IUB Dr. Mustafa H. Chowdhury 1

HF Electromagnetic Waves

Lecture – 12

Spring 2016

Instructor

Dr. Mustafa H. Chowdhury

Curl, Magnetic Flux

&

Maxwell’s Equations

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Curl

Now, our objective is to obtain the point form of Ampere’s

circuital law.

An incremental closed path in Cartesian coordinates is selected

for the application of Ampere’s circuital law in order determine

the spatial rate of change of H as shown in figure.

zzyyxx HHH aaaH 0000

We assume that some current, as

yet unspecified, produces a

reference value for H at the

center of this small rectangle.

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Curl (contd..)

The closed line integral of H about this path is then

approximately the sum of the four values of H • L on each side.

We choose the direction of traverse as 1-2-3-4-1, which

corresponds to a current in the a direction az, and the first

contribution therefore

where Hy,1-2 can be written in terms

of Hy as

yH y 21,21LH

x

x

HHH

y

yy2

1021,

(1)

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Curl (contd..)

Therefore, equation (1) can be written as

Along the next section of the path we have

xyy

HHxH x

xx

2

1032,32 LH

yxx

HH

y

y

2

1021 LH (2)

(3)

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Curl (contd..)

Similarly, for the remaining two segments

Adding, (2), (3), (4), (5), we have

yxx

HH

y

y

2

1043 LH

xyy

HH x

x

2

1014 LH

(4)

(5)

yxy

H

x

Hd xy

LH

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Curl (contd..)

By Ampere’s circuital law, this result must be equal to the current

enclosed by the path, or the current crossing any surface bounded

by the path.

If we assume a general current density Jz, the enclosed current is

then I = Jz x y, and therefore

or

yxJyxy

H

x

Hd z

xy

LH

zxy

Jyxy

H

x

H

yx

d

LH

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Curl (contd..)

If the closed path shrink path tends to zero, the above equation

will be more exact. By taking the limit,

If we choose closed paths which are oriented perpendicularly to

each of the remaining two coordinate axes, analogous processes

lead to expressions for the y and z components of the current

density,

zxy

yxJ

y

H

x

H

yx

d

LH

0,lim

x

yz

zyJ

z

H

y

H

zy

d

LH

0,lim

(6)

(7)

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Curl (contd..)

And

Comparing (6), (7), and (8), we see that a component of the

current density is given by the limit of the quotient of the closed

line integral of H about a small path in a plane normal to that

component and of the area enclosed as the path shrinks to zero.

This limit has its counterpart in other fields of science and long

ago received the name of curl.

The curl of any vector is a vector, and any component of the curl

is given by the limit of the quotient of the closed line integral of

the vector about a small path in a plane normal to that component

desired and the area enclosed, as the path shrinks to zero.

yzx

xzJ

x

H

z

H

xz

d

LH

0,lim (8)

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Curl (contd..)

It should be noted that the above definition of curl does not refer

specifically to a particular coordinate system.

The mathematical form of the definition is

where SN is the planar area enclosed by the closed line integral.

The subscript N indicates that the component of the curl is that

component which is normal to the surface enclosed by the closed

path. It may represent any component in any coordinate system.

N

SN

S

d

N

LHH

0limcurl (9)

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Curl (contd..)

In Cartesian coordinates the definition (9) shows that the x, y, and

z components of the curl of H are given by (6), (7), and (8), and

therefore

This result may be written in the form of a determinant,

zxy

yzx

x

yz

y

H

x

H

x

H

z

H

z

H

y

HaaaHH

curl

zyx

zyx

HHH

zyx

aaa

HHcurl

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Curl (contd..)

Therefore, the curl in different coordinate systems are

Cartesiany

H

x

H

x

H

z

H

z

H

y

Hz

xy

yzx

x

yz aaaH

lCylindrica

y

HHH

z

H

z

HHz

xzz aaaH

11

Spherical

H

r

rH

r

r

rHH

rz

HHH

r

r

rr

z

a

aaH

1

sin

11sin

sin

1

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Curl (contd..)

Now in Cartesian coordinate system

Therefore, combining (6), (7), (8) and (10)

So

This is the point form of Ampere’s circuital law and the 2nd

Maxwell’s equation.

zxy

yzx

x

yz

y

H

x

H

x

H

z

H

z

H

y

HaaaH

(10)

zyx JJJ H

JH

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Maxwell’s 2nd equation

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Curl (contd..)

According to Kirchhoff’s voltage law

Using Stroke’s theorem

or

Finally,

0 LE d

0 E Maxwell’s 3rd equation

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SELE ddS

0SE dS

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Magnetic flux density

In free space, the magnetic flux density is defined as

where 0 is the permeability of free space and is given by

The unit of B in mks unit is Wb/m2 or in SI unit is Tesla (T).

HB 0

H/m104 7

0

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Magnetic flux

Magnetic flux is defined as the flux passing through any

designated area,

Our analogy should now remind us of the electric flux E,

measured coulombs, and of Gauss's law, which states that the

total flux passing through any closed surface is equal to the

charge enclosed,

The charge Q is the source of the lines of electric flux and these

lines begin and terminate on positive and negative charge,

respectively.

S

dSB

QdS

SD

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Magnetic flux (contd..)

But, no such source has ever been discovered for the lines of

magnetic flux. In the example of the infinitely long straight

filament carrying a direct current I, the H field formed concentric

circles about the filament. Since B = µ0H, the B field is of the

same form. The magnetic flux lines are closed and do not

terminate on “magnetic charge”. For this reason Gauss's, law for

the magnetic field is

Using the divergence theorem

0S dSB

volS

dvdS BB

0 B Maxwell’s 4th equation

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Maxwell’s equations

The point form of Maxwell’s equations for static electric and

magnetic fields are

And the corresponding 4 sets of Maxwell’s equations in integral

form

0

0

B

JH

E

D v

SJLH

SB

LE

SD

dId

d

d

dvQd

S

S

vvolS

0

0

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Example 1

Find the magnetic flux contained between the conductors in a

length d is the flux crossing any radial plane extending from =

a to = b and from z = 0 to z = d.

Solution :

aHB2

00

I

a

bId

dzdI

db

a

d

S

ln2

2

0

0

0

aaSB

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Assignment-10

A solid nonmagnetic conductor of circular cross section has its

axis on the z axis and carries a uniformly distributed total current

of 60 A in the z direction. If the radius is 4 mm, find: (a) B at =

5 mm; (b) B at = 3 mm; (c) the total magnetic flux per unit

length inside the conductor; (d) the total flux per unit length

outside the conductor.

Ans : 2.40 mT; 2.25 mT; 6 µWb;

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Stroke’s theorem

Consider the surface S as shown in Fig. Below, which is broken

up into incremental surface of area S. The sum of the closed

line integrals about the perimeter of every S is the same as the

closed line integral about the perimeter of S.

If we apply the

definition of the curl to

one of these incremental

surfaces, then

NN

S

S

daHH

LH

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Therefore,

where aN is a unit vector in the direction of the right-hand normal

to S.

Finally, we have

This equation is known as the Stroke’s theorem.

Stroke’s theorem (contd..)

SHaHLH Sd NS

SHLH ddS

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An alternating way of deriving the Stroke’s theorem is

Therefore,

Stroke’s theorem (contd..)

SH

SJ

LH

d

d

Id

S

S

SHLH ddS

SJ dIS

JH

since

since

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Next class

Next Class

On

Magnetic

Materials

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Spring 2016