MAGNETS AND MAGNETIC CIRCUITSPermeability and Relative Permeability

The Magnetic Flux Density at a point in a vacuum due to electric current is

EQUATION 1 Bo=∑μo I ∆l sinӨ4 π r2

Where Bois used here to indicate the vacuum. If a homogeneous medium is present instead of a vacuum, it is found that the flux density increases to a value given by:

EQUATION 2 B=∑ μr μo I ∆l sinӨ4 π r2

Whereμr is called the relative permeability of the medium. The product μr μo is often denoted by μand called the permeability of the medium so:

μ (permeability )=μ (relative permeability) *μo

Where μo= 4π * 10−7Wb/Am or N/mThe relative permeability μrhas for most materials a value very close to 1 and only for

ferromagnetic materials it is appreciably greater the behavior of such materials is very complicated in general and it is not possible to specify a single value of μr or even to give a sensible meaning to itself when a hard magnetic material like steel is present. For the purposes of the problems in this given chapter it is assumed that only soft magnetic materials are present, for which all the flux density disappears when the magnetizing currents are reduced to zero (pure iron is an example). In these materials, the approximation will be made that μr is constant.

The formula given for the flux density is due to various currents were obtained from the formula (1) above. When a medium is present everywhere, expression (2) must be used and all the formulas multiplied by μr.

Thus the flux density in a toroid or solenoid with a ferromagnetic core is

B=μr μo∋¿l¿

MAGNETIC FIELD STRENGHTIt is convenient to have a quantity that represents the magnetic field that is produced by the

electric current only, in the absence of any medium. The quantity Bo in (1) above is used by some but more usual to define a new quantity, the magnetic field strength denoted by H . Provided the μr of any material present is single constant number, H can be define by:

H=Bμ =

Bμr μo

This means, for instance, that for toroid of N turns in length l of its circumference

H=¿l

The SI unit for H is A/m.THE MAGNETIC MOMENT OF COIL

The torque L on a coil of N turns carrying a current I in a field of magnetic flux density B was shown to be L=BIN A, where the plane of the coil is parallel to the field. We define magnetic moment m of the coil by the equation :

m= LB =

BINAB

=¿INA

The SI unit of magnetic moment is A m2.

MAGNETIC POLESAlthough magnetic poles do not actually exist, the concept is often useful in simplifying

calculations involving magnets.A bar magnet of length l placed with its axis normal to the field B has a torque L acting on it

which tends to rotate the axis a position parallel to B and its magnetic momentum m =L/B. We now define the pole strength P of each end face of the magnet as:

P= ml

Where P is in A m2since m is Am2 , the force exerted on each pole is:F(N)=B(T)*P(Am)

The force on north pole being on the same direction as B and the force on the south pole being directed opposite to B. Note that 1 T= 1 N/Am

FORCE BETWEEN TWO MAGNETIC POLES: If two poles of strength p and P’ are separated in a free space by a distance r, the force between

them is:

MAGNETIC FIELD OF A POLE:The force exerted by a pole P’ to another P’ at a distance r from it in free space is F= UPP’/4π r2.

Also the magnetic flux density at P is B=P/P’. Then B = (μoPP'

4 π r2)

P ' or

B=μoP

4 π r2

Where B is the magnetic flux density at a distance r from a pole of strength P.

THE MAGNETIC CIRCUIT:The magnetic flux inside a closed ring solenoid of a uniform cross section area A and mean

circumference is Ф=B A=μHA=μ ¿

lA

Where μis the permeability of the core material. Rearranging this expression gives the law of the magnetic circuit

Ф=¿lμA

or flux=magnetomotive force (mmf )

reluctance(β )

Magnetomotive force (NI) causes the establishment of flux in the circuit. Flux is analogus to current in the electric circuit, magnetomotive force(mmf) to emf, and reluctance to resistance. Flux is given in weber Bb, magnetomotive force in (A), and reluctance in

lμA =

NTФ in A/Wb