Electromagnetic Fields
Lecture 6
Magnetostatics 1
History
●
Lodestone: natural magnets
●
Magnetic compasses
René Descartes, 1644.
History
●
Hans Christian Ørsted (Oersted) 1777-1851
●
André-Marie Ampère 1775-1836)
Magnetic Field of the Earth
[Wikipedia]
North geographic pole = South magnetic pole
Right hand grip rule
[Wikipedia]
Magnetic field
●
Magnetic field intensity
H
●
[A/m] - Amper per metre
Magnetic flux density
B= H
B=0  H  M 
B=0 1m  H = H
[T] - Tesla
magnetic permeability
magnetic susceptibility
Ampere's Law
●
Ampère's Circuital Law
The integrated magnetic field around a closed loop is equal the electric
current passing through the loop.
∮C B⋅d l=0 I
∮C H⋅d l= I
∮C H⋅d l=∫S J⋅d S
S
I
H
Gauss's law for magnetism
Magnetic monopoles does not exist.
∇⋅B=0
∮ B⋅d S=0
There isn't any point where magnetic isolines starts.
All of them are closed loops.
Biot-Savart Law
The Biot–Savart law is used to compute the magnetic field generated by
a steady current flowing in the wire.
0 I d l ×r
B=∫
4  ∣r∣3
I - electric current,
dl - a vector, whose magnitude is the
length of the differential element of
the wire, and whose direction is the
direction of current,
B - magnetic field,
u0 - magnetic constant,
r - displacement vector,
|r| - magnitude of r,
dl
I
r
Example 1: long wire
Find value of magnetic field intensity around
long wire with current I.
r
Ampere's law:
H
∮C H⋅d l= I
I
Because of symmetry, H is constant for given r, so:
H ∮circle 1 dl = I
H 2  r= I
I
H=
2r
Example 2: single loop
Find value of magnetic field intensity in the center of circular
wire loop with current I.
dl
Biot-Savarte law:
1 I d l ×r
H =∫
4  ∣r∣3
For the center of circle, r and
angles are constant, so:
I
H=
1d l
2 ∫circle
4r
r
I
I
H=
2r
H
Energy in magnetics
B⋅H
u=
2
●
Energy density:
●
Total energy stored in the field:
U =∫V u dv
●
For coil:
1
2
U= LI
2
Example 3: energy in wire
Find energy stored in the magnetic field in one meter of
long, straight coaxial cable.
2
B⋅H
u=
2
2 2
0 J r
u1 =
4
2 2
4
 0 J  R1
u2=
2 2
8 r
J  R1
H 2=
2r
Jr
H 1=
2
R1
R2
R1
U =U 1 U 2= ∫ u1 d v
r=0
R2
∫
r= R1
u2 d v
Example 3: cont.
0  J
U 1=
2
2 R1
∫r
r=0
2
0  J
4
U 1=
R1
8
Total energy:
2
3
dr
2
0 J  R 1
U 2=
4
2
4
4
R2
∫
r=R1
1
dr
r
 
0 J  R1
R2
U 2=
ln
4
R1
U =U 1 U 2
References
References:
Deventra K. Mistra: Practical Electromagnetics, From Biomedical Science to Wireless
Communication, Wiley-Interscience, 2007
Joseph F. Becker: Physics 51 - Electricity & Magnetis, Califonia State University
http://www.physics.sjsu.edu/becker/physics51/
some figures were taken from Wikipedia.
Licence:
This work is published under the Creative Commons Attribution-ShareAlike Licence.