magnetostatics (1)
TRANSCRIPT
MagnetostaticsMagnetostatics
If charges are moving with constant velocity, a staticmagnetic (or magnetostatic) field is produced. Thus, magnetostatic fields originate from currents (for instance, direct currents in current-carrying wires).
Most of the equations we have derived for the electric fields may be readily used to obtain corresponding equations for magnetic fields if the equivalent analogous quantities are substituted.
MagnetostaticsMagnetostatics
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Biot – Savart’s Law The magnetic field intensity dH produced at a point P by the differential current element Idl is proportional to the product of Idl and the sine of the angle between the element and the line joining P to the element and is inversely proportional to the square of the distance R between P and the element.
IN SI units, , so
2RsinIdl
kdH
2R4sinIdl
kdH
41
k
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Using the definition of cross product we can represent the previous equation in vector form as
The direction of can be determined by the right-hand rule or by the right-handed screw rule.
If the position of the field point is specified by and the position of the source point by
where is the unit vector directed from the source point tothe field point, and is the distance between these twopoints.
)sin( nABaABBA
32 44 R
RdlI
R
adlIHd r
RR RRar
r
rre
rr
r
324
)]([
4
)(
rr
rrdlI
rr
adlIHd r
Hd
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The field produced by a wire can be found adding up the fields produced by a large number of current elements placed head to tail along the wire (the principle of superposition).
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Types of Current Distributions
• Line current density (current) - occurs for infinitesimally thin filamentary bodies (i.e., wires of negligible diameter).
• Surface current density (current per unit width) - occurs when body is perfectly conducting.
• Volume current density (current per unit cross sectional area) - most general.
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Ampere’s Circuital Law in Integral Form
• Ampere’s Circuit Law states that the line integral of H around a closed path proportional to the total current through the surface bounding the path
enc
C
IldH
Ampere’s Circuital Law in Integral Form (Cont’d)
By convention, dS is taken to be in the
direction defined by the right-hand rule applied
to dl.
S
encl sdJI
Since volume currentdensity is the most
general, we can write Iencl in this way.
S
dl
dS
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Applying Stokes’s Theorem to Ampere’s Law
S
encl
SC
sdJI
sdHldH
Because the above must hold for any surface S, we must have
JH Differential formof Ampere’s Law
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Ampere’s Law in Differential Form
• Ampere’s law in differential form implies that the B-field is conservative outside of regions where current is flowing.
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Applications of Ampere’s Law
• Ampere’s law in integral form is an integral equation for the unknown magnetic flux density resulting from a given current distribution.
encl
C
IldH known
unknown
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Applications of Ampere’s Law (Cont’d)
• In general, solutions to integral equations must be obtained using numerical techniques.
• However, for certain symmetric current distributions closed form solutions to Ampere’s law can be obtained.
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Applications of Ampere’s Law (Cont’d)
• Closed form solution to Ampere’s law relies on our ability to construct a suitable family of Amperian paths.
• An Amperian path is a closed contour to which the magnetic flux density is tangential and over which equal to a constant value.
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Magnetic Flux Density of an Infinite Line Current Using Ampere’s Law
Consider an infinite line current along the z-axis carrying current in the +z-direction:
I
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Magnetic Flux Density of an Infinite Line Current Using Ampere’s Law (Cont’d)
(1) Assume from symmetry and the right-hand rule the form of the field
(2) Construct a family of Amperian paths
HaH ˆ
circles of radius where
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Magnetic Flux Density of an Infinite Line Current Using Ampere’s Law (Cont’d)
Amperian path
IIencl
I
x
y
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Magnetic Flux Density of an Infinite Line Current Using Ampere’s Law (Cont’d)
(4) For each Amperian path, evaluate the integral
HlldHC
2HldHC
magnitude of Hon Amperian
path.
lengthof Amperian
path.
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Magnetic Flux Density of an Infinite Line Current Using Ampere’s Law (Cont’d)
(5) Solve for B on each Amperian path
l
IH encl
2ˆ
IaH
Magnetic Flux Density
The magnetic flux density vector is related to the magnetic field intensity by the following equation
T (tesla) or Wb/m2
where is the permeability of the medium. Except for ferromagnetic materials ( such as cobalt, nickel, and iron),most materials have values of very nearly equal to that for vacuum,
H/m (henry per meter)The magnetic flux through a given surface S is given by
Wb (weber)
H
,HB
7o 104
S
,sdB
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0s
sdB
Law of conservation of magnetic flux
Law of conservation of magnetic flux or Gauss’s law for magnetostatic field
JB
B
0
0
0 J
There are no magnetic flow sources, and the magnetic flux lines always close upon themselves
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Forces due to magnetic field
There are three ways in which force due to magnetic fields can be experienced
1.Due to moving charge particle in a B field2.On a current element in an external field 3.Between two current elements
The force is perpendicular to the direction of charge motion, and is also perpendicular to .
Force on a Moving Charge
• A moving point charge placed in a magnetic field experiences a force given by
BvQ
The force experienced by the point charge is in the direction into the paper.
B
BuqFm
BuqF
EqF
m
e
FFF me
)( BuEqF
Lorentz’s Force equation
Note: Magnetic force is zero for q moving in the direction of the magnetic field (sin0=0)
If a point charge is moving in a region where both electric and magnetic fields exist, then it experiences a total force given by
Magnetic Force on a current elementOne can tell if a magneto-static field is present because it exerts forces on moving charges and on currents. If a currentelement is located in a magnetic field , it experiences a force
This force, acting on current element , is in a direction perpendicular to the current element and also perpendicularto . It is largest when and the wire are perpendicular.
Idl B
dF Idl B
B
Idl
B
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__
uQlId
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FORCE ON A CURRENT ELEMENT
• The total force exerted on a circuit C carrying current I that is immersed in a magnetic flux density B is given by
C
BldIF
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• Experimental facts:– Two parallel wires
carrying current in the same direction attract.
– Two parallel wires carrying current in the opposite directions repel.
I1 I2
F12F21
I1 I2
F12F21
Force between two current elements
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• Experimental facts:– A short current-
carrying wire oriented perpendicular to a long current-carrying wire experiences no force.
I1
F12 = 0
I2
Force between two current elements
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• Experimental facts:– The magnitude of the force is inversely
proportional to the distance squared.– The magnitude of the force is proportional to
the product of the currents carried by the two wires.
Force between two current elements
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• The force acting on a current element I2 dl2 by a current element I1 dl1 is given by
2
12
1122012
12ˆ
4)(
R
aldIldIFdd R
Permeability of free space0 = 4 10-7 F/m
Force between two current elements
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• The total force acting on a circuit C2 having a current I2 by a circuit C1 having current I1 is given by
2 1
12
212
1221012
ˆ
4 C C
R
R
aldldIIF
Force between two current elements
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• The force on C1 due to C2 is equal in magnitude but opposite in direction to the force on C2 due to C1.
1221 FF
Force between two current elements