MAGNETIC FIELDSUBJECT : ELECTROMAGNETICS THEORY

INTRODUCTION

• Definition

A magnetic field is the magnetic effect of electric currents and magnetic materials.

• The magnetic field at any given point is specified by both a direction and a magnitude as such it is a vector field.

• Denoted by : B or H

• Unit : A·m−1 or A/m or teslas (T)

UNDERSTANDING

• “Whenever electron moves Magnetic Field is generated associated with movement.”

• Now we are very well aware of magnetic field created by electric currents. We can easily relate previous statement with this phenomenon.

• However, magnetic field of permanent magnets can create confusion with that statement. In permanent magnets, examining on atomic level we can notice the movement of electrons with magnetic moments in one direction causing magnetic field.

• In non magnetic fields magnetic moments of electrons in every orbit is canceled by each other.

• Iron has two electrons that can be aligned, so it is the "strongest" magnetic material.

MAGNETIC FILED LINES

• A pictorial representation of magnetic field lines is very useful in visualizing the strength and direction of the magnetic field . The direction of magnetic field lines is defined to be the direction in which the north end of a compass needle points.

• There are many types of magnets differing by shape creating different patterns of magnetic filed lines as below :

Magnetic field by Bar magnet

Magnetic field by Horseshoe magnet

Magnetic field by Horseshoe magnet

Toroidal Magnetic Field

PROPERTIES

• They seek the path of least resistance between opposite magnetic poles. In a single bar magnet as shown to the right, they attempt to form closed loops from pole to pole.

• They never cross one another.

• They all have the same strength.

• Their density decreases (they spread out) when they move from an area of higher permeability to an area of lower permeability.

• Their density decreases with increasing distance from the poles.

• They are considered to have direction as if flowing, though no actual movement occurs.

• They flow from the south pole to the north pole within a material and north pole to south pole in air.

DIFFERENTIATING ‘B’ AND ‘H’ • It's all to do with the difference between free charge and bound charge (which together make total charge).

• E and B are the total electric and magnetic fields.

• D and H are the free electric and magnetic fields.

• P and M are the bound electric and magnetic fields.

• So E = D + P (except that for historical reasons E is defined differently, so we need to multiply it by the permittivity, and for some reason P is multiplied by minus-one ).

• And B = H + M (except that for the same historical reasons B is defined like E, so we need to divide it by the permeability).

• The latter equation says that the total magnetic field equals the free magnetic field plus the bound magnetic field (the bound magnetic field is all those little loopy currents that make things magnetic). When an electric current produces a magnetic field, it couldn't care what the field is going to be used for (i.e. for bold sweeping field lines or pokey little loops inside matter), so it produces a total field, which it's sensible to measure as B. But once we put matter in the way, we can only measure the free field, H.

• Basically, (apart from the permeability factor, of course) B and H are the same away from matter, but in or near matter the matter soaks up some of the B, and all we measure is what's left, the H.

DIFFERENTIATING ‘B’ AND ‘H’ (CONT.)

• Based on Maxwell's equations, electric fields are generated by changing B fields, while H fields are generated by changing electric fields.

• In dc fields, static electric E fields create currents (magnetization currents) I which in turn produce static H fields. A static B field cannot produce an electric field E.

• Even if we used natural units where μ0 = ε0 = 1, this distinction between B and H remains. In magnetic materials, B is not linearly related to H due to the magnetization term M. B = H in space, then B ≠ H in magnetic materials.

• If voltage = d/dt ∫B·n dA, then how can curl H= σ·E = σ·voltage/length if B = H? So even if μ0 = ε0 = 1, they have to have units.

Important note : “Magnetic Field” term is historically assigned to symbol ‘H’ and ‘B’ to all other terms. But generally in many books it is also used for ‘B’ as well and also used usually by physicists and professors.

BIOT-SAVART'S LAW

• Biot-Savart's law states that, “The magnetic field dH produced at a point P, as shown in Figure, by the differential current element Idl is proportional to the product Idl and the sine of the angle a 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.”

• The direction of the current and magnetic field can be determined by right hand thumb rule.

i.e.

or

BIOT-SAVART'S LAW (CONT.)

• If we define K as the surface current density (in amperes/meter) and J as the volume current density (in amperes/meter square), the source elements are related as

• Thus in terms of the distributed current sources, the Biot-Savart law as in eq. in previous slide becomes,

AMPERE'S CIRCUIT LAW - MAXWELL'S EQUATION

• Ampere's circuit 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,

• By applying Stoke's theorem to the left-hand side of equation, we obtain

= =

but,

MAGNETIC FLUX DENSITY - MAXWELL'S EQUATION

• The magnetic flux density B is similar to the electric flux density D. As D = E in free space, the magnetic flux density B is related to the magnetic field intensity H according to

where is a constant known as the permeability of free space.

The constant is in henrys/meter(H/m) and

has the value of

MAGNETIC FLUX DENSITY - MAXWELL'S EQUATION• The magnetic flux through a surface S is given by

where the magnetic flux f is in webers (Wb) and the magnetic flux

density is in webers/square meter (Wb/) or teslas.

• The magnetic flux line is the path to which B is tangential at every point in a magnetic field. It is the line along which the needle of a magnetic compass will orient itself if placed in the magnetic field. The direction of B is taken as that indicated as "north" by the needle of the magnetic compass. Notice that each flux line is closed and has no beginning or end.

• In an electrostatic field, the flux passing through a closed surface is the same as the charge enclosed; that is, . Thus it is possible to have an isolated electric charge as shown in Figure, which also reveals that electric flux lines are not necessarily closed. Unlike electric flux lines, magnetic flux lines always close upon themselves as in Figure. This is due to the fact that it is not possible to have isolated magnetic poles (or magnetic charges).

MAGNETIC FLUX DENSITY - MAXWELL'S EQUATION• Thus the total flux through a closed surface in a magnetic field must be zero; that is,

This equation is referred to as the law of conservation of magnetic flux or Gauss's law for magnetostatic fields just as is Gauss's law for electrostatic fields. Although the magnetostatic field is not conservative, magnetic flux is conserved.

• By applying the divergence theorem to equation, we obtain

=

or • This equation is the fourth Maxwell's equation to be derived. This equations shows that

magnetostatic fields have no sources or sinks. Last equation suggests that magnetic field lines are always continuous.

PRESENTATION BY…

• Arjun Dedaniya (130170111016)

• Asif Faruki (130170111020)

• Poorn Mehta (130170111052)

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“No man really becomes a fool until he stops asking questions.

- CHARLES P. STEINMETZ

Thank You!