Lecture 9

Vector Magnetic Potential

Biot Savart Law

Prof. Viviana Vladutescu

Figure 1: The magnetic (H-field) streamlines inside and outside a

single thick wire.

Figure 2: The H-field magnitude inside and outside the thick wire

with uniform current density

Figure 3: The H-field magnitude inside and outside the thick

conductors of a coaxial line.

0

0

A

B)( TAB

A - vector magnetic potential (Wb/m)

Vector Magnetic Potential

Figure 1: The vector potential in the cross-section of a wire with

uniform current distribution.

Figure 2: Comparison between the magnetic vector potential component  of a wire with uniformly distributed current and the

electric potential V of the equivalent cylinder with uniformly

distributed charge.

JAA

AAAAA

JA

02

2

0

)(

)()()(

JAA 020

Vector Poisson’s equation

Laplacian Operator (Divergence of a gradient)

Poisson’s Equation

In electrostatics

ED

VE

E

D

0

V

EE

V2 Poisson’s Equationin electrostatics

4

4

1

00

2

00

2

dvR

JAJA

dvR

VV

v

v

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

Biot Savart Law and Applications

The Biot-Savart Law relates magnetic fields to the currents which are their sources. In a similar manner, Coulomb’s Law relates electric fields to the point charges which are their sources. Finding the magnetic field resulting from a current distribution involves the vector product, and is inherently a calculus problem when the distance from the current to the field point is continuously changing.

)( TAB

4

0 c R

ldIA

4

0

c R

ldIB

GfGfGf

11

40

c

ldR

ldR

IB

2

11

Ra

R R

By using

(T) 4 2

0

c

R

R

aldIB

(see eq 6.31)

Biot-Savart Law

20

4

R

aldIBd

BdB

R

c

In two steps

Illustration of the law of Biot–Savart showing magnetic field arising from a differential segment of current.

212

12112

4 R

aLdIHd

rzR arazaR

Example1Component 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

23222

322

Example 2We want to find H at height h above a ring of current centered in the x – y plane.

The component values shown for use in the Biot–Savart equation.

2

0 2322 )(4

)(

ah

aaahaIadH rz

The radial components of H cancel by symmetry.

23

22

2

2

023

22

2

2

4

ah

aIaH

dah

aIaH

z

z

Solenoid

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

For a set N number of loops around a ferrite core, the flux generated is the same even when the loops are bunched together.

Example : A simple toroid wrapped with N turns modeled by a magnetic circuit. Determine B inside the closely wound toroidal coil.

b

a

)()(,2

2

0

0

abrabr

NIaaBB

NIrBldB

Ampere’s Law

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

Electromagnets

Levitated trains: Maglev prototype

Electromagnet supporting a bar of mass m.

Applications

Wilhelm Weber (1804-1891). Electromagnetism.