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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 l.jpg

ENTC 3331

RF Fundamentals

Dr. Hugh Blanton

ENTC 3331



Magnetostatics3 l.jpg

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


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+

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


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+

+

I

I

Dr. Blanton - ENTC 3331 - Magnetostatics 5


Current density l.jpg
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


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volume

Dr. Blanton - ENTC 3331 - Magnetostatics 7


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projection of

  • Generalization:

projection of onto the surface normal

Dr. Blanton - ENTC 3331 - Magnetostatics 8


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Current Density

  • The definition of current density is:

    • Therefore,

Dr. Blanton - ENTC 3331 - Magnetostatics 9


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Electrical Current

Electrical Currents

Dr. Blanton - ENTC 3331 - Magnetostatics 10


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+

-

Conducting Media

  • Two types of charge carriers:

    • Negative charges

    • Positive charges

Dr. Blanton - ENTC 3331 - Magnetostatics 11


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constant of proportionality (mobility)

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  • In semiconductors: the charge velocity.

    • electron mobility

      • electrons move against the direction

    • hole mobility

      • holes move in the same direction as

Dr. Blanton - ENTC 3331 - Magnetostatics 15


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  • Since the charge velocity.

Ohm’s law

conductivity

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  • It follows that for the charge velocity.

    • a perfect dielectric

      • s = 0 

    • and for a perfect conductor

      • s 

      • since current is finite.

  • inside all conductors.

Dr. Blanton - ENTC 3331 - Magnetostatics 17


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  • Since the charge velocity.

    • for all conductors.

  • All conductors are equipotential, but may have surface charge.

Dr. Blanton - ENTC 3331 - Magnetostatics 18


Electrical resistance l.jpg
Electrical Resistance the charge velocity.

  • For a conductor

  • Show that for a conductor of

    cylindrical shape.

A2

A1

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The reciprocal of conductivity Resistivity (ohms/meter).

Do not confuse charge distribution!

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

Dr. Blanton - ENTC 3331 - Magnetostatics 24


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field comes out of plane due to the cross product description for the magnetic field.

point of interest

differential section of conductor

contributes to field at

Dr. Blanton - ENTC 3331 - Magnetostatics 25


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  • Total field through integration over . description for the magnetic field.

  • The line integration is not convenient

    • Wires are irregularly bent, but

    • Wires typically have constant cross-sections, Ds.

magnetic field strength

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useful relationship

Biot-Savart Law

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X stationary current?

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


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X stationary current?

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


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X stationary current?

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


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Dr. Blanton - ENTC 3331 - Magnetostatics 32


Important consequences l.jpg
Important Consequences stationary current?

  • The force on a closed, current carrying loop is zero.

closed loop = 0

Dr. Blanton - ENTC 3331 - Magnetostatics 33


Example l.jpg

X stationary current?

Example

  • Linear conductor

    • Determine magnetic field .

    • Determine the force, , on another conductor.

z

Biot-Savart Law

x

Dr. Blanton - ENTC 3331 - Magnetostatics 34


Slide35 l.jpg

X stationary current?

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


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X stationary current?

z

  • Note:

x

Dr. Blanton - ENTC 3331 - Magnetostatics 36


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X stationary current?

z

  • Using the previous transformations:

x

Dr. Blanton - ENTC 3331 - Magnetostatics 37


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z stationary current?

  • Note the following

x

Dr. Blanton - ENTC 3331 - Magnetostatics 38


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Dr. Blanton - ENTC 3331 - Magnetostatics 39


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z

y

field by I1 at location of I2

x

Dr. Blanton - ENTC 3331 - Magnetostatics 40


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z the current,

  • 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 l.jpg

the current,

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


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Dr. Blanton - ENTC 3331 - Magnetostatics 43


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Dr. Blanton - ENTC 3331 - Magnetostatics 44


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X rotational field.

  • 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 l.jpg
Long line rotational field.

  • 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 l.jpg
Long line rotational field.

  • 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


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Dr. Blanton - ENTC 3331 - Magnetostatics 48


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Dr. Blanton - ENTC 3331 - Magnetostatics 49


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z

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Dr. Blanton - ENTC 3331 - Magnetostatics 52


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Dr. Blanton - ENTC 3331 - Magnetostatics 53


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  • Can any potential be defined in magnetostatics? and (b) of this problem shows that for a convective current,

    • 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


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Dr. Blanton - ENTC 3331 - Magnetostatics 55


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Dr. Blanton - ENTC 3331 - magnetostatic field is given byMagnetostatics 56


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y

0.25m

0.25m

x

Dr. Blanton - ENTC 3331 - Magnetostatics 57


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Dr. Blanton - ENTC 3331 - integralMagnetostatics 58


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Dr. Blanton - ENTC 3331 - integralMagnetostatics 59


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0.25m

0.25m

x

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Dr. Blanton - ENTC 3331 - theorem thatMagnetostatics 61


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