## (How Radio Waves Are Born)

Revised September 8, 2003 (E and H field depiction made in-phase in time dimension due to visitor challenge and verification.)
Revised November 20, 2013 (review notice).
Revised November 25, 2013 (Additional information about the reactive Near field at bottom.)
Revised December 3, 2013 (Addition of Faraday's Law in differential form to prove Eand B fields are in-phase.)
Revised December 11, 2013 (Addition of link to MIT study guide chapter 13)
Revised December 16, 2013 (Addition of Wave Illustration as seen from the YZ plane)
Revised December 26, 2013 (Modification of explanation that CURL, not amplitude of one field is proportional to change in other.)
Revised January 25, 2014 (Change "Near Field has faded away (due to 1/r product).." to "Near Field has faded away (due to
1/r4 product)..."
High School level explanation video

## Contents

 A changing magnetic field will create an electric field in space A changing electric field will create a magnetic field in space Definition of radiation and propagation: Explain the transition point between conduction fields around the antenna or radiator to fields in space Show how radio waves propagate via oscillating magnetic fields in space. Radiation Resistance (electronic "friction" and "bootstrap" forces) Bibliography Related Links

A changing magnetic field will
create an electric field in space.

 The whole is the summation of its parts: The voltage, which is induced in the loop does not arise at any single point in the loop, but, rather is the summation of all the infinitesimal bits of voltage that are arising around the loop, distributed along its contour. If the total distance around the contour of the loop is s, the differential element of length along the contour is said to be ds. Hence, the total voltage induced in the loop will be the dot product , summed or integrated once around the contour of the loop. The total voltage around the loop is the sum of all the infinitesimal pieces of voltage around the contour of the loop. Each bit of voltage is represented by the dot product of the electric field intensity volts/meter and the segment length. (The field component, which is tangent to the loop at that point.) Eq. 1

Now, we find the relationship of the total voltage to the magnetic flux density B.

The voltage induced in a loop of wire is directly proportional to the rate of change of the magnetic flux which links or passes through that loop. In other words, it is the time derivative of the flux linkage. Eq. 2

 Since, according to Faraday's law, this total voltage is solely dependent on the rate of change of the total flux , passing through the loop, it is important to know how the total flux is related to the density of the magnetic flux. The differential flux is the dot product of the flux density with the unit vector normal (perpendicular to) the differential area, which says that it is the component of flux which is passing perpendicularly through the element area da. Eq. 3

 The total flux can now be expressed in terms of the flux density as follows: To obtain the total flux passing through the contour area, we sum the differential pieces by integrating both sides of the equation. Eq. 4

 Recalling Faraday's law, stating that the voltage induced in the loop is proportional to the rate of change of the flux, we can differentiate the above integral with respect to time to get the voltage as related to the flux. Eq. 5

 Getting back to Faraday's law, if we were to hold up a ring of wire in front of us and have an expanding (increasing positively) magnetic flux passing through it, but away from us, the induced voltage would cause a current flow in a counterclockwise direction. The counterclockwise current flow, is contrary to the right-hand screw rule, therefore we consider it a negative flow. Following this, Faraday's law equation is rewritten as follows: Eq. 6
 Now, substituting each side of Eq. 6 with Eq. 1 on the left side and Eq. 5 on the right side, yields the first of the field equations as shown in Eq. 7. Eq. 7

 The circle added to the integral symbol indicates that we are integrating around the entire closed contour. This provides a means of computing the induced voltage in a loop of arbitrary size and shape with any distribution of magnetic flux within the enclosed area. The most important implication of this equation is that it relates an electric field with a changing magnetic flux without considering that any current carrying wire is present. In other words, electric fields can exist in space without a conductor, allowing for energy to be transferred from one place to another without a physical conductive connection.

 Correction from previous statement here: This statement says that the induced voltage in a closed loop, which is the CURL in space, is proportional to the change in magnetic flux with respect to time. The CURL actually represents the electric field at a point of maximum change in space. This puts the two orthogonal waves IN PHASE, because the maximum CURL (minimum amplitude of E), occurs at a point of minimum amplitude of the magnetic field. The relationship of the CURL to a line integral of a closed path is proven by Green's Theorem. A full understanding of Green's theorem, line integrals and curls are subjects of higher level of Calculus. If one desires a better understanding of these concepts I would recommend the sequence of lessons given online by Sal Khan of Khan's Academy. If you already know two dimensional calculus and vector operations, such as the cross product and dot product, you should be able to extend your knowledge in a relatively short period of time. Khan Academy sequence of line integrals and Green's Theorem can be found at the following link: https://www.khanacademy.org/math/calculus/line_integrals_topic Or by searching YouTube

## A changing electric field will create a magnetic field in space.

### Biot-Savart law -

If a long, straight conductor carrying a current, it will cause a magnetic field of intensity H to surround it at any radial distance r, hence the Biot-Savart Law equation: Eq. 8

 We can obtain the magnetomotive force with a relation that is analogous to that used to obtain the electromotive force as expressed in Eq. 2. The total magnitomotive force is the sum of the components of magnetic field strength tangent to the contour. Eq. 9

 If we assume that the conductor is long and straight and passes perpendicularly through the center of the contour, the solution simplifies to: Eq. 10

 Substituting equation 8 and the circumference of a circle for H and s respectively give us: Eq. 11

Note that the radius has been canceled in the equation, which means that the magnetomotive force around the contour is independent of the radius or, for that matter, even the shape of the contour. A circle is the special case where r is constant in a curve expressed in polar coordinates and that a non-circle is when r is a function of the angle. So, whether r is constant or variable with the angle, the total magnetomotive force is dependent only on the current through the conductor.

This equation tells us that a current passing through a conductor results in a magnetic field and that the strength of the magnetic field is proportional to the current. It follows that if the magnitude of the current were varying in any way, the resultant magnetic field would also vary. So, if the current were alternating in a sinusoidal fashion at some frequency, the magnetic field would also be caused to alternate at the same frequency.

So far we have shown that an alternating current through a wire or conductor results in alternating magnetic and electric fields around the conductor. So far, we have not shown that these fields radiate or "leave" the conductor on a distant journey in any way nor do they "snap," "jump," or "hop" off the antenna as many texts would have you believe.

In order for radiation to be able to occur, we have to prove that magnetic and electric fields can exist without the presence of a nearby current and that these fields will "move" through space. We will see the idea of movement of these fields through space as radio waves is really an abstraction. What really happens is more of a domino effect, which is a demonstration of "propagation."

In order to begin to understand the transition between the fields we have just described, which are referred to as conduction fields and fields that become free of the wire we must find some way to relate what happens in a conductor to what happens in space. We will start off by stating Kirchoff's law.

### Kirchoff's Law -

The sum of all currents entering and leaving a node or branch point in a circuit must equal zero as illustrated by figure 1.. Fig. 1

 Expanding this to an enclosed volume. If the dashed line in Fig. 1 represents an enclosed volume then, the sum of all the currents entering the boundary of the volume must equal the sum of all the currents passing out of the volume. Mathematically speaking, the integral of the current density over the entire volume regardless of shape or size equals zero. Current cannot be dead ended inside of a volume. What goes in, must come out. Let us now enclose one of the plates of a capacitor with a volume as shown in Fig. 2. Fig. 2

 The boundary of the volume passes between the plates. The arrows indicate a current entering the left capacitor lead and exiting the right capacitor lead. Since there is no conductive connection between the plates, how can we apply Kirchoff's law if current cannot actually flow out of the volume enclosed by the dotted line to get to the right capacitor plate? The answer lies in the idea of displacement current.

### Displacement Current -

First, we introduce dielectric displacement. The dielectric displacement, designated by the letter D, represents the electrical strain which occurs in a dielectric medium, when an electric field is present. D is analogous to magnetic flux density B, hence D is really the electric flux density. It is related to the electric field by , which is the permittivity or dielectric constant of the material between the plates of a capacitor. The magnetic flux density B is related to the magnetic field strength H by or the permeability of the material within the magnetic field. Eq. 12a and 12b

 To explain what is happening between the plates of the capacitor, we first realize that as current flows into the capacitor, an electric charge accumulates on the plates with excess electrons on one plate and a dearth of them on the other. As the charge builds up, an electric E field between the plates, builds up causing the dielectric displacement or electric field density to also increase. In order to satisfy Kirchoff's law, there must be some sort of current flowing in the dielectric, proportional to the rate of change of dielectric displacement. This current has been given the name displacement current and we can think of it as a flowing-in (displacement) of additional flux as required as the electric flux density increases. This total displacement current is equal to the conduction current flowing into and out of the capacitor and, therefore, out of the volume with which we have surrounded the plate. As Maxwell was aware of the displacement current, he had to decide whether it was consistent with conduction current in that it would also have a magnetic field associated with it. Maxwell, then made the assumption that there was, indeed a magnetic field associated with the displacement current. In considering this, he added an additional term into the of Ampere's Rule and the Biot-Savart equation to account for it. Equation 13 shows this new relationship. Eq. 13

 This equation takes into consideration that the area within the contour of H . ds integration might contain both conduction and displacement currents which would both contribute to the intensity of a magnetic field. D . n da is the differential electric flux across the capacitor space normal to a given point and the integral sums it up to the total flux. The rate of change of the total flux is the displacement current flowing through the contour.

 We have finally arrived at the two principal field equations of Maxwell. But, these equations describe the behavior of the two inerleaving fields close to the antenna, which is the near field. This still does not account for the fields "launched" into space for an eternal trip, disconnected from the origin or source. A changing magnetic field normal to a closed surface is proportional to the curl of the electric field around a closed loop. A changing electric field normal to a closed surface is proportional to the curl of the magnetic field, around a closed loop. Eq. 14 and 15 Maxwell's free space equations.

 If we make the appropriate substitutions from Eqs. 12a and 12b to Eq. 14 and 15, we have: Where Mu is the permiability of free space (it should be Mu Naught). Where Epsilon is the permittivity of free space (it should be Epsilon Naught). Eq. 16 and 17 Maxwell's free space induction equations. Faraday's Law in integral form.

E and B relationships can easily be satisfied with the differential
form of the Maxwell Faraday Equation. Differential form of Faraday's Law relating E to B fields in space. With: E field amplitude in x along Z axis B field amplitude in y along Z axis Both equations are expressed as vectors along the unit vectors i, j, k axes. Plugging E and B into Faraday's Law, the solutions is true for the phase difference of zero and k = w. Eq. 18, 19 and 20: Maxwell's free space equations. Faraday's Law says that the curl of the induced electric field is equal to the time rate of change of the magnetic field. The two fields are physically orthogonal or perpendicular in space and they are in phase with each other. OR - The spatial variation of the electric field, E, gives rise to a time-varying magnetic field, B, and visa-versa. Professor Ramamurti Shanker of Yale University proves out the radiated wave using the differential form of Maxwell's Equation which is Faraday's Law. See embedded YouTube video below. The ratio of the magnetude of the E field to the B field is equal to the speed of light 'c'. That is, the magnetude of the magnetic wave is only 3*(10-8) of the amplitude of the electric wave! (Still more details and a full derivation to come!) See Also: 1) Wave Equation Derivation by Lynda Williams, Santa Rosa Junior College Physics Department 2) Maxwell's Equations and Electromagnetic Waves - MIT study guide chapter 13

 These equations are the key to electromagnetic radiation! Most important is the fact that no conduction current need exist and, therefore, no physical conductor.

 Three Fields coexist at the antenna or radiator: There are three fields associated with the antenna. The static field, the Near Field (reactive or stored), and the Far Field (the radiated portion) as a result of acceleration and deceleration of charges. The Near Field (stored) can be stated as follows: If there is an alternating current in a conductor, an alternating magnetic field will be created surrounding the wire. The alternating magnetic field due to the current in the wire will create an alternating electric field in space further out from the conductor. The first transition from conduction fields to near fields has been made. Now, carrying it further, the alternating electric field, will create a magnetic (due to the corresponding displacement current in space) field further away from the conductor (according to Eq.17). The alternating magnetic field will then create another alternating electric field (according to Eq. 16). This process, which continues on away from the conductor is called electromagnetic wave propagation. Transition or Fresnel Region: The transition region is the region where the Near Field Power (inversely proportional to the cube of the distance of the radiator), tapers off, leaving the more sustainable Far Field Power (inversely proportional to the square of the distance from the radiator. The Far Field (radiated field) is stated as follows: It is tempting to believe that the radiated wave that continues away from the antenna is a continuation of the Near Field pattern of change in E begets maximum H and change in H beget maximum E as derivatives of each other. But, in order to radiate power, we must have both an electric E field and H field in phase. Just like the initial electric field E in space, a magnetic field H is created from the acceleration of electric field intensity or displacement current at the surface of the conducting wire. Both E and H fields, perpendicular to each other sustain associated H and E fields. The combination of these, in phase fields give rise to what is called the Poynting vector, which is perpendicular to both E and H fields and in the direction of propagation. The Poynting vector represents the actual power propagated in space. The magnitude of the Poynting vector is equal to the cross product of the E and H fields as in: P = E x H. It is measured in Watts per square meter and is, therefore, a power density. If still convinced of the 90 degree time phase shift, then please consider this: As one type of field changes as in the Near field, it generates the other field in the the direction we are assuming that the waves are propagating in. But, this change also affects the space in the other direction, so you have one type of field generating the other type in BOTH or OPPOSING DIRECTIONS! This is, in fact, very true in the Near Field, which is why the wave does not continue forever in one direction. It is the condition known as STANDING WAVES, which exist between the radiator and the transition region. This is a reactive wave and is also what happens in a transmission line when reactance is not tuned out. The induction E and H fields close to the conductor
are stored and, therefore, not radiated. They are separated
by a 900 time phase with each other. When the fields collapse,
energy is returned to the system.

a plane in space) are IN PHASE because they are
delivering power to space. Energy is lost from the
radiating system in this way and appears as a
H = Ho sin(kx - wt)
E = Eo cos(kx - wt)
This graphic illustration is seen everywhere, but it is misleading. It represents the MAGNITUDE of the E and H fields, but it does not
represent the actual vectors of the fields. Remember that magnetic fields wrap around moving E fields. The time derivative of the H field is
proportional to the CURL of the E field. More changes to this page are coming. Eq. 16 and 17 are misleading for radiation in free
space, far away from the radiating antenna. The figure on the left shows two representations of the propagated E and H fields. The top diagram depicts the magnetude, it's variation and phase relationship between E and H fields. It is not a true picture of the lines of flux. The lower diagram shows the E field as arrows pointing up and down and the H field as dots pointing out of the screen and crosses representing the lines into the screen. It's not a perfect reprentation, but a more accurate representation of the field lines in space. The propagation moves smoothly through space with each field perpendicular and in phase with each other. If the field of each was generated in proportion to the derivative of the other, the propagation would jump in 90 degree steps, which is not the case. This diagram was scanned from "Electromagnetism - Principles and Applications" Second Edition, sixth printing 1997 by Paul Lorrain and Dale R. Corson, Pg. 446.



Got an Hour? Professor Ramamurti Shanker of Yale University proves out the radiated wave using the differential form of Maxwell's Equation which is Faraday's Law, showing Radiated Wave fields are time in-phase. Part I

Professor Ramamurti Shanker of Yale University proves out the radiated wave using the differential form of Maxwell's Equation which is Faraday's Law, showing Radiated Wave fields are time in-phase. Part II

High School level explanation from New Jersey Center for Teaching and Learning Time 6:38

Michel van Biezen Series Physics - "E&M: E&M Radiation (1 of 22) What is Electromagnetic Radiation?" This is probably the best and most easy to understand treatment of the subject I have seen to date. It is a series of 22 videos which do a very thorough job in most aspects. These videos show the connection between the fields discussed here and the antenna that radiates them.

Electromagnetism - Maxwell's Laws A tutorial using the beautiful animations of Eugene Khutoryansky Narrated by Kira Vincent

### Radiation Resistance (electronic "friction") -

When the voltage and current are observed at the terminals of a radiating conductor or antenna, one component of alternating current is 90 degrees out of phase with the alternating voltage and another component is in phase with the alternating voltage. The component of current which is 90 degrees out of phase behaves like and, therefore is considered a reactive component. No power transfer or loss takes place due to the reactive component. The portion of current that is in phase with the alternating voltage is considered current due to power transfer. A portion of the power transferred or lost is due to the ohmic resistance of the antenna or radiator. It will be found, however, that there is an additional amount of power transferred, which the ohmic resistance can not be held accountable for. This power transfer is due to radiation. As far as the power source or transmitter is concerned, all power transfer appears as a total resistance and can be treated as so at the antenna feed point.

The energy lost can be thought of as being due to a "drag" or force acting against the motion of electrons which carry the electric charge. Part of this "drag" is due to energy transferred in collisions between atoms and the moving electrons, setting the atoms in motion, manifesting itself in the form of radiated heat. How, then is a "drag" felt, due to radiation?

In the American Radio Relay League article, "Why an Antenna Radiates" by Dr. Kenneth MacLeish, he describes the notion of Bootstrap Forces. Since the electric fields exist down to the surface of the electron and there is a force against charged as they accelerate through a magnetic field, the electron experiences a force or "drag" as it accelerates through it's own field! That portion of drag, due to the dynamic or radiated electric field is the force of radiation resistance. That is, the electron is pulled by it's own bootstraps! Part of the fields which drag the electron, collapse back into the conductor as it decelerates, thus returning energy to the electron in the opposite direction. This returned force is the back-emf due to inductance, which is part of the reactive component of the antenna impedance.

### CFAs

A Cross-field Antenna (CFA) works on the principle of generating separate E and H fields in phase and reducing the wasteful induction field
which is the reactive field with E and H 90 out of phase.

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