Magnetostatics 3rd 1

Unit-2 Unit-2 MagnetostaticsMagnetostatics

Mr. HIMANSHU DIWAKARAssistant ProfessorDepartment of ECE

If charges are moving with constant velocity, a static magnetic (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.

MagnetostaticsMagnetostatics

4

1k

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

2RsinIdlkdH

2R4sinIdlkdH


Biot – Savart’s Law

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

32 44 RRdlI

RadlIHd r

RR RRar

r

rre

rr

r

32 4)]([

4)(

rrrrdlI

rradlIHd r

Hd


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).


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

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encC

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 sdJISince volume currentdensity is the most

general, we can write Iencl in this way.

S

dldS

Applying Stokes’s Theorem to Ampere’s Law

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Sencl

SC

sdJI

sdHldH

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

JH Differential formof Ampere’s Law

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.

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enclC

IldH known

unknown

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:

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I

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

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HaH ˆ

circles of radius where


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Amperian path

II encl

I

x

y


(4) For each Amperian path, evaluate the integral

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HlldHC

2HldHC

magnitude of Hon Amperian

path.

lengthof Amperian

path.


(5) Solve for B on each Amperian path

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lIH encl

2ˆ

IaH

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

0s

sdB

Law of conservation of magnetic flux

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

JB

B

0

0

0 J

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

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

Force on a Moving Charge

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

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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

One 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.

Idl B

dF Idl B

B

Idl

B

__

uQlId

Magnetic Force on a current element

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

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C

BldIF

Force between two current elements

Experimental facts: Two parallel wires carrying

current in the same direction attract.

Two parallel wires carrying current in the opposite directions repel.

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I1 I2

F12F21

I1 I2

F12F21


Experimental facts: A short current-carrying

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

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I1

F12 = 0I2


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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Magnetostatics 3rd 1

Engineering

Transcript of Magnetostatics 3rd 1