Chapter 8

Magnetic Forces, Materials and Inductance

The magnetic field B is defined from the Lorentz Force Law, and specifically from the magnetic force on a moving charge:

F = qv x B

1. The force is perpendicular to both the velocity v of the charge q and the magnetic field B.

2. The magnitude of the force is F = qvB sin where is the angle < 180 degrees between the velocity and the magnetic field. This implies that the magnetic force on a stationary charge or a charge moving parallel to the magnetic field is zero.

3. The direction of the force is given by the right hand rule. The force relationship above is in the form of a vector product.

From the force relationship above it can be deduced that the units of magnetic field are Newton seconds /(Coulomb meter) or Newton per Ampere meter. This unit is named the Tesla. It is a large unit, and the smaller unit Gauss is used for small fields like the Earth's magnetic field. A Tesla is 10,000 Gauss. The Earth's magnetic field is on the order of half a Gauss.

8.1 Force On A Moving Charge

Lorentz Force Law

Both the electric field and magnetic field can be defined from the Lorentz force law:

The electric force is straightforward, being in the direction of the electric field if the charge q is positive, but the direction of the magnetic part of the

force is given by the right hand rule.

8.2 Force On A Moving Charge

8.3 Force on a Differential Current

dF = dQv x B

J v v

dF J Bdv dQ v dv

dF v dv v B

dF J Bdv

Jdv KdS IdL

dF K BdS

dF IdL B

F vJ B

dvol

F SK B

dS

F LI

d B I LB_x_

d

F IL B

Example 8.1

az axFy1 2 10

3 3 106

1

3

x1

x

d

Fy1 6.592 109

Fx1 2 103 3 10

6

0

2

y1

3

d

1 az ay Fx1 4 109

Fy2 2 103 3 10

6

3

1

x1

x

d

az ax Fy2 6.592 109

az ayFx2 2 10

3 3 106

2

0

y1

d

1 Fx2 1.2 108

F Fx1 Fx2 Fy1 Fy2 F 8 109

The net force on the loop is in the -ax direction

HI

2 xaz

B o H

F I LBx

d

Force Between Differential Current ElementsExample 8.2

The magnetic field at point 2 due to a current element at point 1 was found to be

Now, the differential force on a differential current element is

Example

8.4 Force And Torque On A Closed Circuit

F I LB_x_

d

F IB L1

d

T R F

• a) given a meter arm R extending from an origin O to a point P where force F is applied, the torque about O is T = RxF

• b) if F2 = -F1, then the torque T=R21 x F1 is independent of the choice of origin for R1 and R2

Force And Torque On A Closed Circuit

dT IdS B

Magnetic Dipole Moment dm

dm IdS

dT dm B

T IS B m B

• A differential current loop in a magnetic field B. The torque is dT=I(dxdy az)x Bo = IdS x B.

• The total force on the loop is therefore 0, and we are free to choose the origin for the torque at the center loop

Force And Torque On A Closed Circuit

DC Motor - Illustration

Example 8.3

• A rectangular loop is located in a uniform magnetic flux density Bo

8.5 The Nature of Magnetic Materials

Magnetic Materials

Magnetic Materials may be classified as diamagnetic, paramagnetic, or ferromagnetic on the basis of their susceptibilities. Diamagnetic materials, such as bismuth, when placed in an external magnetic field, partly expel the external field from within themselves and, if shaped like a rod, line up at right angles to a non-uniform magnetic field. Diamagnetic materials are characterized by constant, small negative susceptibilities, only slightly affected by changes in temperature.

Paramagnetic materials, such as platinum, increase a magnetic field in which they are placed because their atoms have small magnetic dipole moments that partly line up with the external field. Paramagnetic materials have constant, small positive susceptibilities, less than 1/1,000 at room temperature, which means that the enhancement of the magnetic field caused by the alignment of magnetic dipoles is relatively small compared with the applied field. Paramagnetic susceptibility is inversely proportional to the value of the absolute temperature. Temperature increases cause greater thermal vibration of atoms, which interferes with alignment of magnetic dipoles.

Ferromagnetic materials, such as iron and cobalt, do not have constant susceptibilities; the magnetization is not usually proportional to the applied field strength. Measured ferromagnetic susceptibilities have relatively large positive values, sometimes in excess of 1,000. Thus, within ferromagnetic materials, the magnetization may be more than 1,000 times larger than the external magnetizing field, because such materials are composed of highly magnetized clusters of atomic magnets (ferromagnetic domains) that are more easily lined up by the external field.

Characteristics of Magnetic Materials

Magnetization and Permeability

Magnetization and Permeability

• A section dL of a closed path along which magnetic dipoles have been partially aligned by some external magnetic field. The alignment has caused the bound current crossing thesurface defined by the closed path to increase by nlbds.dLA

• Where

And I is the total free current enclosed path.

Current enclosed

Define H in terms of B and M

Where

Example

Magnetization and Permeability sample representation

• An orbiting electron is shown having a magnetic moment m in the same direction as an applied Bo

Magnetic Boundary Conditions

• A gaussian surface and a closed path are constructed at the boundary between media 1 and 2, having permeabilities of µ1 and µ2, respectively. From this we determine the boundary conditions BN1 = BN2 and Ht1 – Ht2 = K, the component of the surface current density directed into the page.

Thus

Example

The Magnetic Circuit

• In this section we shall digress briefly to discuss the fundamental techniques involved in solving a class of magnetic problems known as magnetic circuits. The name arises from the great similarity to the dc resistive-circuit analysis with which it is assumed we are all facile. The only difference lies in the nonlinear nature of the ferromagnetic portions of the magnetic circuit; the methods which must be adopted are similar to those required in nonlinear electric circuits which contain diodes, thermistors, incandescent filaments, and other nonlinear elements.

• In an electric circuit the voltage source is a part of the closed path; in the magnetic circuit the current-carrying coil will surround or link the magnetic circuit.

• For example

The Magnetic Circuit

 Magnetic circuit is made up of one or more closed loop paths containing a magnetic flux. The flux is usually generated by permanent magnets or electromagnets and confined to the path by magnetic cores consisting of ferromagnetic materials like iron, although there may be air gaps or other materials in the path.

Example

Sources:

• Engineering Electromagnetics, Hayt, 8th ed.