Magnetostatics

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,

α

2

sin

R

IdlkdH

α=

2R4sinIdl

kdHπ

α=π

=41

k

3

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.

)sin( nABaABBA θ=×

32 44 R

RdlI

R

adlIHd r

ππ×=×= RR = R

Rar =

r′

rre ′

rr ′−

r

324

)]([

4

)(

rr

rrdlI

rr

adlIHd r

′−′−×=

′−×=

ππ

Hd

MagnetostaticsMagnetostatics

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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).

MagnetostaticsMagnetostatics

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Types of Current Distributions

• Line current density (current) - occurs for infinitesimally thin filamentary bodies (i.e., wires of negligible diameter).

• Surface current density (current per unit width) - occurs when body is perfectly conducting.

• Volume current density (current per unit cross sectional area) - most general.

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Ampere’s Circuital Law in Integral Form

• Ampere’s Circuit Law states that the line integral of H around a closed path proportional to the total current through the surface bounding the path

enc

C

IldH =⋅∫

Ampere’s Circuital Law in Integral Form (Cont’d)

By convention, dS is taken to be in the

direction defined by the right-hand rule applied

to dl.

∫ ⋅=S

encl sdJI

Since volume currentdensity is the most

general, we can write Iencl in this way.

S

dl

dS

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Applying Stokes’s Theorem to Ampere’s Law

∫

∫∫⋅==

⋅×∇=⋅

S

encl

SC

sdJI

sdHldH

⇒ Because the above must hold for any surface S, we must have

JH =×∇ Differential formof Ampere’s Law

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Ampere’s Law in Differential Form

• Ampere’s law in differential form implies that the B-field is conservative outside of regions where current is flowing.

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Applications of Ampere’s Law

• Ampere’s law in integral form is an integral equation for the unknown magnetic flux density resulting from a given current distribution.

encl

C

IldH =⋅∫known

unknown

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Applications of Ampere’s Law (Cont’d)

• In general, solutions to integral equations must be obtained using numerical techniques.

• However, for certain symmetric current distributions closed form solutions to Ampere’s law can be obtained.

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Applications of Ampere’s Law (Cont’d)

• Closed form solution to Ampere’s law relies on our ability to construct a suitable family of Amperian paths.

• An Amperian path is a closed contour to which the magnetic flux density is tangential and over which equal to a constant value.

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Magnetic Flux Density of an Infinite Line Current Using Ampere’s Law

Consider an infinite line current along the z-axis carrying current in the +z-direction:

I

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Magnetic Flux Density of an Infinite Line Current Using Ampere’s Law (Cont’d)

(1) Assume from symmetry and the right-hand rule the form of the field

(2) Construct a family of Amperian paths

φφ HaH ˆ=

circles of radius ρ where

∞≤≤ ρ0

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Amperian path

IIencl =I

ρx

y

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(4) For each Amperian path, evaluate the integral

HlldHC

=⋅∫

ρπφ 2HldHC

=⋅∫

magnitude of Hon Amperian

path.

lengthof Amperian

path.

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(5) Solve for B on each Amperian path

l

IH encl=

ρπφ 2ˆ

IaH =

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 µ=

µ

7o 104 −⋅π=µ

∫ ⋅=ΨS

,sdB

µ

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0=•∫s

sdB

Law of conservation of magnetic flux

Law of conservation of magnetic flux or Gauss’s law for magnetostatic field

JB

B

0

0

µ=×∇

=•∇

0=•∇ J

There are no magnetic flow sources, and the magnetic flux lines always close upon themselves

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Forces due to magnetic field

There are three ways in which force due to magnetic fields can be experienced

1.Due to moving charge particle in a B field2.On a current element in an external field 3.Between two current elements

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

Force on a Moving Charge

• A moving point charge placed in a magnetic field experiences a force given by

BvQ

The force experienced by the point charge is in the direction into the paper.

B

BuqFm ×=

BuqF

EqF

m

e

×=

= FFF me =+

)( BuEqF ×+=

Lorentz’s Force equation

Note: Magnetic force is zero for q moving in the direction of the magnetic field (sin0=0)

If a point charge is moving in a region where both electric and magnetic fields exist, then it experiences a total force given by

Magnetic Force on a current elementOne can tell if a magneto-static 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.

€

I dl B

€

dF =Idl×B B €

Idl

B

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__

uQlId ⇔

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FORCE ON A CURRENT ELEMENT

• The total force exerted on a circuit C carrying current I that is immersed in a magnetic flux density B is given by

∫ ×=C

BldIF

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• Experimental facts:– Two parallel wires

carrying current in the same direction attract.

– Two parallel wires carrying current in the opposite directions repel.

× ×

I1 I2

F12F21

× •

I1 I2

F12F21

Force between two current elements

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• Experimental facts:– A short current-

carrying wire oriented perpendicular to a long current-carrying wire experiences no force.

×

I1

F12 = 0

I2


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• Experimental facts:– The magnitude of the force is inversely

proportional to the distance squared.– The magnitude of the force is proportional to

the product of the currents carried by the two wires.


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• The force acting on a current element I2 dl2 by a current element I1 dl1 is given by

( )2

12

1122012

12ˆ

4)(

R

aldIldIFdd R××

=π

µ

Permeability of free spaceµ0 = 4π × 10-7 F/m


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• The total force acting on a circuit C2 having a current I2 by a circuit C1 having current I1 is given by

( )∫ ∫

××=

2 1

12

212

1221012

ˆ

4 C C

R

R

aldldIIF

πµ


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• The force on C1 due to C2 is equal in magnitude but opposite in direction to the force on C2 due to C1.

1221 FF −=


EEL 3472EEL 347234

Magnetostatics

Engineering

Transcript of Magnetostatics