(Equation 3.6)

where is a unit vector along a straight line
from Q to q and painting toward q, and R is the distance
between the two charges, as shown in Figure 3.7. In the SI
system of units, _{o} is a constant
called the *permittivity of free space*. The units of charge
are coulombs, and the units of permittivity are farads per
meter (see Section 3.1). When both q and Q have the same
sign, the force in Equation 3.6 is repulsive. When the
charges have opposite signs,the force is attractive. When
more than one charge is present, the force on one charge is
the summation of all forces acting on it due to each of the
other individual charges. Keeping track of all the charges in
a complicated electrical system is not always convenient, so
we use a quantity called *electric-field strength vector*
(**E**-field) to account for the forces exerted on charges
by each other.

Force on a charge, q, due to the presence of another charge, Q.

where it is understood that q is infinitesimally small, so it does not affect the measurement. The units ofE=F/q (Equation 3.7)

F= qE(Equation 3.8)

Thus if **E** is known, the force on any
charge placed in **E** can easily be found.

B = F_{m}/qv (Equation 3.9)

where F_{m} is the maximum
force on q in any direction, and **v** is the velocity of q. The
units of **B** are webers per square meter. The
**B**-field is more complicated than the **E**-field in
that the direction of force exerted on q by the
**B-**field is always perpendicular to both the velocity
of the particle and to the **B**-field. This force is
given by

F= q(vxB) (Equation 3.10)

(which is analogous to Equation 3.7). The
quantity in parentheses is called a vector cross product. The
direction of the vector cross product is perpendicular to
both **v** and **B** and is in the direction that a right-handed
screw would travel if **v** were turned into **B** (see Section
3.1.3). When a moving charge, q, is placed in a space where
both an **E**-field and a **B**-field exist, the total force exerted
on the charge is given by the sum of Equations 3.8 and
3.10:

F= q(E+vxB) (Equation 3.11)

Equation 3.11 is called the Lorentz force equation.

- Time variation complicates the description of the fields.
- Static
**E**- and**B**-fields are independent of each other and can be treated separately, but time-varying**E**- and**B**-fields are coupled together and must be analyzed by simultaneous solution of equations.

(Equation 3.12)

(a)

(b)

Field lines between infinite parallel conducting plates. Solid lines are E-field lines. Dashed lines are equipotential surfaces.

(a)

(b)

Figure 3.11.

- Fields produced by a point charge
- Fields between two infinite parallel conducting plates

Potential scalar fields (a) for a point charge and (b) between infinite parallel conducting plates. Solid lines are E-field lines; dashed lines are equipotential surfaces.

- Polarization of bound charges
- Orientation of permanent dipoles
- Drift of conduction charges (both electronic and ionic)

(a) Polarization of bound charges. (b) Orientation of permanent dipoles.

(Equation 3.13)

where o is the permittivity of free space; ' - j", the *complex relative permittivity*; ', the real part of the complex relative permittivity (' is also called the *dielectric constant*); and ", the imaginary part of the complex relative permittivity. This notation is used when the time variation of the electromagnetic fields is described by e^{jwt}, where j = and is the radian frequency. Another common practice is to describe the time variation of the fields by e^{-iwt}, where i =. For this case complex permittivity is defined by * = _{o} (' + i"). " is related to the *effective conductivity* by

(Equation 3.14)

where is the effective conductivity, _{o} is the permittivity of free space, and

(Equation 3.15)is the radian frequency of the applied fields. The ' of a material is primarily a measure of the relative amount of polarization that occurs for a given applied

(Equation 3.16)The loss tangent usually varies with frequency. For example, the loss tangent of distilled water is about 0.040 at 1 MHz and 0.2650 at 25 GHz. Sometimes the loss factor is called the

(Equation 3.17)

where |**E**| is the root-mean-square (rms) magnitude of the **E**-field vector at that point inside the
material. If the peak value of the **E**-field vector is used, a factor of 1/2 must be included on the right-hand side of
Equation 3.17.' The rms and peak values are explained in Section 3.2.8. Unless otherwise noted, rms values are usually given. To find the total power absorbed by an object, the power density given by Equation 3.17 must be calculated at each point inside the body and summed (integrated) over the entire volume of the body. This is usually a very complicated calculation.

(Equation 3.18)

An important property of **D **is that its integral over any closed surface (that is, the total flux passing through the closed surface) is equal to the total free charge (not including polarization or conduction charge in materials)
inside the closed surface. This relationship is called Gauss's law. Figure 3.15 shows an example of this. The total flux passing out through the closed mathematical surface, S, is equal to the total charge, Q, inside S, regardless of what the permittivity of the spherical shell is. Electric-flux density is a convenient quantity because it is independent of the charges in materials.

Charge Q inside a dielectric spherical shell. S is a closed mathematical surface.

(Equation 3.19)

where ' - j" is the *complex relative permeability* and _{o} is the permeability of free space. For the general case, permeability is usually designated by .

(Equation 3.20)The magnetic-field intensity is a useful quantity because it is independent of magnetic currents in materials. The term "magnetic field" is often applied to both

(Equation 3.21) (Equation 3.22) (Equation 3.23) (Equation 3.24)

where

The other quantities have been defined previously.

**Go to Chapter 3.2.8**

**Return to Table of Contents.**

Last modified: June 14, 1997

© October 1986, USAF School of Aerospace Medicine, Aerospace Medical Division (AFSC), Brooks Air Force Base, TX 78235-5301