Static magnetic Field

Static magnetic FieldMagnetic filed The region around a magnet in which magnetic fore are present.The source of the Magnetic field: 1. Permanent magnet2. An electric field changing linearly with time3. Direct currentBiot savarts lawIt states that the differential magnetic field intensity produced at a point P by the differential current element (dl) is:1. Directly Proportional to the product of: (i)the current ( ) (ii)differential length ( ) (iii)the sine of the angle ( ) between the current element and the line joining point P to the current element2.Inversely proportional to the square of the distance ( )between the current element and the point P

PdlIdH

;

Direction of the magnetic field intensity is normal to the plain containing ,the current element and ,loin joining the element and point P.In vector formEqn.2 in Eqn.1

.1

.2- Unit vector along the line joining point P to the current element

Consider a conductor from the point A to B along the z axis and it carries current I as shown in Fig.1Consider a small length in the conductorThe perpendicular distance from the point P to Z axis is

The magnetic field at a point P due to small length ( ) of conductor :

Magnetic Field intensity due to a finite and infinite wire carrying a current I

1

5Eqn.2,Eqn.3,Eqn.4,Eqn.5 in Eqn.1

2

3

4

6

Fig.1

Fig.2

From the Fig.2Diff. w.r.t.

A / B

.9

.10 Eqn.9 Eqn.10 Eqn.11 Eqn.12 in Eqn.8

.11

.12

Fig.2

A

B

Magnetic field at the point P due to the conductor from A to B:.7

Eqn.6 in Eqn.7

.8For infinite conductor ; .

Magnetic filed due to infinite conductor:

Fig.1Consider a circular conductor of radius It lies in a xy plane.Let the current flowing through the loop is I Its center is origin of co-ordinate systemThe point P is present in the Z axisConsider a small length ( ) in the circular loop

Magnetic field (Magnetic field intensity) on the axis of a circular loop carrying a current I

Using Biot-Savart law , the magnetic field at a point P due to small current carrying conductor isWe know that small length dl in cylindrical co-ordinate system

Fig.2From the Fig.2

Magnetic field due to total ring :

The redial components ( )produced by current element pair, 180 degree apart, cancel each other

Magnetic flux densityMagnetic flux density is defined as the magnetic flux passing per unit surface areaIts unit is or Tesla.

- PermeabilityLorentz force equation for moving charge

By coulombs law the electric force on a moving or stationary charge ,when it present in an electric filed is

The Magnetic force on a moving charge ,when it present in an magnetic filed with flux density is

By superposition principle the force on moving charge in the presence of both electric and magnetic field is

This equation is known as Lorentz force equation

Force on a wire, carrying current I, which is placed in a magnetic field:

We can write as, the small current element ,

This shows that a small charge dQ moving with velocity V is equal to a small current element

1 2We know that, the small force (dF) acting on the small charge dQ when it present in a magnetic field BEqn.2 in Eqn.1

The force on the straight conductor carrying the current I

The magnitude of the force is given by

Torque on a current carrying curve Torque is defined as rotating force or tangential force multiplied by a radial distance at which the force is acting Consider the rectangular loop of length l and breath b. It carry the current I in the uniform magnetic field of flux density BThe force acting on the current carrying conductor placed in the magnetic filed is

Magnetic filed intensity at the centre of the current carrying rectangular loopThe magnetic field due to finite wire, carrying current I

The magnetic filed due to rectangular loop = M.F due to segment ab + M.F due to segment bc M.F due to segment cd+ M.F due to segment da Magnetic filed due to segment ab

Similar due to the segment cd

Magnetic filed due to segment bc

Similar due to the segment da

Total magnetic field at point p field due to rectangular loop

Total magnetic at point p field due to square loop

Magnetic PotentialThe magnetic Potential could be a scalar denoted as or a vector denoted as

The vector identity are

The scalar magnetic Potential should satisfy the equation 1 .. 1.. 2The vector magnetic Potential should satisfy the equation 2 Just as we can define the magnetic scalar Potential related with as

It should satisfy the equation 1

But we know that Thus the required condition is

The Gausss law for the magnetic field is

By using Divergence theorem

Consider the similarity equation between the equation and

We can conclude that