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ENE 325ENE 325Electromagnetic Electromagnetic Fields and WavesFields and Waves

Lecture 9Lecture 9 Magnetic Boundary Magnetic Boundary Conditions, Inductance and Mutual Conditions, Inductance and Mutual InductanceInductance

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Scalar magnetic potential (Vm ) is different from the e lectric potential in that the scalar magnetic potential

is not a function of positions and there is no physicalinterpretation.

Vector magnetic potential (A ) is useful to find a magn etic field for antenna and waveguide. We can transfo

- rm Bio Savart’s law and can show that

Review (1)Review (1)

for the current line 0

4Id L

AR

for current sheet 0

4SKdS

AR

for current volume 0 .4vol

JdvA

R

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Review (2)Review (2) Magnetic force on a moving charge N.

Magnetic force on a current element N.

For a straight conductor in a uniform magnetic field or F = ILBsin N.

Torque on a closed circuit in which current is unifor m can be expressed as Nm.

For current loop that has uniform current and magn etic field, torque can be expressed as

F qv B

F Id L B

F I L B

T R F

.T I S B

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Magnetic boundary Magnetic boundary conditions (1)conditions (1)

Gauss’s law for magnetostatics

0S

B d S

1 2 0n nB S B S

1 2n nB B

12 1

2n nH H

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Magnetic boundary Magnetic boundary conditions (2)conditions (2)

Use Ampere’s circuital law

H d L I

1 2t tH L H L K L

1 2t tH H K

1 2 12( ) nH H a K

or 1 2 12 .nH H a K

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ExEx11 From the interface shown, given From the interface shown, given mT, determine and the angle th mT, determine and the angle th

at it makes with the interface. at it makes with the interface.

1 2 10x yB a a

2B

y-z

r1 1 r2 500

x

1 2

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1Ex1Ex The interface between two magnetic The interface between two magnetic materials is defined by the equation sho materials is defined by the equation sho

wn. Given H wn. Given H11 30= 30= A/mA/m , determine the , determine the following following

a) a)

xa

3x-4y > 0

3x-4y < 0

r1=3

r2=5

1B

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b)b)

c)c)

d) d)

1nB

1tH

2H

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33 Let the permeability be Let the permeability be 5 5 H/m in reg H/m in reg ion ion AA where where x x << 0 0 , and , and 2020 H/m in region H/m in region BB

where where xx > >00 . If there is a surface current d . If there is a surface current d ensity A/m at ensity A/m at xx = =00 , and if , and if

A/m, find A/m, find

a)a)

b) b)

150 200y zK a a

300 400 500A x y zH a a a

tAH

nAH

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c)c)

d) d)

tBH

nBH

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Potential energy of magneti Potential energy of magneti c materials c materials

12H

volw B Hdv

221 1

2 2vol vol

BH dv dv

J/m3.

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Duality of magnetostatics a Duality of magnetostatics a nd electrostatics nd electrostatics

Electrostatics Magnetostatics

E V

mH V

b

aba

V E d L ,

b

m aba

V H d L

J E

B H

I J d S B d S

V = IR Vm = dRS

= dS

0E d L totalH d L I NI

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Inductance and mutual indu Inductance and mutual inductancectance

FF lux linkage is the total flux passing through the surface lux linkage is the total flux passing through the surface bounded by the contour of the circuit carrying the curre bounded by the contour of the circuit carrying the curre

ntnt.. Inductane L Inductane L is defined as the ratio of flux linkage to the c is defined as the ratio of flux linkage to the c

urrent generating the flux, urrent generating the flux,

henrys or Wb/A henrys or Wb/A..totalNL

I

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A procedure for finding the ind A procedure for finding the inductanceuctance

1. Assume a current I ii iii iiiiiiiii

- 2. ,Determine using the law of Bio Savart or Ampere’ s circuit al law if there is sufficient symmetry.

3. Calculate the total flux linking all the loops.

4. Multiply the total flux by the number of loops to get the flux liiiiii.

5. Divide the flux linkage by I iii iii iiiiiiiiiii iii iiiii.ii iiiiiii i iii ii iiiiiii iii.

B

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Inductance for a coaxial cab Inductance for a coaxial cab le le

total flux 0 ln2

bIda

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Inductance for a solenoid wi Inductance for a solenoid wi th N turns th N turns

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Inductance for a toroid that Inductance for a toroid that has has NN turns and current turns and current II . .

If the wired is tightly wounded, the flux linkage will be the same for the adjacent turns of toroid. If the adjacent turns are

separated by some finite distance, the total flux must be calculated from the flux from each turn.

1 2( ) .......totalN

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More about inductanceMore about inductance The definition of the inductance can be written in

t he f ormof magnet i c energy as

T hecurrent i nsi de conduct or creat es t he magnet ic flux inside the material texture. This flux causes an internal inductance which combines with the e

xternal inductance to get the total inductance. No rmally, the internal inductance can be neglected d

ue to its small value compared to the external one.

22 .Hw NL

II

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Mutual inductanceMutual inductance Mutual inductance M is the inductance that is cau

sed by the flux linking to the different circuit. The mutual inductance between circuit1 and circuit2

can be expressed as2 12

121

1 2121

2

NMI

NMI

where M12 = M21 12 = flux produced by current I1 that is linked to current I2

21 = flux produced by current I2 that is linked to current I1N1 , N2 = number of loops in circuit1 and circuit2 respectively.

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ExEx44 Calculate the inductance for th Calculate the inductance for th e following configuration: e following configuration: a) A coaxial cable with the length l = 10 m, the inner

radius a =1 mm, and the outer radius b =4 mm. The inserted magnetic material has r = 18 and r

= 80.

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b) A toroid with a number of turns N = 5000 t urns wi t h in =3 cm, out =5 cm, and t he l engt h l = 1.5 cm. The i nse

rt ed mat eri al has r =6.

c) A solenoid has the radius r =2 cm, t he l engt h l =8 cm, a nd N = 900 turns. The i nsert ed mat eri al has r = 100.

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ExEx55 Two solenoids with the square cores are placed Two solenoids with the square cores are placed as shown below. The inner dimension is as shown below. The inner dimension is 1.21.2xx1.21.2 cm. cm.

The outer dimension is The outer dimension is33xx33 cm. The solenoid has cm. The solenoid has12001200 turns and the length is turns and the length is 25 25 cm. Given cm. Given rr11 = = 6.256.25 , ,rr22 = = 1,1, determine the following determine the following

a) Gi ven t he current I =1 A, determine H inside th e solenoid when the inner solenoid is removed.

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b) Determine the resulting self inductance.

c) Determine the mutual inductance between two solenoids.