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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

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ENTC 3331

RF Fundamentals

Dr. Hugh Blanton

ENTC 3331


Magnetostatics


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


+

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


+

+

I

I

Dr. Blanton - ENTC 3331 - Magnetostatics 5


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


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

volume

Dr. Blanton - ENTC 3331 - Magnetostatics 7


projection of

  • Generalization:

projection of onto the surface normal

Dr. Blanton - ENTC 3331 - Magnetostatics 8


Current Density

  • The definition of current density is:

    • Therefore,

Dr. Blanton - ENTC 3331 - Magnetostatics 9


Electrical Current

Electrical Currents

Dr. Blanton - ENTC 3331 - Magnetostatics 10


+

-

Conducting Media

  • Two types of charge carriers:

    • Negative charges

    • Positive charges

Dr. Blanton - ENTC 3331 - Magnetostatics 11


Dr. Blanton - ENTC 3331 - Magnetostatics 12


  • 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


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

constant of proportionality (mobility)

Dr. Blanton - ENTC 3331 - Magnetostatics 14


  • In semiconductors:

    • electron mobility

      • electrons move against the direction

    • hole mobility

      • holes move in the same direction as

Dr. Blanton - ENTC 3331 - Magnetostatics 15


  • Since

Ohm’s law

conductivity

Dr. Blanton - ENTC 3331 - Magnetostatics 16


  • 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


  • Since

    • for all conductors.

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

Dr. Blanton - ENTC 3331 - Magnetostatics 18


Electrical Resistance

  • For a conductor

  • Show that for a conductor of

    cylindrical shape.

A2

A1

Dr. Blanton - ENTC 3331 - Magnetostatics 19


  • Potential difference between A1 and A2.

  • Current through A1 and A2.

Dr. Blanton - ENTC 3331 - Magnetostatics 20


The reciprocal of conductivity Resistivity (ohms/meter).

Do not confuse charge distribution!

Dr. Blanton - ENTC 3331 - Magnetostatics 21


  • 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


  • Qualitatively,

    • circular field lines

Dr. Blanton - ENTC 3331 - Magnetostatics 23


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

Dr. Blanton - ENTC 3331 - Magnetostatics 24


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


  • 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


  • Take advantage of:

useful relationship

Biot-Savart Law

Dr. Blanton - ENTC 3331 - Magnetostatics 27


  • What force does such a field exert onto a stationary current?

    • What is equivalent to:

Dr. Blanton - ENTC 3331 - Magnetostatics 28


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


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


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


  • The experimental facts also show that:

    • and

      • Thus, the magnetic force for a straight conductor is:

Dr. Blanton - ENTC 3331 - Magnetostatics 32


Important Consequences

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

closed loop = 0

Dr. Blanton - ENTC 3331 - Magnetostatics 33


X

Example

  • Linear conductor

    • Determine magnetic field .

    • Determine the force, , on another conductor.

z

Biot-Savart Law

x

Dr. Blanton - ENTC 3331 - Magnetostatics 34


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


X

z

  • Note:

x

Dr. Blanton - ENTC 3331 - Magnetostatics 36


X

z

  • Using the previous transformations:

x

Dr. Blanton - ENTC 3331 - Magnetostatics 37


z

  • Note the following

x

Dr. Blanton - ENTC 3331 - Magnetostatics 38


  • For an infinitely long wire where

Dr. Blanton - ENTC 3331 - Magnetostatics 39


  • 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


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

  • 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


  • 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


  • 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


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

  • 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 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


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

    • This implies that

Dr. Blanton - ENTC 3331 - Magnetostatics 48


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

Dr. Blanton - ENTC 3331 - Magnetostatics 49


  • What is inside the wire?

    • Again, use an Ampere’s circuital law.

z

Dr. Blanton - ENTC 3331 - Magnetostatics 50


  • is current through the Amperian surface

  • The magnitude of increases linearly inside the conductor.

Dr. Blanton - ENTC 3331 - Magnetostatics 51


  • 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


  • The magnetostatic field is rotational without sources

  • In electrostatics

    • A scaler potential, V, exists, so that

Dr. Blanton - ENTC 3331 - Magnetostatics 53


  • 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


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

    • Determine

Dr. Blanton - ENTC 3331 - Magnetostatics 55


Dr. Blanton - ENTC 3331 - Magnetostatics 56


  • 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


Dr. Blanton - ENTC 3331 - Magnetostatics 58


Dr. Blanton - ENTC 3331 - Magnetostatics 59


  • Note that since , it follows from Stoke’s theorem that

    • Calculate again using

0.25m

0.25m

x

Dr. Blanton - ENTC 3331 - Magnetostatics 60


Dr. Blanton - ENTC 3331 - Magnetostatics 61


Dr. Blanton - ENTC 3331 - Magnetostatics 62


Dr. Blanton - ENTC 3331 - Magnetostatics 63


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