MAGNETOSTATIC FIELD

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CHAPTER 7
MAGNETOSTATIC FIELD (STEADY MAGNETIC)
7.1 INTRODUCTION - SOURCE OF MAGNETOSTATIC FIELD
7.2 ELECTRIC CURRENT CONFIGURATIONS 7.3 BIOT
SAVART LAW 7.4 AMPERES CIRCUITAL LAW 7.5 CURL
(IKAL) 7.6 STOKES THEOREM 7.7 MAGNETIC FLUX
DENSITY 7.8 MAXWELLS EQUATIONS 7.9 VECTOR
MAGNETIC POTENTIAL

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7.1 INTRODUCTION - SOURCE OF MAGNETOSTATIC FIELD

• Originate from
• constant current
• permanent magnet
• electric field changing linearly with time

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Analogous between electrostatic and magnetostatic
fields
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Two important laws for solving magnetostatic
field

• Biot Savart Law general case
• Amperes Circuital Law cases of symmetrical
  current distributions

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7.2 ELECTRIC CURRENT CONFIGURATIONS
Three basic current configurations or
distributions
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7.3 BIOT SAVART LAW
Consider the diagram as shown
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Total magnetic field can be obtained by
integrating
Similarly for surface current and volume current
elements the magnetic field intensities can be
written as
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Ex. 7.1 For a filamentary current distribution
of finite length and along the z axis, find (a)
and (b) when the current extends from -?
to ?.
Solution
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(b) When a - ? and b ?, we see that ?1
?/2, and ?2 ?/2
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Solution
Using Biot Savart Law
and
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Hence
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Fig. 7.3 (a) Closely wound solenoid (b) Cross
section (c) surface current, NI (A).
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Solution

• Total surface current NI Ampere
• Surface current density, Js NI / l Am-1
• View the dz length as a thin current loop that
• carries a current of Jsdz (NI / l )dz

Solution from Ex. 7.2
Hence
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at the center of the solenoid
which is one half the value at the center.
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Ex. 7.4 Find at point (-3,4,0) due to the
filamentary current as shown in the Fig. below.
Solution
Total magnetic field intensity is given by
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Unit vector
Hence
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Unit vector
Hence
Hence
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7.4 AMPERES CIRCUITAL LAW

• Solving magnetostaic problems for cases of
  symmetrical current distributions.

Definition
The line integral of the tangential component of
the magnetic field strength around a closed path
is equal to the current enclosed by the path
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Graphical display for Amperes Circuital Law
interpretation of Ien
Path (loop) (a) and (b) enclose the total current
I , path c encloses only part of the current I
and path d encloses zero current.
I

(a)

(b)

(c)

(d)
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Ex. 7.5 Using Amperes circuital law, find
field for the filamentary current I of infinite
length as shown in Fig. 7.6.
z
to ?
I
y
x
to -?
Fig. 7.6
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Ex. 7.6 Find field above and below a surface
current distribution of infinite extent with a
surface current density
Solution
Graphical display for finding and using
Amperes circuital law
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where
Therefore
Similarly if we takes on the path 3-3'-2'-2-3,
the equation becomes
Hence
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In vector form
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z
h
0
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7.5 CURL (IKAL)
The curl of a vector field, is another
vector field.
For example in Cartesian coordinate, combining
the three components, curl can be written as
And can be simplified as
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Expression for curl in cyclindrical and spherical
coordinates
cyclindrical
spherical
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(a) for a
filamentary current
(b) in an
infinite current carrying conductor with radius
a meter
(c) for infinite sheet
of uniformly surface current Js
(d)
in outer conductor of coaxial cable
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Solution
(a)
Cyclindrical coordinate
Hence
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Solution
(b)
Cyclindrical coordinate
Hence
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Solution
(c)
Cartesian coordinate
because Hx constant and Hy Hz 0.
Hence
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Solution
(d)
Cyclindrical coordinate
Hence
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7.6 STOKES THEOREM
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It can be shown as follow
Consider an open surface S whose boundary is a
closed surface l
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Hence
where loop l is the path that enclosed surface S
and this equation is called Stokes Theorem.
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Solution
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7.7 MAGNETIC FLUX DENSITY
Magnetic field intensity
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In magnetics, magnet poles have not been isolated
4th. Maxwells equation for static fields.
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Solution
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7.8 MAXWELLS EQUATIONS
Electrostatic fields
Magnetostatic fields
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7.9 VECTOR MAGNETIC POTENTIAL
To define vector magnetic potential, we start
with
magnet poles have not been isolated
gt
Using divergence theorem
ltgt
From vector identity
Therefore from Maxwell and identity vector, we
can defined if is a vector magnetic
potential, hence
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