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Magnetostatics

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If charges are moving with constant velocity, a staticmagnetic (or magnetostatic) field is produced. Thus, magnetostatic fields originate from currents (for instance, direct currents in current-carrying wires).

Most of the equations we have derived for the electric fields may be readily used to obtain corresponding equations for magnetic fields if the equivalent analogous quantities are substituted.

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Biot – Savart’s Law The magnetic field intensity dH produced at a point P by the differential current element Idl is proportional to the product of Idl and the sine of the angle between the element and the line joining P to the element and is inversely proportional to the square of the distance R between P and the element.

IN SI units, , so

2RsinIdl

kdH

2R4sinIdl

kdH

41

k

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Using the definition of cross product we can represent the previous equation in vector form as

The direction of can be determined by the right-hand rule or by the right-handed screw rule.

If the position of the field point is specified by and the position of the source point by

where is the unit vector directed from the source point tothe field point, and is the distance between these twopoints.

)esinABBA( nAB

32r

R4RdlI

R4edlI

Hd

RR RRer

r

rre

rr

r

dH I(dl e r r )

4 r r 2

I[dl (r r )]

4 r r 3

Hd

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Example. A current element is located at x=2 cm,y=0, and z=0. The current has a magnitude of 150 mA and flows in the +y direction. The length of the current element is 1mm. Find the contribution of this element to the magnetic field at x=0, y=3cm, z=0

r 310 2e y x2e10*2r

r r 13 10 2

(r r ) ( 2e x 3e y ) 10 2

dH I[dl (r r )]

4 r r 3

0.15(210-5)e z4( 13 10 2)3

5.0910-3e z A m

dl (r r ) 10 3e y ( 2e x 3e y ) 10 2

10-5(2e z)

e y e x e z e y e y 0

y3e10dl

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The field produced by a wire can be found adding up the fields produced by a large number of current elements placed head to tail along the wire (the principle of superposition).

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Consider an infinitely long straight wire located on the z axisand carrying a current I in the +z direction. Let a field point be located at z=0 at a radial distance R from the wire.

)ZR(4

esinIdz

rr4

)ee(Idz

rr4

)edl(IHd 22

)ZR(

R

2rrz

2rr

2/122

2/122 )ZR(rr

R2I

)11(R4

I

222

z2/3222/12222

ZRR

Z4IR

dz)ZR(

1)ZR(

R)ZR(4

IdzH

Magnetostatics

z 0

ze

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constbarr 22

addl

22

0

ba

a)90sin(cos

H Iad4 (a2 b2)

a

(a2 b2)1/ 20

2

Ia2

4 (a2 b2)3 / 2 2 Ia2

2(a2 b2)3 / 2

H H e z

The horizontal components of (x-components and y-components add to zero

H

r're

dl(perpendicular to both

and )

dH I(dl e r'r )

4 r r 2

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Magnetic Flux Density

The magnetic flux density vector is related to the magnetic field intensity by the following equation

T (tesla) or Wb/m2

where is the permeability of the medium. Except for ferromagnetic materials ( such as cobalt, nickel, and iron),most materials have values of very nearly equal to that for vacuum,

H/m (henry per meter)The magnetic flux through a given surface S is given by

Wb (weber)

H

,HB

7o 104

S

,sdB

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Magnetic ForceOne can tell if a magnetostatic field is present because it exerts forces on moving charges and on currents. If a currentelement is located in a magnetic field , it experiences a force

This force, acting on current element , is in a direction perpendicular to the current element and also perpendicularto . It is largest when and the wire are perpendicular.

The force acting on a charge q moving through a magnetic field with velocity is given by

The force is perpendicular to the direction of charge motion, and is also perpendicular to .

Idl B

dF Idl B

F q(v B )

B

Idl

B

v

B

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The Curl OperatorThis operator acts on a vector field to produce another vectorfield. Let be a vector field. Then the expression for the curl of in rectangular coordinates is

where , , and are the rectangular components of

The curl operator can also be written in the form of a determinant:

)z,y,x(B

zxy

yzx

xyz e

yB

x

Be

xB

zB

ez

B

yB

B

B

xB yB zB

zyx

zyx

BBB

zyx

eee

B

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The physical significance of the curl operator is that it describes the “rotation” or “vorticity” of the field at the pointin question. It may be regarded as a measure of how much the field curls around that point.

The curl of is defined as an axial ( or rotational) vectorwhose magnitude is the maximum circulation of per unitarea as the area tends to zero and whose direction is thenormal direction of the area when the area is oriented so asto make the circulation maximum.

where the area is bounded by the curve L, and is the unit vector normal to the surface and is determined using the right – hand rule.

n

max

L

0sa

s

dlBlimBBrotBcurl

BB

snas

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zxy

yzx

xyz e

yB

x

Be

xB

zB

ez

B

yB

B

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Classification of Vector FieldsA vector field is uniquely characterized by its divergence and curl (Helmholtz’s theorem). The divergence of a vector field is a measure of the strength of its flow source and the curl of the field is a measure of the strength of its vortex source.

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If then is said to be solenoidal or divergenceless. Such a field has neither source nor sink offlux.

Since (for any ), a solenoidal field can always be expressed in terms of another vector :

If then is said to be irrotational (or potential, or conservative). The circulation of around a closed path is identically zero.

Since (for any scalar V), an irrotational field can always be expressed in terms of a scalar field V:

0A A

0)F( F

0A

FA

0V

VA

AA

A

A

F

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Magnetic Vector PotentialSome electrostatic field problems can be simplified by relating the electric potential V to the electric field intensity

. Similarly, we can define a potential associated with

the magnetostatic field :

where is the magnetic vector potential.

Just as we defined in electrostatics (electric scalar

potential)

we can define

(for line current)

)VE(E

AB

'rr4)'r(dq

)r(V

L 'rr4

'dl)'r(I)r(A

B

A

mWb

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The contribution to of each differential current element points in the same direction as the current element thatproduces it.The use of provides a powerful, elegant approach tosolving EM problems (it is more convenient to find by firstfinding in antenna problems).

The Magnetostatic Curl EquationBasic equation of magnetostatics, that allows one to find thecurrent when the magnetic field is known is

where is the magnetic field intensity, and is the currentdensity (current per unit area passing through a planeperpendicular to the flow).The magnetostatic curl equation is analogous to the electricfield source equation

A 'dlI

B

JH

D

A

A

Magnetostatics

H

J

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Stokes’ Theorem

This theorem will be used to derive Ampere’s circuital law which is similar to Gauss’s law in electrostatics.According to Stokes’ theorem, the circulation of around a closed path L is equal to the surface integral of the curl of over the open surface S bounded by L.

The direction of integration around L is related to the direction of by the right-hand rule.

AA

ds

A dl A dsS

Magnetostatics

L

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Stokes’ theorem converts a surface integral of the curl of a vector to a line integral of the vector, and vice versa.(The divergence theorem relates a volume integral of the divergence of a vector to a surface integral of the vector, and vice versa).

Sv

dsAdvA

Determining the sense of dl and dS involved in Stokes’s theorem

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Magnetostatics

S

dsAdlA

1

Surface S2 (circular)

22

L

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Ampere’s Circuital Law

Choosing any surface S bounded by the border line L andapplying Stokes’ theorem to the magnetic field intensity vector , we have

Substituting the magnetostatic curl equation

we obtain

which is Ampere’s Circuital Law. It states that the circulation of around a closed path is equal to the current enclosed by the path.

S

L

dlHsdHH

JH

L

I

S

dlHsdJ

enc

H

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Ampere’s law is very useful in determining when there is a closed path L around the current I such that the magnitude of is constant over the path.H

H

Magnetostatics

(Amperian Path)

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Boundary conditions are the rules that relate fields on opposite sides of a boundary.

T= tangential components

N= normal components

We will make use of Gauss’s law for magnetic fields

and Ampere’s circuital law

BN1

BN 20dsB

1TH 2THencIdlH

for and

for and

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In the limit as

we have

where is the current on the boundary surface

(since the integration path in the limit is infinitely narrow).

When the conductivities of both media are finite,

Boundary condition on the tangential component of

0L2

0L4

H dlL

HT1l HT 2l Ienc

encI

2T1T HH

H

The tangential component of is continuous across the boundary, while that of is discontinuous

H

B

0I enc and

22T11T /B/B

Magnetostatics

HT1

HT 2

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Gaussian surface (cylinder with its plane faces parallel to the boundary)

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Applying the divergence theorem to and noting that(magnetic field is solenoidal) we have

In the limit as , the surface integral over the curved surface of the cylinder vanishes. Thus we have

(the normal component of is continuous at the boundary)

(the normal component of is discontinuous at the boundary)

B 0B

VS

0BdsB

h 0

BN 2A BN1A 0

BN1 BN 2

1HN1 2HN 2

B H

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Chapter 9

Magnetic Forces, Materials and Inductance

The magnetic field B is defined from the Lorentz Force Law, and specifically from the magnetic force on a moving charge:

F = qv x B

1. The force is perpendicular to both the velocity v of the charge q and the magnetic field B.

2. The magnitude of the force is F = qvB sin where is the angle < 180 degrees between the velocity and the magnetic field. This implies that the magnetic force on a stationary charge or a charge moving parallel to the magnetic field is zero.

3. The direction of the force is given by the right hand rule. The force relationship above is in the form of a vector product.

From the force relationship above it can be deduced that the units of magnetic field are Newton seconds /(Coulomb meter) or Newton per Ampere meter. This unit is named the Tesla. It is a large unit, and the smaller unit Gauss is used for small fields like the Earth's magnetic field. A Tesla is 10,000 Gauss. The Earth's magnetic field is on the order of half a Gauss.

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Force On A Moving Charge

Lorentz Force Law

Both the electric field and magnetic field can be defined from the Lorentz force law:

The electric force is straightforward, being in the direction of the electric field if the charge q is positive, but the direction of the magnetic part of the force is given by the right hand rule.

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Force On A Moving Charge

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Force on a Differential Current

dF = dQv x B

J v v

dF J Bdv dQ v dv

dF v dv v B

dF J Bdv

Jdv KdS IdL

dF K BdS

dF IdL B

F vJ B

dvol

F SK B

dS

F LI

d B I LB_x_

d

F IL B

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Example 9.1

az axFy1 2 10

3 3 106

1

3

x1

x

d

Fy1 6.592 109

Fx1 2 103 3 10

6

0

2

y1

3

d

1 az ay Fx1 4 109

Fy2 2 103 3 10

6

3

1

x1

x

d

az ax Fy2 6.592 109

az ayFx2 2 10

3 3 106

2

0

y1

d

1 Fx2 1.2 108

F Fx1 Fx2 Fy1 Fy2 F 8 109

The net force on the loop is in the -ax direction

HI

2 xaz

B o H

F I LBx

d

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Force Between Differential Current ElementsExample 9.2

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D9.4

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Force And Torque On A Closed Circuit

F I LB_x_

d

F IB L1

d

T R F

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Force And Torque On A Closed Circuit

dT IdS B

Magnetic Dipole Moment dm

dm IdS

dT dm B

T IS B m B

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Force And Torque On A Closed Circuit

Build a DC Motor – Hands On

Explain Operation

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Force And Torque On A Closed Circuit

DC Motor - Illustration

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Example 9.3 and 9.4

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The Nature of Magnetic Materials

Magnetic MaterialsMagnetic Materials may be classified as diamagnetic, paramagnetic, or

ferromagnetic on the basis of their susceptibilities. Diamagnetic materials, such as bismuth, when placed in an external magnetic field, partly expel the external field from within themselves and, if shaped like a rod, line up at right angles to a non-uniform magnetic field. Diamagnetic materials are characterized by constant, small negative susceptibilities, only slightly affected by changes in temperature.

Paramagnetic materials, such as platinum, increase a magnetic field in which they are placed because their atoms have small magnetic dipole moments that partly line up with the external field. Paramagnetic materials have constant, small positive susceptibilities, less than 1/1,000 at room temperature, which means that the enhancement of the magnetic field caused by the alignment of magnetic dipoles is relatively small compared with the applied field. Paramagnetic susceptibility is inversely proportional to the value of the absolute temperature. Temperature increases cause greater thermal vibration of atoms, which interferes with alignment of magnetic dipoles.

Ferromagnetic materials, such as iron and cobalt, do not have constant susceptibilities; the magnetization is not usually proportional to the applied field strength. Measured ferromagnetic susceptibilities have relatively large positive values, sometimes in excess of 1,000. Thus, within ferromagnetic materials, the magnetization may be more than 1,000 times larger than the external magnetizing field, because such materials are composed of highly magnetized clusters of atomic magnets (ferromagnetic domains) that are more easily lined up by the external field.

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Magnetization and Permeability

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Magnetization and Permeability

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Example 9.5

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Magnetic Boundary Conditions

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The Magnetic Circuit

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The Magnetic Circuit

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The Magnetic Circuit

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Inductance and Mutual Inductance

A self-induced electromotive force opposes the change that brings it about. Consequently, when a current begins to flow through a coil of wire, it undergoes an opposition to its flow in addition to the resistance of the metal wire. On the other hand, when an electric circuit carrying a steady current and containing a coil is suddenly opened, the collapsing, and hence diminishing, magnetic field causes an induced electromotive force that tends to maintain the current and the magnetic field and may cause a spark between the contacts of the switch. The self-inductance of a coil, or simply its inductance, may thus be thought of as electromagnetic inertia, a property that opposes changes both in currents and in magnetic fields.