8/14/2019 Lecture9 (2).ppt

1/29

Lecture 9

Vector Magnetic Potential

Biot Savart Law

Prof. Viviana Vladutescu

8/14/2019 Lecture9 (2).ppt

2/29

Figure 1: The magnetic (H-field)

streamlines inside and outside a

single thick wire.

8/14/2019 Lecture9 (2).ppt

3/29

Figure 2: The H-field magnitude

inside and outside the thick wire

with uniform current density

8/14/2019 Lecture9 (2).ppt

4/29

Figure 3: The H-field magnitude

inside and outside the thick

conductors of a coaxial line.

8/14/2019 Lecture9 (2).ppt

5/29

8/14/2019 Lecture9 (2).ppt

6/29

Figure 1: The vector potential in

the cross-section of a wire with

uniform current distribution.

8/14/2019 Lecture9 (2).ppt

7/29

Figure 2: Comparison between the magnetic vector potential

component of a wire with uniformly distributed current and the

electric potential Vof the equivalent cylinder with uniformly

distributed charge.

8/14/2019 Lecture9 (2).ppt

8/29

8/14/2019 Lecture9 (2).ppt

9/29

JAA

AAAAA

JA

0

2

20

)(

)()()(

JAA 020 Vector Poissons equation

Laplacian Operator (Divergence of a gradient)

Poissons Equation

8/14/2019 Lecture9 (2).ppt

10/29

In electrostatics

ED

VE

E

D

0

V

EE

V2 Poissons Equationin electrostatics

8/14/2019 Lecture9 (2).ppt

11/29

4

4

1

00

2

00

2

dv

R

JAJA

dvR

VV

v

v

8/14/2019 Lecture9 (2).ppt

12/29

Magnetic Flux

(Wb))(

cs

s

ldAdsA

dsB

The line integral of the vector magnetic potential A around

any closed path equals the total magnetic flux passing

through area enclosed by the path

8/14/2019 Lecture9 (2).ppt

13/29

8/14/2019 Lecture9 (2).ppt

14/29

The Biot-Savart Law relates magnetic fields to the currents

which are their sources. In a similar manner, CoulombsLaw

relates electric fields to the point charges which are their

sources. Finding the magnetic field resulting from a currentdistribution involves the vector product, and is inherently a

calculus problem when the distance from the current to the

field point is continuously changing.

8/14/2019 Lecture9 (2).ppt

15/29

)( TAB

40

c RldIA

4

0

c R

ldIB

GfGfGf

8/14/2019 Lecture9 (2).ppt

16/29

1140

c

ldR

ldR

IB

211R

aR

R

By using

(T)4 20

c

R

RaldIB

(see eq 6.31)

Biot-Savart Law

8/14/2019 Lecture9 (2).ppt

17/29

2

0

4

R

aldIBd

BdB

R

c

In two steps

8/14/2019 Lecture9 (2).ppt

18/29

Illustration of the law of BiotSavart showing

magnetic field arising from a differential segment of

current.

2

12

12112

4 R

aLdIHd

8/14/2019 Lecture9 (2).ppt

19/29

rzR arazaR

Example1

Component values for the equation to find the

magnetic field intensity resulting from an infinite

length line of current on the z-axis. (ex 6-4)

r

aIH

rzr

zaIr

rz

dzaIr

rz

arazaIdzH rzz

24

)(4)(4

)(

222

23

2223

22

8/14/2019 Lecture9 (2).ppt

20/29

Example 2

We want to find Hat height habove

a ring of current centered in the xyplane.

8/14/2019 Lecture9 (2).ppt

21/29

The component values shown for use in the BiotSavart

equation.

2

02

322 )(4

)(

ah

aaahaIad

H rz

8/14/2019 Lecture9 (2).ppt

22/29

The radial components of Hcancel

by symmetry.

2

322

2

2

02

322

2

2

4

ah

aIaH

dah

aIaH

z

z

8/14/2019 Lecture9 (2).ppt

23/29

Solenoid

Many turns of insulated wire coiled in the shape of a cylinder.

8/14/2019 Lecture9 (2).ppt

24/29

For a set N number of loops around a ferrite

core, the flux generated is the same even when

the loops are bunched together.

8/14/2019 Lecture9 (2).ppt

25/29

Example : A simple toroid wrapped with N turns modeled

by a magnetic circuit. Determine B inside the closely wound

toroidal coil.

b

a

8/14/2019 Lecture9 (2).ppt

26/29

)()(,2

2

0

0

abrabr

NIaaBB

NIrBldB

Amperes Law

8/14/2019 Lecture9 (2).ppt

27/29

a) An iron bar attached to an electromagnet.b) The bar displaced by a differential length d.

Electromagnets

8/14/2019 Lecture9 (2).ppt

28/29

Levitated trains: Maglev prototype

Electromagnet supporting a

bar of mass m.

Applications

8/14/2019 Lecture9 (2).ppt

29/29

Wilhelm Weber (1804-1891). Electromagnetism.