnetic Field
A) An Uniformly Magnetized Sphere
Consider a sphere of radius “a”,
having uniform permanent magnetization ,
placed in vacuum parallel to z axis.
Its field can be found in terms of
the scalar magnetic potential Φm
in spherical co-ordinates as follows.
A) An Uniformly Magnetized Sphere
As there is zero volume magnetic charge density in a
vacuum or a uniformly magnetized magnetic medium,
Φm satisfies the Laplace's equation
However, there is a magnetic surface charge density σm
on the surface of the sphere,
A) An Uniformly Magnetized Sphere
One of the matching conditions at the surface of the sphere
is that the tangential component of H must be continuous.
The scalar magnetic potential must be continuous at
Integrating over a Gaussian pill-box straddling
the surface of the sphere yields
A) An Uniformly Magnetized Sphere
The magnetic charge sheet on the surface of the sphere
gives rise to a discontinuity in the gradient of
the magnetic scalar potential along the radius at
The most general axisymmetric solution which satisfies
and
is
physical boundary conditions at
and
for
for
A) An Uniformly Magnetized Sphere
The boundary conditions yield
for all ℓ
for all ℓ
As
For ℓ ≠ 1
A) An Uniformly Magnetized Sphere
A1 = 1/3 M0
B1 =1/3 (M0 a3)
for
for
A) An Uniformly Magnetized Sphere
According to the uniqueness theorem associated with
Poisson's equation, this axisymmetric potential is
the only solution to the problem satisfying
physical boundary conditions at r =0 and r = ∞ is
in the vacuum region outside the sphere
A) An Uniformly Magnetized Sphere
Outside the sphere, the potential φm is the potential of
a dipole with magnetic moment m
Sphere is a special geometry in the sense that
field is dipole in character and
there are no higher multipoles for spherical geometry.
The net dipole moment of the sphere is equal to
the integral of the magnetization M
(i.e. the dipole moment per unit volume)
over the volume of the sphere.
A) An Uniformly Magnetized Sphere
The lines of B field are continuous closed curves while
those of H field originate and terminate on the surface
due to the effective surface charge density σm
Inside the sphere
and
and
Thus, both the H and B fields are uniform
inside the sphere. Bin is parallel to M.
The magnetic intensity H is antiparallel to magnetization.
H field acts to demagnetize the sphere.
B) A Magnetized Sphere in a Uniform Magnetic Field
Consider an ideal situation that a sphere
with negligible remnant magnetization
(of soft ferromagnetic material like soft iron)
is placed in uniform magnetic field
B) A Magnetized Sphere in a Uniform Magnetic Field
Then there is no hysteresis.
B-H relation is linear as
where μ(B) is a single valued function.
A uniform magnetic induction
B0 = μ0H0 can be superposed.
Then μ0H and B fields inside the sphere are obtained as
μ0H = B0 – 1/3(μ0M )
B = B0 +2/3(μ0M )
B) A Magnetized Sphere in a Uniform Magnetic Field
with
where, in general, μ=μ(B)
The relation shows that magnetization vanishes
when the external magnetic field vanishes.
This happens in the case of
paramagnetic or diamagnetic substances.
The relation for M is analogues to the polarization P
of a dielectric sphere in a uniform electric field
C) A Soft Iron Sphere in a Uniform Magnetic Field
Soft iron has high permeability μ i.e.
then above relation for Bin becomes
Bin = 2 μ0Hin = 3B0
i.e. Bin the magnetic field inside the sphere
is approximately three times
that of the externally applied field B0.
It implies the magnetic field
is amplified inside the sphere.
C) A Soft Iron Sphere in a Uniform Magnetic Field
Bin = 3B0
This relation is specific to a sphere.
Other relations apply for other geometries.
For elongated objects (e.g., rods),
aligned along the direction of the external field,
the amplification factor
can be considerably larger than 3.
C) A Soft Iron Sphere in a Uniform Magnetic Field
As shown in part A,
H field acts to demagnetize the sphere.
The amount of demagnetization depends on
the shape of the demagnetization curve of
the magnetic material of the sphere i.e.
the Hysteresis curve in the II quadrant
(H negative and B positive).
C) A Soft Iron Sphere in a Uniform Magnetic Field
The demagnetization curve gives
two characteristic quantities:
the retentivity BR and the coercivity μ0Hc
The values of B and μ0H inside the sphere
are obtained from the operating point
which is the intersection of
the demagnetization curve and
the curve B = μH .
A permanent magnet
A permanent magnet is an object
made from a material that is magnetized
(i.e. becomes a material or object
having a magnetic field in its vicinity
where it exerts force of attraction on
iron / other ferromagnetic materials
(e.g. Nickel, Cobalt and naturally occurring loadstone etc.)
and attracts or repels other magnets.
Its own magnetic field is persistent i.e.
once magnetized, it retains its magnetization
even after removal of imposed magnetizing field.
A permanent magnet
Permanent magnets produce
a high magnetic field with
a low mass and
are stable against the influences
responsible for their demagnetization.
Permanent magnets are made from
magnetically hard ferromagnetic materials.
A permanent magnet
The quality factors
characterizing these materials are high values of
1) Coercivity (a measure of the reverse field
required to attain zero magnetization after saturation) ,
2) Remanence (a measure of retaining magnetization
even after removal of driving field ) and
3) (BB0/μ0)max.
(i.e. property to obtain magnetic flux with smaller volume).
A permanent magnet
Ampère model
Applications may be divided in different categories
according to the principles as follows
1)Applications using attractive
and/or repelling force of the magnet:
Magnetic separators,
magnetic holding devices, such as magnetic latches;
Magnetic torque drives, Magnetic bearing devices
A permanent magnet
Ampère model
2)Applications using magnet
to convert mechanical energy to electrical energy:
Magnetos, Generators and alternators,
Eddy current brakes
3) Applications using magnet
to convert electrical energy to mechanical energy:
Motors , Meters, Loudspeakers,
Relays, Actuators: linear and rotational
A permanent magnet
Ampère model
4) Applications using magnet to
direct, shape and control electron or ion beams:
Magnetically focused cathode-ray tubes,
Traveling Wave Tubes, Magnetrons,
BWO’s, Klystrons, Ion Pumps etc.
Electromagnets
An electromagnet is a type of magnet in which
the flow of electric current produces magnetic field.
The magnetic field lasts
till only the current flows i.e.
Its magnetic field is not permanent
(in contrast to a permanent magnet).
Electromagnets
The most simple electromagnet is
a wire / conductor carrying electric current.
Magnetic field is generated around the wire
proportional to the amount of current.
The direction of the magnetic field is given by
the right-hand rule.
Electromagnets
A coil wound in several turns of wire
placed side by side creates
a strong magnetic field passing through
the center of the coil.
The quantity “Number of turns x Current”
is called as the magneto-motive force (MMF)
needed to establish a magnetic field in space
Electromagnets
Electromagnets are of two types
depending on the type of core:
Air core electromagnets have coils available
in different shapes e.g.
helical solenoid, donut shaped toroid etc.
They have low inductance,
low magnetic field strengths and
high frequency inputs.
Electromagnets
By making the core of the coil
of a ferromagnetic material like iron
the magnetic filed can be increased tremendously
due to its high magnetic permeability.
The magnetic field of
ferromagnetic-core electromagnet is given by
where k is the relative permeability of iron,
n is number of turns, I is the current.
Electromagnets
An Electrmagnet with an Air core has weak magnetic field
Electromagnets
An Electrmagnet with an Iron core has strong magnetic field
Electromagnets
Electromagnets are classified into two categories
depending upon the type of input current:
DC (Direct-Current) and AC (Alternating-Current).
DC electromagnets are like permanent magnets.
They are principally used to pick up or hold objects.
The polarity of the AC electromagnets changes
as the current reverses direction every half cycle.
They can be used to demagnetize objects
(like TV screens, audio tapes, VCR tapes).
Electromagnets
The cores of electromagnets are prone to
eddy-current and hysteresis losses.
To reduce these losses cores are made from
powdered irons, ferrites and laminated structures.
Resistive losses due to Ohmic heating in the windings
occur in both DC and AC electromagnets.
Forced means of cooling
are provided for thermal relief.
Electromagnets
Electromagnets enable to control /manipulate
the magnetic circuit design simply
by adjusting input power characteristics.
They can control
the strength of the magnetic flux density by
the magnitude of the current flowing in the coil,
can determine the polarity of the field by
the direction of the current flow,
and the shape of the field by
the shape of the iron core
around which the coil is wound.
Electromagnets
Electromagnets have numerous applications
in simple devices like
electric bells, magnetic locks, relays, loud speakers etc.;
data storage instruments like
tape recorders, computer hard disks, VCRs etc.;
machines like
electric generators, motors, particle accelerators etc.
They are used extensively
In homes, industries, institutions, commercial organizations
For entertainment, education, research etc.
Electromagnets
High field electromagnets made of
superconducting windings are used in
Magnetic Resonance Imaging (MRI)
in the medical field for Diagnosis,
Nuclear Magnetic Resonance (NMR) machines,
Mass spectrometers etc for materials research.