Lecture 9 Vector Magnetic Potential Biot Savart Law

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Lecture 9 Vector Magnetic PotentialBiot
Savart Law

• Prof. Viviana Vladutescu

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Figure 1 The magnetic (H-field) streamlines
inside and outside a single thick wire.
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Figure 2 The H-field magnitude inside and
outside the thick wire with uniform current
density
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Figure 3 The H-field magnitude inside and
outside the thick conductors of a coaxial line.
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Vector Magnetic Potential
A - vector magnetic potential (Wb/m)
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Figure 1 The vector potential in the
cross-section of a wire with uniform current
distribution.
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Figure 2 Comparison between the magnetic vector
potential component  of a wire with uniformly
distributed current and the electric potential V
of the equivalent cylinder with uniformly
distributed charge.
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Poissons Equation
Laplacian Operator (Divergence of a gradient)
Vector Poissons equation
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In electrostatics
Poissons Equation in electrostatics
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Magnetic Flux
The line integral of the vector magnetic
potential A around any closed path equals the
total magnetic flux passing through area enclosed
by the path
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Biot Savart Law and Applications
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The Biot-Savart Law relates magnetic fields to
the currents which are their sources. In a
similar manner, Coulombs Law relates electric
fields to the point charges which are their
sources. Finding the magnetic field resulting
from a current distribution involves the vector
product, and is inherently a calculus problem
when the distance from the current to the field
point is continuously changing.
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Biot-Savart Law
By using
(see eq 6.31)
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In two steps
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Illustration of the law of BiotSavart showing
magnetic field arising from a differential
segment of current.
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Example1 Component values for the equation to
find the magnetic field intensity resulting from
an infinite length line of current on the z-axis.
(ex 6-4)
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Example 2 We want to find H at height h above a
ring of current centered in the x y plane.
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The component values shown for use in the
BiotSavart equation.
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The radial components of H cancel by symmetry.
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Solenoid
Many turns of insulated wire coiled in the shape
of a cylinder.
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For a set N number of loops around a ferrite
core, the flux generated is the same even when
the loops are bunched together.
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Example A simple toroid wrapped with N turns
modeled by a magnetic circuit. Determine B inside
the closely wound toroidal coil.
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Amperes Law
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Electromagnets
a) An iron bar attached to an electromagnet. b)
The bar displaced by a differential length d?.
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Applications
Levitated trains Maglev prototype
Electromagnet supporting a bar of mass m.
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Wilhelm Weber (1804-1891). Electromagnetism.