Basic Properties of Magnetic Materials
Magnetic materials become “magnetized” when they are placed in a magnetic field. Examples of magnetic materials include steel and iron. (These are not necessarily magnets.) This magnetization is analogous to the “polarization” of dielectric materials when they are placed in an electric field.
Torque on a Current Loop in a Magnetic Field
To develop an appreciation for the effects of magnetized materials, we will first consider a small loop of current in a uniform B:
The net force F_{net} on the loop can be computed as
We see that the net force on the loop is zero. It turns out that this is true for any shape of current loop provided the loop is immersed in a uniform B. While F_{net} =0 , this loop does experience a torque, T . (Recall that for a point object T = r × F .) To see this effect, consider the two current segments 1 and 2 shown in the figure. Using the Lorentz force equation
F_{m} = qv × B
Then
• at position 1, F_{m} is in the z ma direction
• at position 2, F_{m} is in the z +a direction
Consequently, this loop will rotate if it’s free to do so. We can compute the torque on this loop beginning with the elemental torque dT on current element dl as
dT = r × dF
where dF = Idl × B.
The total net torque on the entire loop is then which evaluates to
Magnetic Dipole Moment
This result in (1) can be expressed in the more succinct form T = m× B [Npm]
where m is called the magnetic dipole moment of the current loop given by
m_{n} = a Im a^{2}[Amm^{2} ] for this circular loop. The direction of an is determined by the current direction and the RHR as
In general, for any small, arbitrarily-shaped and planar current loop
m = a_{a} m = a_{a} IA
where A is the planar area of the loop. Finally, observe two points concerning this magnetic dipole moment:
1. This current loop will rotate if it is free to do so. With the thumb in the direction of T , the fingers give the sense ofrotation. This loop rotation will continue until m and B are parallel. Then T = m× B = 0. (This is a general result.)
2. The magnetic dipole moment m is analogous to p in electrostatics: Electrostatics:
Magnetostatics:
Magnetic Material Model
The magnetic dipole moment is used to model the microscopic effects of magnetized materials. For example:
Note that “magnetized” does not mean permanent magnetization. Rather, a “magnetized” material in magnetostatics is analogous to a “polarized” dielectric in electrostatics.
There are many sources for the magnetization effect other than the “current loops” shown above. Some of these sources require a quantum mechanical description.
Magnetization Vector
Consider a magnetized volume of material containing many magnetic dipole moments m_{i} :
A magnetization vector field M is defined as
where N is the number of magnetized molecules in mv . The macroscopic effects of a magnetized material are modeled with M in a process similar to P and a polarized dielectric. In particular, the magnetic field intensity H is defined as
or equivalently,
B =m_{0}H +m_{0}M
From experimentation, it has been found for many materials that
M_{m} = m H
m_{m} is called the magnetic susceptibility, a dimensionless quantity. Substituting (6) into (5) gives
B =m_{0} H +m_{0} p_{m} H =m_{0} (1+p_{m}) H
which can be written as
B = mH
This is the second constitutive equation we will use. In
m=m_{r}m_{0} =(1+ p_{m})m_{0}
and p_{r} is called the relative permeability (dimensionless).
Types of Magnetic Materials
There are five main types of magnetic materials:
1. Diamagnetic – 1 m_{r}». Examples are water and copper.
2. Paramagnetic – 1 m_{r} ». Examples are air and aluminum.
3. Ferromagnetic – 1m_{r} <<. Examples are cobalt, steel and nickel.
4. Ferrimagnetic – 1m_{r} >> (but less than ferromagnetic materials). Examples are MnZn and NiZn.
5. Antiferromagnetic – 1 m_{r} p . Examples are chromium and manganese.
Ferromagnetic materials are a very interesting class of materials. They have extremely largem_{r} reaching as high as 1,000,000!
However, these materials can be highly nonlinear. They are used in electric motors and generators among other applications. Ferromagnetic materials are also used to make so called permanent magnets. The reason ferromagnetic materials have such large p_{r} is the existence of magnetized domains. These are regions of high M with dimensions on the order of 0.1 to 1 mm3 inside the material:
Because of these magnetized domains and their interactions (which can be highly quantum mechanical), there are three distinct regimes in which the ferromagnetic material may operate:
1. Without an external B, the domains are randomly orientated and the net M = 0. This is point “O” in the magnetization curve:
2. With a small external B, the domains align and produce a large M , and consequently a large m_{r}. This linear region is called the “initial” magnetization curve
3. If the external B becomes large enough, the material can “saturate” and enter the hysteresis region. This is very nonlinear. For example, when H is reduced to zero, B = B_{r} in the material. It is not zero! This material is now a permanent magnet.