18/10/58 1 ENE 325 Electromagnetic Fields and Waves Lecture 7 Amp é re ’ s circuital law.

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ENE 325ENE 325Electromagnetic Electromagnetic Fields and WavesFields and Waves

Lecture 7Lecture 7 Amp Ampéérere’’s circuital laws circuital law

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Review (1)Review (1)

Capacitance Static magnetic field (time invariant) Source of magnetic field

permanent magnet electric field changing linearly with time direct current

Bio-Savart law is a method to determine the magnetic field intensity. It is an analogy to Coulomb’s law of Electrostatics.

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Magnetic field intensity from an infinite length line of current ( -line is located on z axis)

Magnetic field intensity from a ring of current (a ri - ng is located on x y plane and the observation point i

-s on z axis.

2 34 4rId L a Id L R

dHr r

��

��

2

I aH

��

2

3/ 22 22z

IaH a

h a

��

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Magnetic field intensity from a rectangular loop of current (a wire loop is located on x-y plane and the observation point is at the origin.

Right hand rule

2 2z

IH a

L

��

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OutlineOutline

Ampé re’s circuital law Curl and point form of Ampé re’s circuital law Magnetic flux density

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AmpAmpéé re’s circuital law re’s circuital law

Analogy to Gauss’s law Use for magnetostatic’s problems with sufficient symme

try. Ampere’s circuital law – the integration of around any c

losed path is equal to the net current enclosed by that p ath.

To find , choose the proper Amperian path that is every where either tangential or normal to and over which is

constant.

encH dL I��

A

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Use Ampere’s circuital law to d Use Ampere’s circuital law to d etermine etermine

from the infinite line of current from the infinite line of current . .H��

From encH dL I��

H H a��

dL d a ��

then 2

0.

�� encH dL H a d a I

�� IH a

2A/m.

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Magnetic field of Magnetic field of the the uniforunifor m sheet of current m sheet of current (1)(1)

xK Ka

��

Create path a-b-c-d and perform the integration along the path.

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Magnetic field of Magnetic field of the the uniform s uniform s heet of current heet of current (2)(2)


( ) ( ) ( ) ( ) ,y z y z xH w H h H w H h K w

divide the sheet into small line segments along x-axis, by symmetry Hz is cancelled.

y x2H K .

Because of the symmetry, the magnetic field intensity on one side of the current sheet is the negative of that on the other.

.xyK

H2

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where is a unit vector normal to the current sheet.

Magnetic field of Magnetic field of the the uniform s uniform s heet of current heet of current (3)(3)

.

Above the sheet,

y1 x1

H K2

(z > 0)

and y2 x1

H K2

(z < 0)

or we can write A/m n

1H K a

2

��

na

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Magnetic field inside the sol Magnetic field inside the sol enoid enoid

a

b c

d

x

y z

h

w


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Magnetic field inside the Magnetic field inside the toroid that has a circular cross toroid that has a circular cross section (1)section (1)

consisting of a circular ring-shaped magnetic core of iron powder, ferrite, or other material around which wire is coiled to make an inductor.

The magnetic flux in a toroid is largely confined to the core, preventing its energy from being absorbed by nearby objects, making toroidal cores essentially self-shielding.

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Magnetic field inside the Magnetic field inside the toroid that has a circular cross toroid that has a circular cross section (2)section (2) From encH dL I

��

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Ex1Ex1 Determine Determine at point P at point P (0.01, 0, 0) m from two current (0.01, 0, 0) m from two current filaments as shown. filaments as shown.

H��

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2Ex2Ex Determine for the coaxial c Determine for the coaxial c able that has a inner radius able that has a inner radius a a = 3 m = 3 m

m, m, bb 9= mm, and 9= mm, and cc 12= mm. Giv 12= mm. Giv en en II AA08 AA08

H��

a) at < a

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b) at a < < b

c) at b < < c

d) at > c

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33 Determine at point (10, 0, 0) Determine at point (10, 0, 0) mm resulted from three current sh mm resulted from three current sh

eets: eets: KK11

15 15 A/m at A/m at x x A AAA 6 A AAA 6 KK22

A A -3-3 A/m at A/m at xx A A A A AAA 9 A A A A AAA 9 KK

33 A A1 A A1

55 A/m at A/m at xx AAA12 AAA12

H��

ya

yaya

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Curl and the point form of Curl and the point form ofAmpAmpéé re’s circuital law re’s circuital law (1)(1) ‘Curl ’ is employed to find the point form Ampèr

e’s circuital law, analogous to ‘Divergence’ to f ind the point form of Gauss’s law.

Curl of or is the maximum circulation of per unit area as the area shrinks to zero.

It gives a measure of the degree to which a field curls around a particular point.

limS 0

H dLH J

S

��

H��

H��

H��

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Curl and the point form of Curl and the point form ofAmpAmpéé re’s circuital law re’s circuital law (2)(2) ‘‘Curl´Curl´ operator perform a derivative of vec operator perform a derivative of vec

tor and returns a vector quantity tor and returns a vector quantity. . For Cart For Cart esian coordinates, can be esian coordinates, can be written aswritten as

. ��

x y z

x y z

a a a

H x y z

H H H

H��

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Physical view of curlPhysical view of curl

a) Field lines indicating divergence A simple way to see the b) Field lines indicating curl direction of curl using

right hand rule

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Stokes’s TheoremStokes’s Theorem Stokes’s Theorem Stokes’s Theorem relates a closed line inte relates a closed line inte

gral into a surface integral gral into a surface integral

SdHLdH

)(

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Magnetic flux density, BMagnetic flux density, B

Magnetic flux density is related to the magnetic field intensity in the free space by

Magnetic flux (units of Webers) passing through a surface is found by

B��

H��

0B H��

Weber/m2 or Tesla (T)

1 Tesla = 10,000 Gauss. The earth’s B is about 0.5 G.where 0 is the free space permeability, given in units of

henrys per meter, or 0 = 410-7 H/m.

dSB

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A fundamental feature of magnetic fields that distinguishes them from electric fields is that the field lines form closed loops.

Gaussian Surface

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N

S

N

S

N

S

N

S

N

S

N

S

You cannot saw magnet in half to isolate the north and the south poles; if you saw magnet in half you get two magnets. Hence you cannot isolate a magnetic pole.

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Gauss’s law for magnetic Gauss’s law for magnetic fieldsfields

or

The net magnetic flux passing through a Gaussian surface (a closed surface) must be zero.This is also referred to as the law of conservation of magnetic flux.

0dSB

0 B

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EX4EX4 A solid conductor of circular cross A solid conductor of circular cross section is made of a homogeneous section is made of a homogeneous nonmagnetic material. If the radius nonmagnetic material. If the radius aa = = 1 mm, the conductor axis lies on the 1 mm, the conductor axis lies on the zz axis, and the total current in the axis, and the total current in the direction is 20 A, finddirection is 20 A, finda) a) HH at at = 0.5 mm = 0.5 mm

b) b) BB at at = 0.8 mm = 0.8 mm

c) The total magnetic flux per unit length inside the c) The total magnetic flux per unit length inside the conductorconductor

za

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Maxwell’s equations for Maxwell’s equations for statstaticic fields fields

Integral form Differential form

��

��

��

��

enc

enc

D dS Q

B dS 0

E dL 0

H dL I

vD

B 0

E 0

H J

��

��

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A parallel plate capacitor with a 1.0 A parallel plate capacitor with a 1.0 mm22 surface area for each plate, a 2.0 surface area for each plate, a 2.0 mm plate separation, and a dielectric mm plate separation, and a dielectric with relative permittivity of 1200 has with relative permittivity of 1200 has a 12. V potential difference across the a 12. V potential difference across the plates. Calculate (a) the capacitance, plates. Calculate (a) the capacitance, and (b) the magnitude of the charge and (b) the magnitude of the charge density on one of the plates.density on one of the plates.

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18/10/58 1 ENE 325 Electromagnetic Fields and Waves Lecture 7 Amp é re ’ s circuital law.

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Transcript of 18/10/58 1 ENE 325 Electromagnetic Fields and Waves Lecture 7 Amp é re ’ s circuital law.