Magnetostatic fields originate from steady currents like direct currents in wires. Most equations for electric fields can also describe magnetic fields by substituting analogous quantities. According to Biot-Savart's law, the magnetic field produced by a current element is proportional to the current and inversely proportional to the distance squared. Ampere's circuital law relates the line integral of magnetic field around a closed loop to the current passing through the enclosed surface. Forces can act on moving charges and on current-carrying wires placed in magnetic fields.

1. Unit-2Unit-2
MagnetostaticsMagnetostatics
Mr. HIMANSHU DIWAKAR
Assistant Professor
Department of ECE

2. If charges are moving with constant velocity, a static
magnetic (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

3. π
=
4
1
k
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
α
2
R
sinIdl
kdH
α
=
2
R4
sinIdl
kdH
π
α
=
MagnetostaticsMagnetostatics
Biot – Savart’s Law

4. 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 to
the field point, and is the distance between these two
points.
)sin( nABaABBA θ=×
32
44 R
RdlI
R
adlI
Hd r
ππ
×
=
×
= RR = R
Rar =
r′
rre ′
rr ′−
r
32
4
)]([
4
)(
rr
rrdlI
rr
adlI
Hd r
′−
′−×
=
′−
×
=
ππ
Hd
MagnetostaticsMagnetostatics

5. 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).
MagnetostaticsMagnetostatics

6. 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.
GETGI

7. 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
GETGI
enc
C
IldH =⋅∫

8. 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 current
density is the most
general, we can write
Iencl in this way.
S
dl
dS

9. Applying Stokes’s Theorem to Ampere’s
Law
GETGI
∫
∫∫
⋅==
⋅×∇=⋅
S
encl
SC
sdJI
sdHldH
⇒ Because the above must hold for any
surface S, we must have
JH =×∇ Differential form
of Ampere’s Law

10. 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.
GETGI

11. 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.
GETGI
encl
C
IldH =⋅∫
known
unknown

12. 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.
GETGI

13. 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.
GETGI

14. 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:
GETGI
I

15. 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
GETGI
φφ HaH ˆ=
circles of radius ρ where
∞≤≤ ρ0

16. Magnetic Flux Density of an Infinite Line
Current Using Ampere’s Law (Cont’d)
GETGI
Amperian path
IIencl =
I
ρ
x
y

17. Magnetic Flux Density of an Infinite Line Current
Using Ampere’s Law (Cont’d)
(4) For each Amperian path, evaluate the integral
GETGI
HlldH
C
=⋅∫
ρπφ 2HldH
C
=⋅∫
magnitude of H
on Amperian
path.
length
of Amperian
path.

18. Magnetic Flux Density of an Infinite Line Current
Using Ampere’s Law (Cont’d)
(5) Solve for B on each Amperian path
GETGI
l
I
H encl
=
ρπ
φ
2
ˆ
I
aH =

19. 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 µ=
µ
7
o 104 −
⋅π=µ
∫ ⋅=Ψ
S
,sdB
µ

20. 0=•∫s
sdB
Law of conservation of magnetic flux
Law of conservation of
magnetic flux or Gauss’s
law for magnetostatic field

21. JB
B
0
0
µ=×∇
=•∇
0=•∇ J

22. There are no magnetic flow sources, and
the magnetic flux lines always close upon
themselves

23. 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 field
2.On a current element in an external field
3.Between two current elements

24. Force on a Moving Charge
 A moving point charge placed in a magnetic field
experiences a force given by
GETGI
BvQ
The force experienced
by the point charge is
in the direction into the
paper.
B
BuqFm ×=

25. 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

26. One can tell if a magneto-static field is present because it
exerts forces on moving charges and on currents. If a current
element 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 perpendicular
to . It is largest when and the wire are perpendicular.
Idl B
dF=Idl×B
B
Idl
B
__
uQlId ⇔
Magnetic Force on a current element

27. 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
GETGI
∫ ×=
C
BldIF

28. Force between two current
elements
 Experimental facts:
 Two parallel wires carrying
current in the same
direction attract.
 Two parallel wires carrying
current in the opposite
directions repel.
GETGI
× ×
I1 I2
F12F21
× •
I1 I2
F12F21

29. Force between two current
elements
 Experimental facts:
 A short current-carrying wire
oriented perpendicular to a
long current-carrying wire
experiences no force.
GETGI
×
I1
F12 = 0
I2

30. Force between two current
elements
 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.
GETGI