Transcript EE3321 ELECTROMAGENTIC FIELD THEORY

Week 6
Magnetic Fields
Inverse Square Law
Gauss’ Law
Biot-Savart’s Law
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A magnetic field is a vector
field that exerts a magnetic
force on
◦ moving electric charges and
◦ magnetic dipoles (such as
permanent magnets).
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When placed in a magnetic
field, magnetic dipoles
tend to align their axes to
be parallel with the
magnetic field
iron filings in the presence of a magnet
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The direction of the
magnetic field near the
poles of a magnet is
revealed by placing
compasses nearby.
The magnetic field points
towards a magnet's south
pole and away from its
north pole.
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Like poles repel each other
Opposite poles attract each other
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First measured by Gauss
The locations are not static
Poles wander as much as
15 km every year
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There is a magnetic field in deep space
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Earth’s magnetic field deflects most of the
radiation from solar flares
Solar flares have a period of 11 years
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The magnetic south pole is about 11o from
true north
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The force between two magnetic poles is given
by:
where
◦ qm1 and qm2 are the magnitudes of magnetic poles (A m)
◦ μ is the permeability of the intervening medium (N / A2)
◦ r is the separation (m)
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Suppose that two bar magnets of equal
length L are placed end-to-end along the x
axis.
L
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x
Show that
F = μ (qm)2 / (4 π)
–1
[x
–2
+ (x + 2L) –2 – 2(x +L) –2 ]
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In theory, a magnet can be viewed as a
ferromagnetic body with “bound” surface
currents that generate the magnetic field
much like a solenoid does.
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Magnetic fields are created by
◦ magnetic dipoles,
◦ electric currents, and
◦ changing electric fields.
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The magnetic field is characterized by the B
and H vectors. Both are related by the
permeability (or magnetic constant) μ of the
medium
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The vector field H is known as the magnetic
field intensity or magnetic field strength:
◦ H is measured in Amperes per meter (A/m).
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The vector field B is known as magnetic flux
density or magnetic induction or simply
magnetic field:
◦ B has the units of Teslas (T), equivalent to Webers
per square meter (Wb/m²) or volt-seconds per
square meter (V s/m²).
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Magnetic sources are inherently dipole
sources
So far we know, one cannot isolate north or
south "monopoles“
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An electric charge e in motion generates a
magnetic field B
A monopole g in motion would generate an
electric field E
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Magnetic field lines always form continuous
closed loops. The differential form for Gauss'
law for magnetism is the following:
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The integral form of Gauss' law for magnetism states that
where
◦ S is any closed surface (a "closed surface" is the boundary of some
three-dimensional volume)
◦ dA is a vector, whose magnitude is the area of an infinitesimal
piece of the surface S, and whose direction is the outwardpointing surface normal.
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The solar wind exerts
pressure on the magnetic
field of the Earth.
As a result the field is
compressed on the side
toward the sun and is
dragged into space on the
side away from the sun.
On the extended side, the
magnetic field lines extend
beyond 100 Earth radii.
What is the divergence of
the field on either side of
the Earth?
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The left-hand side of this
equation is called the net
flux of the magnetic field Φ
out of the surface, and
Gauss' law for magnetism
states that it is always zero.
The integral and differential
forms of Gauss' law for
magnetism are
mathematically equivalent,
due to the divergence
theorem.
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The magnetic field intensity generated by a
small current loop of radius a located at the
center of the x-y plane is given by
H = Ia2 (aR 2 cos θ + aθ sin θ) / 4R3
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Show that net flux coming out of a Gaussian
spherical surface is zero.
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Magnetostatics is the study of
static magnetic fields produced
by direct currents.
If all the currents in a system are
known, then the magnetic field
can be determined from the
currents by the Biot-Savart
equation.
The Biot–Savart law is used to
compute the magnetic field
generated by a steady current,
for example through a wire,
which is constant in time and in
which charge is neither building
up nor depleting at any point.
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The corresponding integral form is
where
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I is the current
dl is the differential element of the wire in the
direction of conventional current
r is the distance between the element and the
observation point P
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The application of this law
implicitly relies on the
superposition principle for
magnetic fields:
◦ the magnetic field is a vector
sum of the field created by each
infinitesimal section I dl of the
wire individually.
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Choose a point P in space at
which you want to compute
the magnetic field.
Holding that point fixed,
integrate over the path of the
current to find the total
magnetic field at that point.
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In 1820 Biot and
Savart
announced that
the magnetic
force exerted by
a long
conductor on a
magnetic pole
falls off with the
reciprocal of the
distance and is
orientated
perpendicular to
the wire.
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Magnet Lab
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http://www.magnet.fsu.edu/education/tutorials/java/magwire/index.html
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Determine the magnetic flux density B around
an infinite straight wire carrying a steady
current Io z.
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Consider a current loop of radius a centered
on the xy plane. Determine B at the center of
the loop.
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Assume that the loop is on the xy-plane and
the observation point is at the origin of the
coordinate system.
Set up the integral to find B at the center of
the loop.
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Read Sections 5-2, 5-4, and 5-5
Solve end-of-chapter problems 5.7, 5.9,
5.12, 5.20, and 5.21
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HW counts 10% of your final grade!
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600 BC - 1599 – Humans discover the magnetic lodestone as well as the attracting properties of amber.
Advanced societies, in particular the Chinese and the Europeans, exploit the properties of magnets in
compasses, a tool that makes possible exploration of the seas, “new worlds” and the nature of Earth’s magnetic
poles.
1600 - 1699 – The Scientific Revolution takes hold, facilitating the groundbreaking work of luminaries such as
William Gilbert, who took the first truly scientific approach to the study of magnetism and electricity and wrote
extensively of his findings.
1700 - 1749 – Aided by tools such as static electricity machines and leyden jars, scientists continue their
experiments into the fundamentals of magnetism and electricity.
1750 - 1774 – With his famous kite experiment and other forays into science, Benjamin Franklin advances
knowledge of electricity, inspiring his English friend Joseph Priestley to do the same.
1775 - 1799 – Scientists take important steps toward a fuller understanding of electricity, as well as some
fruitful missteps, including an elaborate but incorrect theory on animal magnetism that sets the stage for a
groundbreaking invention.
1800 - 1819 – Alessandro Volta invents the first primitive battery, discovering that electricity can be generated
through chemical processes; scientists quickly seize on the new tool to invent electric lighting. Meanwhile, a
profound insight into the relationship between electricity and magnetism goes largely unnoticed.
1820 - 1829 – Hans Christian Ørsted’s accidental discovery that an electrical current moves a compass needle
rocks the scientific world; a spate of experiments follows, immediately leading to the first electromagnet and
electric motor.
1830 - 1839 – The first telegraphs are constructed and Michael Faraday produces much of his brilliant and
enduring research into electricity and magnetism, inventing the first primitive transformer and generator.
1840 - 1849 – The legendary Faraday forges on with his prolific research and the telegraph reaches a milestone
when a message is sent between Washington, DC, and Baltimore, MD.
1850 - 1869 – The Industrial Revolution is in full force, Gramme invents his dynamo and James Clerk Maxwell
formulates his series of equations on electrodynamics.
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1870 - 1879 – The telephone and first practical incandescent light bulb are invented while the
word “electron” enters the scientific lexicon.
1880 - 1889 – Nikola Tesla and Thomas Edison duke it out over the best way to transmit
electricity and Heinrich Hertz is the first person (unbeknownst to him) to broadcast and receive
radio waves.
1890 - 1899 – Scientists discover and probe x-rays and radioactivity, while inventors compete
to build the first radio.
1900 - 1909 – Albert Einstein publishes his special theory of relativity and his theory on the
quantum nature of light, which he identified as both a particle and a wave. With ever new
appliances, electricity begins to transform everyday life.
1910 - 1929 – Scientists’ understanding of the structure of the atom and of its component
particles grows, the phone and radio become common, and the modern television is born.
1930 - 1939 – New tools such as special microscopes and the cyclotron take research to
higher levels, while average citizens enjoy novel amenities such as the FM radio.
1940 - 1959 – Defense-related research leads to the computer, the world enters the atomic
age and TV conquers America.
1960 - 1979 – Computers evolve into PCs, researchers discover one new subatomic particle
after another and the space age gives our psyches and science a new context.
1980 - 2003 – Scientists explore new energy sources, the World Wide Web spins a vast network
and nanotechnology is born.