 # Radiofrequency Radiation Dosimetry Handbook

### (Fourth Edition) ## Chapter 3. Some Basics of Electromagnetics

A number of concepts are important to understanding any work that involves electromagnetic (EM) fields. The purpose of this chapter is to summarize the most important of these concepts as background for the specific applications described in this handbook. So that they can be understood by readers without an extensive background in electrical engineering or physics, the concepts are explained without complicated mathematical expressions where practical. This material is intended not to encompass all of EM theory but to provide a convenient summary.

### 3.1.1. Glossary

The following terms are used in this section and throughout this handbook. The list is more an explanation of terms than precise definitions. Boldface symbols indicate vector quantities (see Section 3.1.3 for an explanation of vectors and vector notation).

antenna: A structure that is designed to radiate or pick up electromagnetic fields efficiently. Individual antennas are often used in combinations called antenna arrays.

dielectric constant: Another name for relative permittivity.

electric dipole: Two equal charges of opposite sign separated by an infinitesimally small distance.

electric field: A term often used to mean the same as E-field intensity, or strength.

electric-field intensity: Another term for E-field strength.

electric-field strength: A vector-force field used to represent the forces between electric charges. E-field strength is defined as the vector force per unit charge on an infinitesimal charge at a given place in space.

electric-flux density (displacement): The electric flux passing through a surface, divided by the area of the surface. The total electric flux passing through a closed surface is equal to the total charge enclosed inside the surface, also equal to the E-field intensity times the permittivity.

electric polarization: Separation of charges in a material to form electric dipoles or alignment of existing electric dipoles in a material when an E-field is applied. Usually designated P, the units of polarization are dipole moments per cubic meter.

energy density: Electromagnetic energy in a given volume of space divided by the volume. The units are joules per cubic meter (J/m3).

far fields: Electromagnetic fields far enough away from the source producing them that the fields are approximately planewave in nature.

field: A correspondence between a set of points and a set of values. That is, a value is assigned to each of the points. If the value is a scalar, the field is a scalar field; if the value is a vector, the field is a vector field. The temperature at all points in a room is an example of a scalar field. The velocity of the air at all points in a room is an example of a vector field.

field point: A point at which the electric or magnetic field is being evaluated.

frequency: The time rate at which a quantity, such as electric field, oscillates. Frequency is equal to the number of cycles through which the quantity changes per second.

impedance, wave: The ratio of the electric field to magnetic field in a wave. For a planewave in free space, the wave impedance is 377 ohms. For a planewave in a material, the wave impedance is equal to 377 times the square root of the permeability divided by the square root of the permittivity.

magnetic field: A term often used to mean the same as magnetic-flux density, also commonly used to mean the same as magnetic-field intensity. The term has no clear definition or pattern of usage.

magnetic-field intensity: A vector field equal to the magnetic-flux density divided by the permeability. H is a useful designation because it is independent of the magnetization current in materials.

magnetic-flux density: A vector-force field used to describe the force on a moving charged particle, and perpendicular to the velocity of the particle. Magnetic-flux density is defined as the force per unit charge on an infinitesimal charge at a given point in space: F/q = v x B, where F is the vector force acting on the particle, q is the particle's charge, v is its velocity, and B is the magnetic-flux density.

near fields: Electromagnetic fields close enough to a source that the fields are not planewave in nature. Near fields usually vary more rapidly with space than far fields do.

nodes: Positions at which the amplitude is always zero in a standing wave.

permeability: A property of material that indicates how much magnetization occurs when a magnetic field is applied.

permittivity: A property of material that indicates how much polarization occurs when an electric field is applied. Complex permittivity is a property that describes both polarization and absorption of energy. The real part is related to polarization; the imaginary part, to energy absorption.

planewave: A wave in which the wave fronts are planar. The E and H vectors are uniform in the planes of the wave fronts; and E, H, and the direction of propagation (k) are all mutually perpendicular.

polarization: Orientation of the incident E- and H-field vectors with respect to the absorbing object.

Poynting vector: A vector equal to the cross product of E and H. The Poynting vector represents the instantaneous power transmitted through a surface per unit surface area. it is usually designated as S, is also known as energy-flux (power) density, and has units of watts per square meter
(W/m2 ).

propagation constant: A quantity that describes the propagation of a wave. Usually designated k, it is equal to the radian frequency divided by the phase velocity, and has units of per meter (m-1 ). A complex propagation constant describes both propagation and attenuation. The real part describes attenuation; the imaginary part, propagation.

radian frequency: Number of radians per second at which a quantity is oscillating. The radian frequency is equal to 2 f, where f is the frequency.

radiation: Electromagnetic fields emitted by a source.

reflection coefficient: Ratio of reflected-wave magnitude to incident-wave magnitude.

relative permittivity: Permittivity of a material divided by the permittivity of free space.

scalar field: See field.

specific absorption rate (SAR): Time rate of energy absorbed in an incremental mass, divided by that mass. Average SAR in a body is the time rate of the total energy absorbed divided by the total mass of the body. The units are watts per kilogram (W/kg).

spherical wave: A wave in which the wave fronts are spheres. An idealized point source radiates spherical waves.

standing wave: The wave pattern that results from two waves of the same frequency and amplitude propagating in opposite directions. Destructive interference produces nodes at regularly spaced positions.

standing-wave ratio: Ratio of Emax to Emin where E max is the maximum value, and Emin the minimum, of the magnitude of the E-field intensity anywhere along the path of the wave. A similar definition holds for other quantities that have wave properties.

vector: A quantity having both a magnitude and a direction. Velocity is an example of a vector: Direction of motion is the direction of the velocity vector, and speed is its magnitude.

vector field: See field.

velocity of propagation: Velocity at which a wave propagates. Units are meters per second (m/s). It is equal to how far one point on the wave, such as the crest or trough, travels in 1s.

wave impedance: (See impedance, wave).

wave length: The distance between two crests of the wave (or between two troughs or other corresponding points). Units are meters (m).

### 3.1.2 Measurement Units

The SI system of units was adopted by the Eleventh General Conference on Weights and Measures, held in Paris in 1970. SI is an internationally agreed-upon abbreviation for Système International d'Unitès (International System of Units). Some units we use are listed in Tables 3.1 and 3.2.

### Table 3.1 The SI Basic Units

 Quantity Unit Symbol Length meter m Mass kilogram kg Time second s Electric current ampere A Temperature Kelvin K Luminous intensity candela cd

### 3.1.3 Vectors and Fields

Vector Algebra--Vectors are used extensively in descriptions of electric and magnetic fields, so in this section we briefly explain vectors and vector notation. A scalar is a quantity that has only a magnitude; in contrast, a vector is a quantity that has a direction and a magnitude. A familiar example of a vector quantity is velocity of a particle. The direction of movement of the particle is the vector's direction, and the speed of the particle is the vector's magnitude. Vectors are represented graphically by directed line segments, as illustrated in Figure 3.1. The length of the line represents the vector's magnitude, and the direction of the line represents its direction.

#### Figure 3.1. A vector quantity represented by a directed line segment.

In this handbook, vectors are represented by boldface type; e.g., A. The magnitude of a vector is represented by the same symbol in plain type; thus A is the magnitude of vector A.

A summary of vector calculus, or even vector algebra, is beyond the scope of this handbook, but we will describe the basic vector addition and multiplication operations because they are important in understanding electromagnetic-field characteristics described later. Because vectors have the two properties, magnitude and direction, algebraic vector operations are more complicated than algebraic scalar operations.

Addition of any two vectors A and B is defined as

```				A + B = C		(Equation 3.1)
```

where C is the vector along the parallelogram shown in Figure 3.2. The negative of a vector A is defined as a vector having the same magnitude as A but opposite direction. Subtraction of any two vectors A and B is defined as

`			A - B = A + (-B)		(Equation 3.2)`

where -B is the negative of B.

#### Figure 3.2. Vector addition.

There are two kinds of vector multiplication. One is called the vector dot product. If A and B are any two vectors, their vector dot product is defined as
`			A · B = A B cos (Equation 3.3)`

where is the angle between A and B, as shown in Figure 3.3. The dot product of two vectors is a scalar. As indicated in Figure 3.3, A ·B is also equal to the projection of A on B, times B. This interpretation is often very useful. When two vectors are perpendicular, their dot product is zero because the cosine of 90° is zero (the projection of one along the other is zero).

#### Figure 3.3. Vector dot product A · B

The other kind of vector multiplication is called the vector cross product and is defined as

```			A x B = C		  (Equation 3.4)
```

where C is a vector whose direction is perpendicular to both A and B and whose magnitude is given by

`			C = A B sin (Equation 3.5)`

As shown in Figure 3.4, the direction of C is the direction a right-handed screw would travel if turned in the direction of A turned into B. The cross product of two parallel vectors is always zero because the sine of zero is zero.

#### Figure 3.4. Vector cross product A x B.

Fields--Two kinds of fields are used extensively in electromagnetic field theory, scalar fields and vector fields. A field is a correspondence between a set of points and a set of values; that is, in a set of points a value is assigned to each point. When the value assigned is a scalar, the field is called a scalar field. Temperature at all points in a room is an example of a scalar field. When the value assigned to each point is a vector, the field is called a vector field. Air velocity at all points in a room is an example of a vector field. Electric potential is a scalar field. Electric and magnetic fields are vector fields.

Scalar fields are usually represented graphically by connecting points of equal value by lines, as illustrated in Figure 3.5. In a temperature field, these lines are called isotherms. In a potential field, the lines are called equipotential lines. In the general three-dimensional field, points of equal potential form equipotential surfaces.

#### Figure 3.5. Graphical representation of a scalar field, such as temperature.Each line represents all points of equal value.

Vector fields are more difficult to represent graphically because both the magnitude and direction of the vector values must be represented. This is done by drawing lines tangent to the direction of the vector field at each point, with arrowheads showing the direction of the vector. The magnitude of the field is represented by the spacing between the lines. When the lines are far apart, the magnitude is small. An example of air velocity for air flowing between two plates is shown in Figure 3.6. Since many vector fields represent a physical flow of particles, such as fluid velocity, the field lines often represent a flux density. Hence, the field lines have come to be called flux lines, even for fields like electric and magnetic fields that do not represent a flow of particles, and fields are said to be a flux density. In electromagnetic-field theory, the flux passing through a surface is often calculated by finding the component of the flux density normal to the surface and integrating (summing) it over the surface.

#### Figure 3.6. Graphical representation of a vector field, such as air velocity between two plates. Go to Chapter 3.2

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Last modified: November 12, 1996
© October 1986, USAF School of Aerospace Medicine, Aerospace Medical Division (AFSC), Brooks Air Force Base, TX 78235-5301