MAGNETOSTATIC FIELD

Dr. Sikder Sunbeam Islam

Associate Professor

Dept. of EEE. IIUC.

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INTRODUCTION

Static Electric field Characterized by E or D.

Similarly, static Magnetic field Characterized

by H or B.

For liner material space according to Electric

filed D= E; while for magnetic field B= H (1).

Like D, B is magnetic flux density and like E,

H is the magnetic flux intensity. Here is the

free space permeability and has the value

The table below shows the analogies between

electric and magnetic field quantities.

0

0

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INTRODUCTION

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MAGNETIC FLUX DENSITY

There are two major laws governing magneto static fields: 1) Biot-Savart’s

Law and 2) Ampere’s Circuit law.

Like Coulomb’s Law, Biot-Savart’s Law is a general law of magneto static.

Likewise, Gauss law is special case of Coulomb’s Law, Ampere’s Circuit

law is special case of Biot-Savart’s Law .

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BIOT-SAVART'S LAW Assume current I following in a differential vector length of

filament dl. Biot-Savart’s Law states that, magnetic field

intensity dH produced at point ‘P’ (as shown in Fig.1) by

differential current element, is proportional to the product of the

current I and differential vector length dl (i.e. Idl) and the sign 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. Here, the proportionality

constant will be 1/4 .

Fig.1

Or,

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AMPERE’S CIRCUIT LAW :MAXWELL'S EQUATION

Ampere’s circuital law states that, the line integral of the

tangential component of H around a closed path is the

same as the net current enclosed by the path.

In other words the circulation of H equals , that is

Ampere’s circuital law is similar to Gauss’s law and it is

easy applied to determine H when the current distribution

is symmetrical. Now applying Stoke’s theorem to the left

side of equ.(2),

encI

encI

------------------------

------------------------- (3)

But ------------------------- (4) 7

MAXWELL'S EQUATION

Now comparing equ.(3) and (4),

This is the third Maxwell’s equation to be derived. Now,

We may now say, curl as circulation per unit area.

Circulation of H per unit area is the current density.

Therefore, magneto-static field is not conservative.

Since divergence of the curl of any vector field is zero. So,

------------------------- (5)

where, Hence we showed earlier,

------------------------- (6)

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AMPERE’S CIRCUIT LAW (FOR MAGNETIC FLUX)

------------------------(2.1)

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IS IT POSSIBLE TO ISOLATE MAGNETIC FIELD?

Therefore,

isolate

magnetic poles

(north or

south)

Is not possible .

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IS IT POSSIBLE TO ISOLATE MAGNETIC FIELD?

So, an isolated magnetic charge does not exist.

Thus total flux through a closed surface in a magnetic field must be

zero.

Now applying Divergence theorem in (7),

------------------------- (7)

So, ------------------------- (8)

This is the 4th Maxwell’s equation to be derived. Equation (7) and (8)

reveals that magneto static fields have no source or sink. This (8)

shows that magnetic field lines are always continuous.

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CHANGING MAGNETIC FIELD AND FARADAY’S LAW

Electrostatic field are usually produced by static electric

charges.

Magneto-static field are produced by the motion of electric

charges with uniform velocity (direct current) or static

magnetic charges (magnetic poles).

Time varying fields (radiation) or waves are usually due to

accelerated charges or time varying currents. Time varying EM

fields are represented by E (x,y,z,t) and H (x,y,z,t).

Any pulsating current will produce radiation (Time varying

fields )

Fig.4. Time varying current (a)sinusoidal and (b) rectangular

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FARADAY’S LAW

Steady electric current produces steady magnetic field .

But steady magnetic field will not produce an electrical current.

However, changing magnetic field will.

Thus, changing magnetic flux (time varying field) ψ through a

closed loop (circuit) produces an EMF (electromotive force) or

voltage Vemf at the terminals which was discovered by Faraday.

So, according to Faraday, induced EMF (in volt) in any closed

circuit is equal to the time rate of change of magnetic flux linkage by

the circuit. This relation is called Faraday’s Law. Therefore,

------------------------- (9)

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TRANSFORMER & MOTIONAL EMF’S

In the Equ.(9) the voltage is the integral of electric filed E

around the loop. Now for N=1,

Now in terms of E and B from the above equation we find,

Here both electric and magnetic fields are present and are

interrelated. The variation flux in time may caused in 3-ways:

------------------------- (9)

------------------------- (10)

Total magnetic flux,

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A. STATIONARY LOOP IN TIME VARYING B FILED

(TRANSFORMER EMF)

This induces EMF (in equ.11) is due to time rate of change of B (flux

density) in a stationary (fixed) loop is often referred to as

Transformer EMF in power analysis since it is due to transformer

action. The equation is called the Transformer induction

equation. Now applying Stoke’s theorem in the middle term of

equation (10),

------------------------- (11)

------------ (10)

------------ (12)

Therefore, ----------------------- (13)

This is one of the Maxwell’s equation for time varying field.

Fig.5

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B. MOVING LOOP IN STATIC B-FIELD (MOTIONAL EMF)

When a conducting loop is moving in a static B-field an EMF is

induced in the loop (conductor). We know, the force on a charge

moving with uniform velocity u, in a magnetic field B is,

Moving loop

Static B field

Fm= Qu×B ----------------------- (14)

We define, the motional electric field Em as,

----------------------- (15)

If, a conducting loop moving with a uniform velocity u as consisting of

large number of free electrons, the induced EMF in the loop is,

------------------- (16)

This EMF is called motional EMF.

Fig.6

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MOTIONAL EMF (CONTINUES).

Applying Stoke’s theorem in equ.(16),

------------------ (17)

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C. MOVING LOOP IN A TIME VARYING FILED

In this case the moving loop in a time varying field.

Both transformer induction and motional

induction is present.

Therefore, combining equation (11) and (16) gives the

total EMF as,

------------------- (18)

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LORENTZ FORCE (MOTOR EQUATION)

A wire perpendicular to the page with

current flowing inward has a magnetic field

as seen in Fig.7a.

In the presence of uniform magnetic field of

flux density B , the field above the wire is

reinforced and is weakened below the wire

resulting in a downward force on the wire as

seen in Fig. 7b.

This is the Lorentz or motor force

equation as given by,

F=IBL (N, newton)

More generally in vector notation,

F= (I×B)L

------------------- (18) Fig.7

Or, F=IBL

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PROBLEMS:

Prob.1. An infinitely long, straight conductor with a circular cross

section of radius b carries a steady current I. Find the magnetic flux

density both inside and outside of the conductor.

Fig.8

We know,

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PROBLEMS:

Prob.2: Consider a rectangular loop with sliding conductor having

width L and its length x is increasing uniformly with time. The

sliding conductor moves with a uniform velocity u. The flux density

B is normal to the plane of the loop and is uniform everywhere. The

magnitude of flux density B vary with time as given by B=BoCosωt.

Find, the total EMF induced in the loop.

Fig.9

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PROBLEMS:PROB.3

Fig.10

Sol.

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PROBLEMS:

Prob.4. Find the magnetic flux density at a point on the axis of a

circular loop of radius b that carries a direct current I.

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DISPLACEMENT CURRENT

For static EM field we recall equ.(5),

Since divergence of the curl of any vector field is zero. So,

According to continuity of current equation,

Equ. (6) and (19) are incompatible with time

varying condition . So we must modify Equ.(5) to

make agree with Equ.(19). So we add a term in Equ.(5).

Applying Divergence theorem,

------------------(19)

------------------------- (6)

------------------------- (5)

------------------(20) Similarly,

------------------(21)

So, -----------(22)

From equ.(21)

So, From equ.(20) ----------------(22)

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DISPLACEMENT CURRENT

----------------(22)

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PROBLEMS

PROB.5

Solution.

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PROB.6

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REFERENCE

Engineering Electromagnetics; William Hayt &

John Buck, 7th & 8th editions; 2012

Electromagnetics with Applications, Kraus and

Fleisch, 5th edition, 2010

Elements of Electromagnetics ; Matthew N.O.

Sadiku

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