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ENTC 3331 RF Fundamentals. Dr. Hugh Blanton ENTC 3331. Magnetostatics . Magnetism Chinese—100 BC Arabs—1200 AD Magnetite—Fe 3 O 4 Found near Magnesia (now Turkey) Permanent magnet Not fundamental to magnetostatics. A permanent magnet is equivalent to a polar material in electrostatics.

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dr hugh blanton entc 3331
ENTC 3331

RF Fundamentals

Dr. Hugh Blanton

ENTC 3331

magnetostatics3
Magnetism

Chinese—100 BC

Arabs—1200 AD

Magnetite—Fe3O4

Found near Magnesia (now Turkey)

Permanent magnet

Not fundamental to magnetostatics.

A permanent magnet is equivalent to a polar material in electrostatics.

Equivalent to electrostatics

The theoretical structure of magnetostatics is very similar to electrostatics.

But there is one important empirical fact that accounts for all the differences between the theory of magnetostatics and electrostatics.

There is no magnetic monopole!

Magnetostatics

Dr. Blanton - ENTC 3331 - Magnetostatics 3

slide4

+

N

+

S

N

S

S

+

N

A magnetic monopole does not exist—A magnetostatic field has no sources or sinks!

+

Dr. Blanton - ENTC 3331 - Magnetostatics 4

slide5
+

+

I

I

Dr. Blanton - ENTC 3331 - Magnetostatics 5

current density
Current Density
  • Moving chargescurrent.
  • Charges move to the right with constant velocity, u.
    • Over a period of time, the charges move distance, D l.

D s

u

rv

D l

Dr. Blanton - ENTC 3331 - Magnetostatics 6

slide7
The amount of charge through an area, Ds, during Dt:

volume

Dr. Blanton - ENTC 3331 - Magnetostatics 7

slide8
projection of
  • Generalization:

projection of onto the surface normal

Dr. Blanton - ENTC 3331 - Magnetostatics 8

current density9
Current Density
  • The definition of current density is:
    • Therefore,

Dr. Blanton - ENTC 3331 - Magnetostatics 9

electrical current
Electrical Current

Electrical Currents

Dr. Blanton - ENTC 3331 - Magnetostatics 10

conducting media
+

-

Conducting Media
  • Two types of charge carriers:
    • Negative charges
    • Positive charges

Dr. Blanton - ENTC 3331 - Magnetostatics 11

slide13
Like mechanics, there is a resistance to motion.
    • Therefore, an external force is required to maintain a current flow in a resistive conductor.

Dr. Blanton - ENTC 3331 - Magnetostatics 13

slide14
Since in most conductors, the resistance is proportional to the charge velocity.

constant of proportionality (mobility)

Dr. Blanton - ENTC 3331 - Magnetostatics 14

slide15
In semiconductors:
    • electron mobility
      • electrons move against the direction
    • hole mobility
      • holes move in the same direction as

Dr. Blanton - ENTC 3331 - Magnetostatics 15

slide16
Since

Ohm’s law

conductivity

Dr. Blanton - ENTC 3331 - Magnetostatics 16

slide17
It follows that for
    • a perfect dielectric
      • s = 0 
    • and for a perfect conductor
      • s 
      • since current is finite.
  • inside all conductors.

Dr. Blanton - ENTC 3331 - Magnetostatics 17

slide18
Since
    • for all conductors.
  • All conductors are equipotential, but may have surface charge.

Dr. Blanton - ENTC 3331 - Magnetostatics 18

electrical resistance
Electrical Resistance
  • For a conductor
  • Show that for a conductor of

cylindrical shape.

A2

A1

Dr. Blanton - ENTC 3331 - Magnetostatics 19

slide20
Potential difference between A1 and A2.
  • Current through A1 and A2.

Dr. Blanton - ENTC 3331 - Magnetostatics 20

slide21
The reciprocal of conductivity Resistivity (ohms/meter).

Do not confuse charge distribution!

Dr. Blanton - ENTC 3331 - Magnetostatics 21

slide22
The electrical field can be expressed in terms of the charge density, r.
    • What is the equivalent expression for the magnetic field, .

Dr. Blanton - ENTC 3331 - Magnetostatics 22

slide23
Qualitatively,
    • circular field lines

Dr. Blanton - ENTC 3331 - Magnetostatics 23

slide24
Jean-Baptiste Biot & Felix Savart developed the quantitative description for the magnetic field.

Dr. Blanton - ENTC 3331 - Magnetostatics 24

slide25
field comes out of plane due to the cross product

point of interest

differential section of conductor

contributes to field at

Dr. Blanton - ENTC 3331 - Magnetostatics 25

slide26
Total field through integration over .
  • The line integration is not convenient
    • Wires are irregularly bent, but
    • Wires typically have constant cross-sections, Ds.

magnetic field strength

Dr. Blanton - ENTC 3331 - Magnetostatics 26

slide27
Take advantage of:

useful relationship

Biot-Savart Law

Dr. Blanton - ENTC 3331 - Magnetostatics 27

slide28
What force does such a field exert onto a stationary current?
    • What is equivalent to:

Dr. Blanton - ENTC 3331 - Magnetostatics 28

slide29
X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

  • Experimental facts:
    • Flexible wire in a magnetic field, .
      • No current

Dr. Blanton - ENTC 3331 - Magnetostatics 29

slide30
X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

  • Experimental facts:
    • Flexible wire in a magnetic field, .
      • Current up.

right-handed rule

Dr. Blanton - ENTC 3331 - Magnetostatics 30

slide31
X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

  • Experimental facts:
    • Flexible wire in a magnetic field, .
      • Current down.

right-handed rule

Dr. Blanton - ENTC 3331 - Magnetostatics 31

slide32
The experimental facts also show that:
    • and
      • Thus, the magnetic force for a straight conductor is:

Dr. Blanton - ENTC 3331 - Magnetostatics 32

important consequences
Important Consequences
  • The force on a closed, current carrying loop is zero.

closed loop = 0

Dr. Blanton - ENTC 3331 - Magnetostatics 33

example
XExample
  • Linear conductor
    • Determine magnetic field .
    • Determine the force, , on another conductor.

z

Biot-Savart Law

x

Dr. Blanton - ENTC 3331 - Magnetostatics 34

slide35
X

z

  • Substituting

x

at P(x,z), points into the plane

Note that for a small dq, R is approximately unchanged when separated by dq which implies:

Dr. Blanton - ENTC 3331 - Magnetostatics 35

slide36
X

z

  • Note:

x

Dr. Blanton - ENTC 3331 - Magnetostatics 36

slide37
X

z

  • Using the previous transformations:

x

Dr. Blanton - ENTC 3331 - Magnetostatics 37

slide38
z
  • Note the following

x

Dr. Blanton - ENTC 3331 - Magnetostatics 38

slide39
For an infinitely long wire where

Dr. Blanton - ENTC 3331 - Magnetostatics 39

slide40
Now, what is the force on a parallel conductor wire carrying the current, I?

z

y

field by I1 at location of I2

x

Dr. Blanton - ENTC 3331 - Magnetostatics 40

slide41
z
  • I1 attracts I2
    • Similarly I2 attracts I1 with the same force.
  • Attraction is proportional to 1/distance.

y

x

Dr. Blanton - ENTC 3331 - Magnetostatics 41

maxwell s magnetostatic equations
Maxwell’s Magnetostatic Equations
  • Experimental fact: An equivalent to the electrostatic monopole field does not exist for magnetostatics.

Charge is the source of the electrostatic field

No equivalent in magnetostatics

Dr. Blanton - ENTC 3331 - Magnetostatics 42

slide43
Let’s apply Gauss’s theorem to an arbitrary field:
    • Gauss’s law of Magnetostatics
      • Mathematical expression of the experimental fact that a source of the magnetostatic field does not exist.

Dr. Blanton - ENTC 3331 - Magnetostatics 43

slide44
Experimental fact: The magnetostatic field is generally a rotational field.
    • Apply Stoke’s theorem to any arbitrary field:
    • Ampere’s Circuital Law

Dr. Blanton - ENTC 3331 - Magnetostatics 44

slide45
X
  • Mathematical expression of the experimental fact that the line integral of the magnetostatic field around a closed path is equal to the current flowing through the surface bounded by this path.

line differential

field vector of the magnetostatic field

surface

current flowing through the surface

contour

Dr. Blanton - ENTC 3331 - Magnetostatics 45

long line
Long line
  • Suppose we have an infinitely long line of charge:
    • Recall that charge is the fundamental quantity for electrostatics

Dr. Blanton - ENTC 3331 - Magnetostatics 46

long line47
Long line
  • Suppose we have an infinitely long line carrying current,I:
    • What is .
      • Orient wire along the

z-axis

      • Choose a circular Amperian contour about the wire.
        • Ampere circuital law

z

Dr. Blanton - ENTC 3331 - Magnetostatics 47

slide48
Symmetry implies that is constant on the contour and is always tangential to the contour.
    • This implies that

Dr. Blanton - ENTC 3331 - Magnetostatics 48

slide49
is always tangential on circles about the wire and its magnitude decreases with 1/r.

Dr. Blanton - ENTC 3331 - Magnetostatics 49

slide50
What is inside the wire?
    • Again, use an Ampere’s circuital law.

z

Dr. Blanton - ENTC 3331 - Magnetostatics 50

slide51
is current through the Amperian surface
  • The magnitude of increases linearly inside the conductor.

Dr. Blanton - ENTC 3331 - Magnetostatics 51

slide52
It is interesting to note that the comparison of part (a) and (b) of this problem shows that for a convective current, I, the electrostatic and magnetostatic fields are perpendicular to each other.
    • This is generally true in electrodynamics!

Dr. Blanton - ENTC 3331 - Magnetostatics 52

slide53
The magnetostatic field is rotational without sources
  • In electrostatics
      • A scaler potential, V, exists, so that

Dr. Blanton - ENTC 3331 - Magnetostatics 53

slide54
Can any potential be defined in magnetostatics?
    • Let’s take advantage of the general vector identity
    • Define a vector potential, ,so that
      • It follows that in agreement with Maxwell equations

Dr. Blanton - ENTC 3331 - Magnetostatics 54

slide55
In a given region of space, the vector potential of the magnetostatic field is given by
    • Determine

Dr. Blanton - ENTC 3331 - Magnetostatics 55

slide57
Magnetic flux, ,through an area S is given by the surface integral
    • Use this equation and the solution to previous problem to calculate the magnetic flux, , for the field through a square loop.

y

0.25m

0.25m

x

Dr. Blanton - ENTC 3331 - Magnetostatics 57

slide60
Note that since , it follows from Stoke’s theorem that
    • Calculate again using

0.25m

0.25m

x

Dr. Blanton - ENTC 3331 - Magnetostatics 60

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