I AM MAGNETIC MONOPOLE

S HUSSAINSHA

Research scholar of ECE, G.Pullaiah College of Engineering and Technology, Kurnool, Andhra Pradesh, India

Email: ssshaik80@gmail.com Cell: +91 9000390153

Abstract: The principal objective of this paper is, finding a way to discover magnetic mono pole or an isolated magnetic charge. This paper also discusses importance of current element and states Gauss law for magnetic charge. So, we modify Maxwell’s equation by using Gauss law and mathematical procedure called Gauss divergence theorem. At final, we introduce principle of electric charge which explains about nature of electric charge.

Keywords: Magnetic charge, Current element, Maxwell’s 2nd equation, principle of electric charge.

I. INTRODUCTION

Let,

Electric field strength = E

Electric force = Fe

Electric charge = qe

Electric field strength is defined as electric force per unit electric charge.

E=Fe

qe

Let,

Magnetic field strength = H

Magnetic force = FM

Magnetic charge = QM

Magnetic field strength is defined as magnetic force per unit magnetic charge.

H=F M

QM

⇒QM=FM

H (1)

From equation (1) we have one unknown (magnetic charge) and we know magnetic field strength and magnetic force, so we discuses about magnetic field strength and magnetic force.

Unification of electricity and magnetism permits us to take electric charge as magnetic charge as given in equation below.

Magnetic charge = QM=μo qe v

qe = electric charge

v = velocity of electric charge

μ o = Magnetic permeability of free space or air.To get the above equation let continue our discussion

1.1.Magnetic field strength:

1.1.1.Current element[3]

The universal facts:

(1) We unified electricity and magnetism by using current elements.(2) Biot-savart’s law is framed by using current element.(3) Current element plays complete role in magnetostatics.

Magnetostatics deals with magnetic fields produced by steady currents. Steady current flowing through a straight conductor produces magnetic field in the form of concentric circles as shown in figure (1). The intensity of magnetic field is given by Biot-savart’s law. A straight conductor is assumed to be formed by several segments. Such segment is called current element.

Figure 1: magnetic flux lines due to a current coming out of the page.

1.1.2. Biot-savart’s law [3]

Let the magnetic field intensity at ‘P’ due to the current element be thedH . According to the Biot-savart’s law,

magnitude of dH is directly proportional to Idl and is inversely proportional to square of the distance between

current element and the field point ‘P’ as shown in figure (2) . The direction of dH is given by right hand thumb

rule. By thumb rule, direction of dH is into the paper at ‘P’. The vector dH is perpendicular to the plane

containing the vectors Idl andar . Where ar is unit vector in the direction of r . In other words the magnitude of

dH is proportional to the sine angle between the vector Idl andar . Mathematically as shown below in equation (2)

Figure 2: representation of magnetic field dH at ‘P’ due to current element Idl.

d H=Idl× ar

4 πr2

(2)

I dl×ar=I dl ar sinθ an

Take θ=90o

⇒ I dl× ar=I dlar an (3)

Substitute equation (3) in equation (2)

d H=I dl ar

4 πr2 an

⇒H =∮L

Idl4 πr2

an

For our convenience

H=∮L2

I 2 dl2

4 πr212 an

(4)

1.2. Magnetic force and Sources of Magnetic field

There are two types of sources.

1) Magnetic Force between two Current elements [2]

Let us now consider the force between two elements I 1dl1 andI 2 dl 2 . According to Biot-savart’s, both current

elements produce magnetic fields. So we may find the force F1 on current element I 1dl1 due to the field B2

produced by element I 2 dl 2 as shown in figure (3) and mathematical representation is shown below in equation (5).

Figure 3: representation of force between two current loops.d (d F1 )=I 1 dl1×d B 2

(5)From Biot-savart’s,

d B2=μo I 1 dl1×ar21

4 πr212

(6)

Hence,

d (d F1 )=μo I 1 dl1×( I2 dl2×ar21)

4 πr212

(7)

This equation is essentially the law of force between two current elements and is analogous to Coulomb’s law,

which expresses the force between two stationary charges. From equation (7) we obtain the total force F1on current loop 1 due to current loop 2 as shown in figure (3) as

F1=∮L1

∮L2

μo I 1 dl1×( I 2 dl2× ar21)4 πr21

2

(8)

Note that F2=−F1

Now simplify equation (7) to decrease complexity.

Now I 2 dl2× ar21

=I 2dl 2 ar21sin θ an (9)

Take θ=90o

Hence equation (9) written asI 2 dl2× ar21

=I 2dl 2 ar21^ an

(10)Substitute equation (10) in equation (8)

F1=∮L1

∮L2

μo I 1 dl1×( I 2 dl2 ar21an )

4 πr212

(11)

Let dH =dH { an¿

Where an is a unit vector in the direction of dH and it is perpendicular to the plane containing the vectors Idl andar . Where ar is unit vector in the direction of r .

If an is a unit vector in the direction of dH and it is perpendicular to the plane containing the vectors Idl andar .Then equation (11) written as

F1=∮L1

∮L2

μo I 1 dl1× an ( I 2 dl2 ar21)4 πr21

2

(12)

Taking θ=90o

I 1 dl1×an=I 1 dl1an ar21 (13)Substitute equation (13) in equation (12)

F1=∮L1

∮L2

μo I 1 dl1 I 2dl2 an ar21

4 πr212 ^ar 21

⇒ F1=∮L1

∮L2

μo I 1 dl1 I 2dl 2

4 πr212 ar21

(14)

Since magnitude of unit vectors is ‘1’

Substitute equation (4) and equation (14) in equation (1)

QM=F M

H=

∮L1

∮L2

μo I 1 dl1 I2 dl2

4 πr212 ar21

∮L2

I 2dl 2

4 πr212 an

(15)Since the direction of magnetic force and the direction of magnetic field is same. That is, perpendicular to current element. Therefore equation (15) written as

QM=μo I 1 L1

Since

[ I=qe

t ]and

[v= Lt ]

QM=μo qe v

I discovered this equation from existing concepts. This is called true unification of electricity and magnetism. Here I

have done nothing, that is we already discovered magnetic mono pole.

2) Magnetic Force between two Bar Magnets

1.2.1. Inverse square law:

The inverse square law of magnetism states that the force of attraction or repulsion between two magnetic poles is directly proportional to the product of their pole strengths and inversely proportional to the square of the distance between them and acts along the line joining the poles. The force of attraction or repulsion between two magnetic poles is given by equation (16) as shown below.

F=μo M 1 M 2

4 πR2 (16)

Where,M1 = pole strength of bar magnet 1.M2 = pole strength of bar magnet 2.

μ o = Magnetic permeability of free space or air.R = distance of separation between the poles.

Let ar is a unit vector along the directionR .

R=|R|ar

⇒ ar=R

|R|F=F ar

(17)

Substitute equation (16) in equation (17)

F=μo M 1 M 2

4 π|R|2ar

(18)

Equation (14) and equation (18) are equal. Therefore, pole strength is equal to current element. So equate equation (14) and equation (18) and we get,

M 1=I 1 dl1

M 2=I 2 dl2Therefore,

Pole strength M=Idl

1.3. Relation between magnetic charge and electric charge

1.3.1. Magnetic charge

Magnetic charge is directly proportional to electric charge and velocity of electric charge. Magnetic charge is a

scalar quantity. Let QM be the magnetic charge, qe be the electric charge and ‘v’ be the velocity of electric charge. Then the relation between magnetic charge and electric charge is given as

QM ∝qe v

⇒QM=μo qe v (19)

μ o = Magnetic permeability of free space or air.

Let us consider the following cases to support equation (19),

Case 1: Electric charge present inside a closed surface is static.

Only electric flux comes out of that closed surface as shown in figure (4). Gauss law gives the measure of electric flux coming out.

Gauss law:

Statement: electric flux coming out of a electric charged body is equal to the amount of electric charge enclosed.

Figure 4: electric flux coming through closed surface.

Case 2: Magnetic flux due to current carrying conductor is in the form of concentric circles. Consider an infinitesimal volume ‘dV’ as shown in figure (5). The number of flux lines entering and leaving the volume ‘dV’ is equal. Therefore the net outflow of flux per unit volume is zero.

Figure 5: representation of magnetic flux around current carrying conductor and net out flow of magnetic flux per unit volume.

In this case it will be zero. Let’s examine other case shown below and try to understand the source of magnetic flux lines.

Case 3: electric charge present inside a closed surface is changing with time.

Both electric flux and magnetic flux come out of a closed surface as shown in figure (6).

Figure 6: representation of net out flow of electric flux and magnetic flux per unit volume.

1.4. Magnetic displacement density or magnetic flux density ( B )

Yes it is not possible to have isolated magnetic poles or magnetic charges, but unification of electricity and magnetism permits us to take electric charge as magnetic charge as given in equation (19). To overcome confusion

observe figure (7). So, to have isolated magnetic pole divide closed surface into two closed surfaces. Magnetic flux lines start at one point and terminate at another point. I will give more clarity about this (dividing closed surface into two closed surfaces) in the next following section of this paper.

Figure 7: representation of magnetic flux lines due to a permanent magnet and representation of closed surfaces as a sum of two closed surfaces.

The number of magnetic flux lines per unit area is called magnetic flux density and its direction is normal to the

surface area having maximum displacement. For an isolated point magnetic charge, B at any point on a spherical surface of radius ‘r’ centered at the isolated magnetic charge.

B=QM

4 πr2 (20)

1.5. Gauss law for magnetic fluxψ M

Statement: Magnetic flux coming out of a magnetic charged body is equal to the amount of magnetic charge enclosed.

ψ M=QM

Consider a point magnetic charge QM kept at the origin as shown in figure (8). Let us consider infinitesimal area

‘ds’ at a distance ‘r’ meters from the origin. The magnetic flux density and area vector ds are normal to the surface

‘ds’ as shown in figure (8). Let the flux crossing the surface ‘ds’ bedψ M .

Figure 8: representation of Gaussian surface about a point magnetic charge at origin

B=dψ M

ds

⇒dψM=B ds

=Bds cosθ

⇒dψM =Bds (21)

Since B and ds are in same direction

The total flux leaving the entire surface can be obtained by surface integration,

ψ M=∫s

Bds(22)

By substitute equation (20) in equation (22) we get

ψ M=∬QM

4 πr2 ds

⇒ψ M=Qm

4 πr2∬ds

⇒ψ M=Qm

4 πr2 Area

We know that surface area of spherical surface of radius ‘r’ is 4 πr 2

Therefore, ψ M=QM (23)

II. MODIFIED MAXWELL’S EQUATION

Let ρMV be volume magnetic charge density.

If the body is uniformly charged with the magnetic charge density ρMV . Total charge is given as follows.

QM=ρMV volume=ρMV∫V

dV

QM=∫V

ρMV dV

By gauss law or from equation (22) and (23)

∫s

Bds=∫V

ρMV dV(24)

Gauss divergence theorem relates surface integral and volume integral [2]. According to Gauss divergence theorem surface integral of normal component of the magnetic flux density is equal to the volume integral of divergence of ‘B’.

∫s

Bds=∫V

∇⋅BdV(25)

By substitute equation (25) in equation (24) we get

∫V

∇⋅BdV =∫V

ρMV dV

(26)

∇⋅B=ρMV is modified Maxwell’s 2nd equation.

Note: For suppose, if an isolated magnetic mono pole is discovered in our laboratories then also equation (26) cannot be altered. Therefore deriving modified Maxwell’s equation is completely scientific.

Universal facts:

⇒∇⋅B=ρMV

(1) Negative electric charge (-q) has its own electric lines of flux and positive electric charge (+q) has its own electric lines of flux.

Therefore, it is not mean that, electric lines of flux negative electric charge (-q) came from positive electric charge (+q).

(2) Electric lines of flux start from positive electric charge (+q) and terminates at negative electric charge (-q).

If we apply common senesce, electric lines of flux start from positive electric charge (+q) and terminates at negative electric charge (-q) is just our notion. That is practically it never happens. It is the diagram of field lines which forcing us to consider Electric flux of lines start from positive electric charge (+q) and terminates at negative electric charge (-q)

Practically field lines never start and never terminates. Every charge has its own field and the direction of the field depends on type of charge present in that field.

When opposite flux lines meet, attract with each other and when ever same flux lines meet, repel with each other.

The same will be applied to North Pole and South Pole.

2.2. Principle of electric charge

Unification of electricity and magnetism permits us to discuss nature of electric charge.

Whenever electric charge is static will be considered as electric charge, and whenever electric charge is in motion will be considered as, both electric charge and magnetic charge.

III. CONCLUSION

This paper may give new roads to discover an isolated magnetic mono pole in our laboratories. In this paper I discussed what we know already. This paper exactly unifies electricity and magnetism.

ACKNOWLEDGEMENT

Author wants to thank Mr. S.ARIF HUSSAIN (Ph.D) Professor of ECE Dept., Stanley Stephen College Of Engineering & Technology, and S.Md.GHOUSE for their comments and constructive suggestions. And special thanks to S IBRAHIM KALEEL for his cooperation in preparing this paper.

REFERENCESBooks:[1] Stephen hawking, A stubbornly persistent illusion (running press book publishers, Philadelphia, London)[2] Matthew N.O. Sadiku, elements of electromagnetics(oxford university press,oxford)

[3] S.Rama reddy, electromagnetic theory (scitech publications(India) Pvt.Ltd.,India)