25-02-2013, 10:19 AM
Modification of Electric and Magnetic Fields by Materials
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INTRODUCTION
Certain materials influence electric and magnetic fields through bound charges and currents. Their
properties differ from those of metals where electrons are free to move. Dielectric materials
contain polar molecules with spatially displaced positive and negative charge. Applied electric
fields align the molecules. The resulting charge displacement reduces the electric field in the
material and modifies fields in the vicinity of the dielectric. There are corresponding magnetic field
effects in paramagnetic and ferromagnetic materials. These materials contribute to magnetic fields
through orientation of atomic currents rather than a macroscopic flow of charge as in a metal.
Although the responses of materials to fields differ in scale, the general behavior is similar in
form. This is the reason the contributions of dielectric and magnetic materials were singled out in
Section 4.5 as 2 and j2 It is often useful to define new field quantities that automatically
incorporate the contributions of bound charges and currents. These quantities are D (the electric
displacement vector) and H (the magnetic field intensity).
The study of the properties of dielectric and magnetic materials (including subsidiary field
quantities and boundary conditions) is not conceptually exciting. This is especially true for
ferromagnetic materials where there is considerable terminology. Nonetheless, it is essential to
understand the properties of dielectric and ferromagnetic materials since they have extensive uses
in all types of accelerators.
DIELECTRICS
Dielectric materials are composed of polar molecules. Such molecules have spatially separated
positive and negative charge. The molecules may be either bound in one position (solids) or free
to move (liquids and gases). Figure 5.la shows a diagram of a water molecule. The
electronegative oxygen atom attracts the valence electrons of the hydrogen atoms, leaving an
excess of positive charge at the locations of the hydrogen atoms.
BOUNDARY CONDITIONS AT DIELECTRIC SURFACES
Methods for the numerical calculation for vacuum electric fields in the presence of dielectrics
were mentioned in Section 5.1. There are also numerous analytic methods. Many problems
involve uniform regions with different values of /o. It is often possible to find general forms of
the solution in each region by the Laplace equation, and then to determine a general solution by matching field components at the interfaces. In this section, we shall consider how electric fields
vary passing from a region with /o 0 to a vacuum. Extensions to interfaces between two
dielectrics is straightforward.
The electric fields at a dielectric-vacuum interface are divided into components parallel and
perpendicular to the surface (Fig. 5.6). The magnitude of the electric field is different in each
region (Section 5.1); the direction may also change. The relationship between field components
normal to the interface is demonstrated by the construction of Figure 5.6b. A surface integral is
taken over a thin volume that encloses the surface. The main contributions come from integration
over the faces parallel to the surface. Using Eq. (5.3) and the divergence theorem.
FERROMAGNETIC MATERIALS
Some materials modify applied magnetic fields by alignment of bound atomic currents. Depending
on the arrangement of electrons, atoms may have a magnetic moment. This means that the
circulating electrons produce magnetic fields outside the atom. The fields, illustrated in Figure
5.8, have the same form as those outside a circular current loop (Section 4.7); therefore, the
circular loop is often used to visualize magnetic interactions of atoms.
STATIC HYSTERESIS CURVE FOR FERROMAGNETIC
MATERIALS
In this section we shall look in more detail at the response of ferromagnetic materials to an
applied field. In unmagnetized material, the directions of domains are randomized because energy
is required to generate magnetic fields outside the material. If the external magnetic field energy is
supplied by an outside source, magnetic moments may become orientated, resulting in large
amplified flux inside the material. In other words, an applied field tips the energy balance in favor
of macroscopic magnetic moment alignment.
A primary use of ferromagnetic materials in accelerators is to conduct magnetic flux between
vacuum regions in which particles are transported. We shall discuss relationships between fields
inside and outside ferromagnetic materials when we treat magnetic circuits in Section 5.7. In this
section we limit the discussion to fields confined inside ferromagnetic materials. Figure 5.11
illustrates such a case; a ferromagnetic torus is enclosed in a tight uniform magnet wire winding.
We want to measure the net toroidal magnetic field inside the material, B, as a function of the
applied field B1, or the field intensity H. The current in the winding is varied slowly so that applied
field permeates the material uniformly. The current in the winding is related to B1 through Eq.
(4.42). By Eq. (5.1 1), H = NI/L, hence the designation of H in ampere-turns per meter. The
magnetic field inside the material could be measured by a probe inserted in a thin gap. A more
practical method is illustrated in Figure 5.11. The voltage from a loop around the torus in
integrated electronically. According to Section 3.5, the magnetic field enclosed by the loop of area
A can be determined fromB = Vdt/A.