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E, D, B & H: What Do They All Mean?

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E, D, B & H: What Do They All Mean? E~ , D~ , B~ & H~ : What do they all mean? W. T. Hill, III Department of Physics and The Institute for Physical Science and Technology University of Maryland, College Park, Maryland 20742 POLARIZATION OF DIELECTRIC MEDIA THE ELECTRIC DISPLACEMENT, D~ Before we talk about magnetic materials, it is instruc- The total ¯eld due to the polarization is given by tive to go back to dielectric materials for a moment. You adding P~ to the applied ¯eld E~ . However, we cannot learned that dielectric media responds to an applied elec- simply add E~ and P~ because they do not have the same tric ¯eld as shown in Fig. 1. units in MKS or the SI system of units. The polarization 3 2 has units of C-m/m ´ C/m , which means ÂE must 2 2 have units of C /N-m . Since ²o, the vacuum permittiv- 2 2 ity, also has units of C /N-m , we multiply E~ by ²o and add the result to P~ to give a new quantity that bears the name the electric displacement, D~ = ²oE~ + P~ (3) = (²o + ÂE)E~ (4) = ²E;~ (5) FIG. 1: Dielectric materials, which have no free electrons, can be polarized (internal dipole moments are induced) by where ² is the permittivity of the medium. With this an external electric ¯eld E~ resulting in a polarization ¯eld, de¯nition,  ! 0 and ² ! ² in vacuum. ~ E o P (the sum of all the induced dipoles). At internal locations, It is important to recognize that E~ is the fundamental the positive and negative charges cancel leaving a net positive ¯eld! If we write E~ as follows, surface charge on the right and net negative surface charge on the left. The resultant polarization ¯eld, which points in the 1 opposite direction to E~ , is normalized by the volume of the E~ = (D~ ¡ P~ ); (6) material so that P~ has units of electric dipole-moment per ²o unit volume. Figure after Fig. 26.23 of Serway and Beichner, ~ 5th edition. it becomes helpful to relate D with free charges, in the sense of Gauss' Law in electrostatics, When the applied electric ¯eld is weak and the medium I I is isotropic, E~ depends linearly on P~ and are parallel. ²oE~ ¢ dA = D~ ¢ dA = Q; (7) Thus, we can write where Q is the charge enclosed. The polarization, P~ , is P~ = ÂE(E~ )E;~ (1) then related to the bound charges in the medium associ- ated with the dipoles in Fig. 1. where ÂE(E~ ) is a scalar, in this case, and is called the electric susceptibility. When the medium is not isotropic, such as a crystal, P~ and E~ are not parallel necessarily, DIELECTRIC CONSTANT and ÂE will be represented by a matrix (a second rank tensor). In the case were E~ is large, P~ can have a non- For media that responds linearly to applied ¯elds, we linear dependence on E~ , usually do not use ÂE but a dimensionless quantity ·, the ~ (1) ~ (2) ~ ~ dielectric constant, which is related to the permittivity,[1] P = ÂE E + ÂE EE + ¢ ¢ ¢ ; (2) (1) (2) ² = ²o·: (8) where ÂE is the linear susceptibility from Eq. 1 and ÂE is the second-order nonlinear susceptibility. Depending This leads to on the strength of the ¯eld there could be higher terms as well. In what follows, we will look at the simplest D~ = ·²oE~ = ²E;~ (9) case of an isotropic medium that responds linearly to the applied ¯eld. where, for most media, · > 1. 2 TABLE I: The names and units of the six electromagnetic ¯elds: E~ , D~ , P~ , B~ H~ and M~ . Symbol Name Units E~ Electric Field V/m = N/C P~ Polarization C/m2 D~ Electric Displacement C/m2 B~ Magnetic Induction N/A-m M~ Magnetization A/m H~ Magnetic Intensity A/m magnetic case and is related to currents through Am- pere's Law I I 1 ~ ~ d©E B~ ¢ dl = H~ ¢ dl = I + ²o ; (11) ¹o dt where we see that H~ and B~ are related through the mag- netic permeability. To be consistent with the electric FIG. 2: The magnetization produced by an external ¯eld, H~ , case, H~ should be de¯ned as ¹ times B~ . However, the on a material with permanent dipole moments, upper images. This is an example of paramagnetic material where the mag- de¯nition is netic susceptibility is positive and the magnetization, which B~ has units of magnetic-dipole per unit volume, is aligned with H~ = ; (12) the applied ¯eld. The lower left image shows the microscopic ¹ currents producing the permanent magnetic dipoles. As in dielectric material, the internal currents cancel leaving a net where ¹ > 1. In analogy to Eq. 6, H~ can also be written surface current producing the magnetization. Figure after in terms of M~ , the magnetization of the medium, and ¹o, Figs. 33.3 (top) and 33.4 of Ohanian Vol 2. 1 H~ = B~ ¡ M:~ (13) ¹o MAGNETIZATION OF MAGNETIC MEDIA We can see that the two terms on the right have the same Magnetic media is in general more complicated than units by doing a little dimensional analysis. From Table I ~ ~ electronic media. We have three di®erent conditions: fer- we see that B has units of N/A-m while M has units of 2 2 2 ~ romagnetism, paramagnetism and diamagnetism, in or- A-m /m = A/m. Since ¹o has units of N/A , M and ~ der of decreasing strength. Ferromagnetism and param- B=¹o have the same units. agnetism lead to a magnetization of the material upon Putting this all together we can write the application of an external ¯eld due to the alignment ~ ~ ~ of permanent dipoles. Diamagnetism, on the other hand, B = ¹o(H + ÂM H) (14) is similar to polarization in dielectric material in that = ¹o(1 + ÂM )H:~ (15) the a magnetization is induced in the medium. Whereas paramagnetism and diamagnetism can be treated with It is clear when we compare Eqs. 12 and 14 that the magnetic analog to Eq. 1, ¹ = ¹o(1 + ÂM ); (16) M~ = ÂM (H~ )H;~ (10) where paramagnetic materials have positive  and dia- it is more appropriate to describe the M~ for ferromag- magnetic material have negative Â. Thus, M~ points in netic material as a strong function of H~ . In Eq. 10, ÂM the opposite direction in the two cases. is the magnetic susceptibility. In contrast to ÂE, how- ever, ÂM is dimensionless and can be both positive and negative. SUMMARY The six ¯elds along with their names and units are THE FIELDS H~ AND B~ summarized in Table I. The literature can be a little sloppy in referring to B~ and H~ . Often H~ is called the As in the electric case, we have two ¯elds in the mag- magnetic ¯eld but it is not uncommon to see that name netic case. The quantity H~ plays the role of D~ for the used for B~ as well. Regardless of the names, just as E~ 3 is the true mean ¯eld, it should be understood that B~ is talk about electromagnetic radiation and optics. the true mean ¯eld! [1] The quantity · is related to the index of refraction of a material and will be used later in the semester when we.
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