Electromagnetic pulse

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Electromagnetic_pulse

An electromagnetic pulse (EMP), also referred to as a transient electromagnetic disturbance (TED), is a brief burst of electromagnetic energy. The origin of an EMP can be natural or artificial, and can occur as an electromagnetic field, as an electric field, as a magnetic field, or as a conducted electric current. The electromagnetic interference caused by an EMP can disrupt communications and damage electronic equipment. An EMP such as a lightning strike can physically damage objects such as buildings and aircraft. The management of EMP effects is a branch of electromagnetic compatibility (EMC) engineering.

The first recorded damage from an electromagnetic pulse came with the solar storm of August 1859, or the Carrington Event.

In modern warfare, weapons delivering a high energy EMP are designed to disrupt communications equipment, computers needed to operate modern warplanes, or even put the entire electrical network of a target country out of commission.

General characteristics

An electromagnetic pulse is a short surge of electromagnetic energy. Its short duration means that it will be spread over a range of frequencies. Pulses are typically characterized by:

• The mode of energy transfer (radiated, electric, magnetic or conducted).
• The range or spectrum of frequencies present.
• Pulse waveform: shape, duration and amplitude.

The frequency spectrum and the pulse waveform are interrelated via the Fourier transform which describes how component waveforms may sum to the observed frequency spectrum.

Types of energy

Main article: Electromagnetism

EMP energy may be transferred in any of four forms:

• Electric field
• Magnetic field
• Electromagnetic radiation
• Electrical conduction

According to Maxwell's equations, a pulse of electric energy will always be accompanied by a pulse of magnetic energy. In a typical pulse, either the electric or the magnetic form will dominate. It can be shown that the non-linear Maxwell's equations can have time-dependent self-similar electromagnetic shock wave solutions where the electric and the magnetic field components have a discontinuity.

In general, only radiation acts over long distances, with the magnetic and electric fields acting over short distances. There are a few exceptions, such as a solar magnetic flare.

Frequency ranges

A pulse of electromagnetic energy typically comprises many frequencies from very low to some upper limit depending on the source. The range defined as EMP, sometimes referred to as "DC [direct current] to daylight", excludes the highest frequencies comprising the optical (infrared, visible, ultraviolet) and ionizing (X and gamma rays) ranges.

Some types of EMP events can leave an optical trail, such as lightning and sparks, but these are side effects of the current flow through the air and are not part of the EMP itself.

Pulse waveforms

The waveform of a pulse describes how its instantaneous amplitude (field strength or current) changes over time. Real pulses tend to be quite complicated, so simplified models are often used. Such a model is typically described either in a diagram or as a mathematical equation.

Rectangular pulse

Double exponential pulse

Damped sinewave pulse

Most electromagnetic pulses have a very sharp leading edge, building up quickly to their maximum level. The classic model is a double-exponential curve which climbs steeply, quickly reaches a peak and then decays more slowly. However, pulses from a controlled switching circuit often approximate the form of a rectangular or "square" pulse.

EMP events usually induce a corresponding signal in the surrounding environment or material. Coupling usually occurs most strongly over a relatively narrow frequency band, leading to a characteristic damped sine wave. Visually it is shown as a high frequency sine wave growing and decaying within the longer-lived envelope of the double-exponential curve. A damped sinewave typically has much lower energy and a narrower frequency spread than the original pulse, due to the transfer characteristic of the coupling mode. In practice, EMP test equipment often injects these damped sinewaves directly rather than attempting to recreate the high-energy threat pulses.

In a pulse train, such as from a digital clock circuit, the waveform is repeated at regular intervals. A single complete pulse cycle is sufficient to characterise such a regular, repetitive train.

Types

An EMP arises where the source emits a short-duration pulse of energy. The energy is usually broadband by nature, although it often excites a relatively narrow-band damped sine wave response in the surrounding environment. Some types are generated as repetitive and regular pulse trains.

Different types of EMP arise from natural, man-made, and weapons effects.

Types of natural EMP events include:

• Lightning electromagnetic pulse (LEMP). The discharge is typically an initial current flow of perhaps millions of amps, followed by a train of pulses of decreasing energy.
• Electrostatic discharge (ESD), as a result of two charged objects coming into proximity or even contact.
• Meteoric EMP. The discharge of electromagnetic energy resulting from either the impact of a meteoroid with a spacecraft or the explosive breakup of a meteoroid passing through the Earth's atmosphere.
• Coronal mass ejection (CME), sometimes referred to as a solar EMP. A burst of plasma and accompanying magnetic field, ejected from the solar corona and released into the solar wind.

Types of (civil) man-made EMP events include:

• Switching action of electrical circuitry, whether isolated or repetitive (as a pulse train).
• Electric motors can create a train of pulses as the internal electrical contacts make and break connections as the armature rotates.
• Gasoline engine ignition systems can create a train of pulses as the spark plugs are energized or fired.
• Continual switching actions of digital electronic circuitry.
• Power line surges. These can be up to several kilovolts, enough to damage electronic equipment that is insufficiently protected.

Types of military EMP include:

• Nuclear electromagnetic pulse (NEMP), as a result of a nuclear explosion. A variant of this is the high altitude nuclear EMP (HEMP), which produces a secondary pulse due to particle interactions with the Earth's atmosphere and magnetic field.
• Non-nuclear electromagnetic pulse (NNEMP) weapons.

Lightning electromagnetic pulse (LEMP)

Main article: Lightning

Lightning is unusual in that it typically has a preliminary "leader" discharge of low energy building up to the main pulse, which in turn may be followed at intervals by several smaller bursts.

Electrostatic discharge (ESD)

Main article: Electrostatic discharge

ESD events are characterized by high voltages of many kV, but small currents sometimes cause visible sparks. ESD is treated as a small, localized phenomenon, although technically a lightning flash is a very large ESD event. ESD can also be man-made, as in the shock received from a Van de Graaff generator.

An ESD event can damage electronic circuitry by injecting a high-voltage pulse, besides giving people an unpleasant shock. Such an ESD event can also create sparks, which may in turn ignite fires or fuel-vapour explosions. For this reason, before refueling an aircraft or exposing any fuel vapor to the air, the fuel nozzle is first connected to the aircraft to safely discharge any static.

Switching pulses

The switching action of an electrical circuit creates a sharp change in the flow of electricity. This sharp change is a form of EMP.

Simple electrical sources include inductive loads such as relays, solenoids, and brush contacts in electric motors. These typically send a pulse down any electrical connections present, as well as radiating a pulse of energy. The amplitude is usually small and the signal may be treated as "noise" or "interference". The switching off or "opening" of a circuit causes an abrupt change in the current flowing. This can in turn cause a large pulse in the electric field across the open contacts, causing arcing and damage. It is often necessary to incorporate design features to limit such effects.

Electronic devices such as vacuum tubes or valves, transistors, and diodes can also switch on and off very quickly, causing similar issues. One-off pulses may be caused by solid-state switches and other devices used only occasionally. However, the many millions of transistors in a modern computer may switch repeatedly at frequencies above 1  GHz, causing interference that appears to be continuous.

Nuclear electromagnetic pulse (NEMP)

Main article: Nuclear electromagnetic pulse

A nuclear electromagnetic pulse is the abrupt pulse of electromagnetic radiation resulting from a nuclear explosion. The resulting rapidly changing electric fields and magnetic fields may couple with electrical/electronic systems to produce damaging current and voltage surges.

The intense gamma radiation emitted can also ionize the surrounding air, creating a secondary EMP as the atoms of air first lose their electrons and then regain them.

NEMP weapons are designed to maximize such EMP effects as the primary damage mechanism, and some are capable of destroying susceptible electronic equipment over a wide area.

A high-altitude electromagnetic pulse (HEMP) weapon is a NEMP warhead designed to be detonated far above the Earth's surface. The explosion releases a blast of gamma rays into the mid-stratosphere, which ionizes as a secondary effect and the resultant energetic free electrons interact with the Earth's magnetic field to produce a much stronger EMP than is normally produced in the denser air at lower altitudes.

Non-nuclear electromagnetic pulse (NNEMP)

Non-nuclear electromagnetic pulse (NNEMP) is a weapon-generated electromagnetic pulse without use of nuclear technology. Devices that can achieve this objective include a large low-inductance capacitor bank discharged into a single-loop antenna, a microwave generator, and an explosively pumped flux compression generator. To achieve the frequency characteristics of the pulse needed for optimal coupling into the target, wave-shaping circuits or microwave generators are added between the pulse source and the antenna. Vircators are vacuum tubes that are particularly suitable for microwave conversion of high-energy pulses.

NNEMP generators can be carried as a payload of bombs, cruise missiles (such as the CHAMP missile) and drones, with diminished mechanical, thermal and ionizing radiation effects, but without the consequences of deploying nuclear weapons.

The range of NNEMP weapons is much less than nuclear EMP. Nearly all NNEMP devices used as weapons require chemical explosives as their initial energy source, producing only one millionth the energy of nuclear explosives of similar weight. The electromagnetic pulse from NNEMP weapons must come from within the weapon, while nuclear weapons generate EMP as a secondary effect. These facts limit the range of NNEMP weapons, but allow finer target discrimination. The effect of small e-bombs has proven to be sufficient for certain terrorist or military operations. Examples of such operations include the destruction of electronic control systems critical to the operation of many ground vehicles and aircraft.

The concept of the explosively pumped flux compression generator for generating a non-nuclear electromagnetic pulse was conceived as early as 1951 by Andrei Sakharov in the Soviet Union, but nations kept work on non-nuclear EMP classified until similar ideas emerged in other nations.

Effects

Minor EMP events, and especially pulse trains, cause low levels of electrical noise or interference which can affect the operation of susceptible devices. For example, a common problem in the mid-twentieth century was interference emitted by the ignition systems of gasoline engines, which caused radio sets to crackle and TV sets to show stripes on the screen. CISPR 25 was established to set threshold standards that vehicles must meet for electromagnetic interference(EMI) emissions.

A demonstration of how Electromagnetic Radiation powers (and destroys) circuits.

At a high voltage level an EMP can induce a spark, for example from an electrostatic discharge when fuelling a gasoline-engined vehicle. Such sparks have been known to cause fuel-air explosions and precautions must be taken to prevent them.

A large and energetic EMP can induce high currents and voltages in the victim unit, temporarily disrupting its function or even permanently damaging it.

A powerful EMP can also directly affect magnetic materials and corrupt the data stored on media such as magnetic tape and computer hard drives. Hard drives are usually shielded by heavy metal casings. Some IT asset disposal service providers and computer recyclers use a controlled EMP to wipe such magnetic media.

A very large EMP event, such as a lightning strike or an air bursted nuclear weapon, is also capable of damaging objects such as trees, buildings and aircraft directly, either through heating effects or the disruptive effects of the very large magnetic field generated by the current. An indirect effect can be electrical fires caused by heating. Most engineered structures and systems require some form of protection against lightning to be designed in. A good means of protection is a Faraday shield designed to protect certain items from being destroyed.

Control

Main article: Electromagnetic compatibility

EMP simulator HAGII-C testing a Boeing E-4 aircraft.

EMPRESS I (antennas along shoreline) with USS Estocin (FFG-15) moored in the foreground for testing.

Like any electromagnetic interference, the threat from EMP is subject to control measures. This is true whether the threat is natural or man-made.

Therefore, most control measures focus on the susceptibility of equipment to EMP effects, and hardening or protecting it from harm. Man-made sources, other than weapons, are also subject to control measures in order to limit the amount of pulse energy emitted.

The discipline of ensuring correct equipment operation in the presence of EMP and other RF threats is known as electromagnetic compatibility (EMC).

Test simulation

To test the effects of EMP on engineered systems and equipment, an EMP simulator may be used.

Induced pulse simulation

Induced pulses are of much lower energy than threat pulses and so are more practicable to create, but they are less predictable. A common test technique is to use a current clamp in reverse, to inject a range of damped sine wave signals into a cable connected to the equipment under test. The damped sine wave generator is able to reproduce the range of induced signals likely to occur.

Threat pulse simulation

Sometimes the threat pulse itself is simulated in a repeatable way. The pulse may be reproduced at low energy in order to characterise the subject's response prior to damped sinewave injection, or at high energy to recreate the actual threat conditions. A small-scale ESD simulator may be hand-held. Bench- or room-sized simulators come in a range of designs, depending on the type and level of threat to be generated.

At the top end of the scale, large outdoor test facilities incorporating high-energy EMP simulators have been built by several countries. The largest facilities are able to test whole vehicles including ships and aircraft for their susceptibility to EMP. Nearly all of these large EMP simulators used a specialized version of a Marx generator. Examples include the huge wooden-structured ATLAS-I simulator (also known as TRESTLE) at Sandia National Labs, New Mexico, which was at one time the world's largest EMP simulator. Papers on this and other large EMP simulators used by the United States during the latter part of the Cold War, along with more general information about electromagnetic pulses, are now in the care of the SUMMA Foundation, which is hosted at the University of New Mexico. The US Navy also has a large facility called the Electro Magnetic Pulse Radiation Environmental Simulator for Ships I (EMPRESS I).

Safety

High-level EMP signals can pose a threat to human safety. In such circumstances, direct contact with a live electrical conductor should be avoided. Where this occurs, such as when touching a Van de Graaff generator or other highly charged object, care must be taken to release the object and then discharge the body through a high resistance, in order to avoid the risk of a harmful shock pulse when stepping away.

Very high electric field strengths can cause breakdown of the air and a potentially lethal arc current similar to lightning to flow, but electric field strengths of up to 200 kV/m are regarded as safe.

According to research from Edd Gent, a 2019 report by the Electric Power Research Institute, which is funded by utility companies, found that a large EMP attack would probably cause regional blackouts but not a nationwide grid failure and that recovery times would be similar to those of other large-scale outages. It is not known how long these electrical blackouts would last, or what extent of damage would occur across the country. It is possible that neighboring countries of the U.S. could also be affected by such an attack, depending on the targeted area and people.

According to an article from Naureen Malik, with North Korea's increasingly successful missile and warhead tests in mind, Congress moved to renew funding for the Commission to Assess the Threat to the U.S. from Electromagnetic Pulse Attack as part of the National Defense Authorization Act.

According to research from Yoshida Reiji, in a 2016 article for the Tokyo-based nonprofit organization Center for Information and Security Trade Control, Onizuka warned that a high-altitude EMP attack would damage or destroy Japan's power, communications and transport systems as well as disable banks, hospitals and nuclear power plants.

In popular culture

By 1981, a number of articles on electromagnetic pulse in the popular press spread knowledge of the EMP phenomenon into the popular culture. EMP has been subsequently used in a wide variety of fiction and other aspects of popular culture. Popular media often depict EMP effects incorrectly, causing misunderstandings among the public and even professionals. Official efforts have been made in the U.S. to remedy these misconceptions.

The novel One Second After by William R. Forstchen and the following books One Year After, The Final Day and Five Years After portrait the story of a fictional character named John Matherson and his community in Black Mountain, North Carolina that after the US loses a war and an EMP attack "sends our nation [the US] back to the Dark Ages".

Ballistics

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Ballistics

Trajectories of three objects thrown at the same angle (70°).

  Stokes' drag

  Newtonian drag.

  No form of drag and moves along a parabola.

Ballistics is the field of mechanics concerned with the launching, flight behaviour and impact effects of projectiles, especially weapon munitions such as bullets, unguided bombs, rockets and the like; the science or art of designing and accelerating projectiles so as to achieve a desired performance.

A ballistic body is a free-moving body with momentum, which can be subject to forces such as those exerted by pressurized gases from a gun barrel or a propelling nozzle, normal force by rifling, and gravity and air drag during flight.

A ballistic missile is a missile that is guided only during the relatively brief initial phase of powered flight, with the trajectory subsequently governed by the laws of classical mechanics, in contrast to (for example) a cruise missile, which is aerodynamically guided in powered flight like a fixed-wing aircraft.

History and prehistory

The earliest known ballistic projectiles were stones, spears, and the throwing stick.

Gaetano Marzagaglia, Del calcolo balistico, 1748

The oldest evidence of stone-tipped projectiles, which may or may not have been propelled by a bow (cf. atlatl), dating to c. 280,000 years ago, were found in Ethiopia, present day East Africa. The oldest evidence of the use of bows to shoot arrows dates to about 10,000 years ago; it is based on pinewood arrows found in the Ahrensburg valley north of Hamburg. They had shallow grooves on the base, indicating that they were shot from a bow. The oldest bow so far recovered is about 8,000 years old, found in the Holmegård swamp in Denmark.

Archery seems to have arrived in the Americas with the Arctic small tool tradition, about 4,500 years ago.

The first devices identified as guns appeared in China around 1000 AD, and by the 12th century the technology was spreading through the rest of Asia, and into Europe by the 13th century.

After millennia of empirical development, the discipline of ballistics was initially studied and developed by Italian mathematician Niccolò Tartaglia in 1531, although he continued to use segments of straight-line motion, conventions established by the Greek philosopher Aristotle and Albert of Saxony, but with the innovation that he connected the straight lines by a circular arc. Galileo established the principle of compound motion in 1638, using the principle to derive the parabolic form of the ballistic trajectory. Ballistics was put on a solid scientific and mathematical basis by Isaac Newton, with the publication of Philosophiæ Naturalis Principia Mathematica in 1687. This gave mathematical laws of motion and gravity which for the first time made it possible to successfully predict trajectories.

The word ballistics comes from the Greek βάλλειν ballein, meaning "to throw".

Projectiles

Main article: Projectile

A projectile is any object projected into space (empty or not) by the exertion of a force. Although any object in motion through space (for example a thrown baseball) is a projectile, the term most commonly refers to a weapon. Mathematical equations of motion are used to analyze projectile trajectory.

Examples of projectiles include balls, arrows, bullets, artillery shells, wingless rockets, etc.

Projectile launchers

Throwing

Main article: Throwing

Baseball throws can exceed 100 mph.

Throwing is the launching of a projectile by hand. Although some other animals can throw, humans are unusually good throwers due to their high dexterity and good timing capabilities, and it is believed that this is an evolved trait. Evidence of human throwing dates back 2 million years. The 90 mph throwing speed found in many athletes far exceeds the speed at which chimpanzees can throw things, which is about 20 mph. This ability reflects the ability of the human shoulder muscles and tendons to store elasticity until it is needed to propel an object.

Sling

Main article: Sling (weapon)

A sling is a projectile weapon typically used to throw a blunt projectile such as a stone, clay or lead "sling-bullet".

A sling has a small cradle or pouch in the middle of two lengths of cord. The sling stone is placed in the pouch. The middle finger or thumb is placed through a loop on the end of one cord, and a tab at the end of the other cord is placed between the thumb and forefinger. The sling is swung in an arc, and the tab released at a precise moment. This frees the projectile to fly to the target.

Bow

Main article: Bow and arrow

A bow is a flexible piece of material which shoots aerodynamic projectiles called arrows. The arrow is perhaps the first lethal projectile ever described in discussion of ballistics. A string joins the two ends and when the string is drawn back, the ends of the stick are flexed. When the string is released, the potential energy of the flexed stick is transformed into the velocity of the arrow. Archery is the art or sport of shooting arrows from bows.

Catapult

Main article: Catapult

Catapult 1 Mercato San Severino

A catapult is a device used to launch a projectile a great distance without the aid of explosive devices – particularly various types of ancient and medieval siege engines. The catapult has been used since ancient times, because it was proven to be one of the most effective mechanisms during warfare. The word "catapult" comes from the Latin catapulta, which in turn comes from the Greek καταπέλτης (katapeltēs), itself from κατά (kata), "against” and πάλλω (pallō), "to toss, to hurl". Catapults were invented by the ancient Greeks.

Gun

Main article: Gun

USS Iowa (BB-61) fires a full broadside, 1984.

A gun is a normally tubular weapon or other device designed to discharge projectiles or other material. The projectile may be solid, liquid, gas, or energy and may be free, as with bullets and artillery shells, or captive as with Taser probes and whaling harpoons. The means of projection varies according to design but is usually effected by the action of gas pressure, either produced through the rapid combustion of a propellant or compressed and stored by mechanical means, operating on the projectile inside an open-ended tube in the fashion of a piston. The confined gas accelerates the movable projectile down the length of the tube imparting sufficient velocity to sustain the projectile's travel once the action of the gas ceases at the end of the tube or muzzle. Alternatively, acceleration via electromagnetic field generation may be employed in which case the tube may be dispensed with and a guide rail substituted.

A weapons engineer or armourer who applies the scientific principles of ballistics to design cartridges are often called a ballistician.

Rocket

Main article: Rocket

SpaceX's Falcon 9 Full Thrust rocket, 2017

A rocket is a missile, spacecraft, aircraft or other vehicle that obtains thrust from a rocket engine. Rocket engine exhaust is formed entirely from propellants carried within the rocket before use. Rocket engines work by action and reaction. Rocket engines push rockets forward simply by throwing their exhaust backwards extremely fast.

While comparatively inefficient for low speed use, rockets are relatively lightweight and powerful, capable of generating large accelerations and of attaining extremely high speeds with reasonable efficiency. Rockets are not reliant on the atmosphere and work very well in space.

Rockets for military and recreational uses date back to at least 13th century China. Significant scientific, interplanetary and industrial use did not occur until the 20th century, when rocketry was the enabling technology for the Space Age, including setting foot on the Moon. Rockets are now used for fireworks, weaponry, ejection seats, launch vehicles for artificial satellites, human spaceflight, and space exploration.

Chemical rockets are the most common type of high performance rocket and they typically create their exhaust by the combustion of rocket propellant. Chemical rockets store a large amount of energy in an easily released form, and can be very dangerous. However, careful design, testing, construction and use minimizes risks.

Subfields

Ballistics can be studied using high-speed photography or high-speed cameras. A photo of a Smith & Wesson revolver firing, taken with an ultra high speed air-gap flash. Using this sub-microsecond flash, the bullet can be imaged without motion blur.

Ballistics is often broken down into the following four categories:

• Internal ballistics the study of the processes originally accelerating projectiles
• Transition ballistics the study of projectiles as they transition to unpowered flight
• External ballistics the study of the passage of the projectile (the trajectory) in flight
• Terminal ballistics the study of the projectile and its effects as it ends its flight

Internal ballistics

Main article: Internal ballistics

Internal ballistics (also interior ballistics), a sub-field of ballistics, is the study of the propulsion of a projectile.

In guns, internal ballistics covers the time from the propellant's ignition until the projectile exits the gun barrel. The study of internal ballistics is important to designers and users of firearms of all types, from small-bore rifles and pistols, to high-tech artillery.

For rocket propelled projectiles, internal ballistics covers the period during which a rocket engine is providing thrust.

Transitional ballistics

Main article: Transitional ballistics

Transitional ballistics, also known as intermediate ballistics, is the study of a projectile's behavior from the time it leaves the muzzle until the pressure behind the projectile is equalized, so it lies between internal ballistics and external ballistics.

External ballistics

Main article: External ballistics

External ballistics is the part of the science of ballistics that deals with the behaviour of a non-powered projectile in flight.

External ballistics is frequently associated with firearms, and deals with the unpowered free-flight phase of the bullet after it exits the gun barrel and before it hits the target, so it lies between transitional ballistics and terminal ballistics.

However, external ballistics is also concerned with the free-flight of rockets and other projectiles, such as balls, arrows etc.

Terminal ballistics

Main article: Terminal ballistics

Terminal ballistics is the study of the behavior and effects of a projectile when it hits its target.

Terminal ballistics is relevant both for small caliber projectiles as well as for large caliber projectiles (fired from artillery). The study of extremely high velocity impacts is still very new and is as yet mostly applied to spacecraft design.

Applications

Apollo 11 – Astrodynamic calculations have permitted spacecraft to travel to and return from the Moon.

Forensic ballistics

Main article: Forensic firearm examination

Forensic ballistics involves analysis of bullets and bullet impacts to determine information of use to a court or other part of a legal system. Separately from ballistics information, firearm and tool mark examinations ("ballistic fingerprinting") involve analyzing firearm, ammunition, and tool mark evidence in order to establish whether a certain firearm or tool was used in the commission of a crime.

Astrodynamics

Main article: Orbital mechanics

Astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and Newton's law of universal gravitation. It is a core discipline within space mission design and control.

Psychodynamics

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Psychodynamics 

Front row: Sigmund Freud, G. Stanley Hall, Carl Jung; Back row: Abraham A. Brill, Ernest Jones, Sándor Ferenczi, at: Clark University in Worcester, Massachusetts. Date: September 1909.

Psychodynamics, also known as psychodynamic psychology, in its broadest sense, is an approach to psychology that emphasizes systematic study of the psychological forces underlying human behavior, feelings, and emotions and how they might relate to early experience. It is especially interested in the dynamic relations between conscious motivation and unconscious motivation.

The term psychodynamics is sometimes used to refer specifically to the psychoanalytical approach developed by Sigmund Freud (1856–1939) and his followers. Freud was inspired by the theory of thermodynamics and used the term psychodynamics to describe the processes of the mind as flows of psychological energy (libido or psi) in an organically complex brain. However, modern usage differentiates psychoanalytic practice as referring specifically to the earliest forms of psychotherapy, practiced by Freud and his immediate followers, and psychodynamic practice as practice that is informed by psychoanalytic theory, but diverges from the traditional practice model.

In the treatment of psychological distress, psychodynamic psychotherapy tends to be a less intensive (once- or twice-weekly) modality than the classical Freudian psychoanalysis treatment (of 3–5 sessions per week) and typically relies less on the traditional practices of psychoanalytic therapy, such as the patient facing away from the therapist during treatment and free association. Psychodynamic therapies depend upon a psychoanalytic understanding of inner conflict, wherein unconscious thoughts, desires, and memories influence behavior and psychological problems are caused by unconscious or repressed conflicts.

Despite largely falling out of favor as the primary modality of psychotherapy and facing criticism as being "non-empirical", psychodynamic treatment has been shown to be effective at treating a number of psychological conditions in randomized controlled trials, more effectively than controls and to the same degree as other psychotherapy modalities.

Overview

In general, psychodynamics is the study of the interrelationship of various parts of the mind, personality, or psyche as they relate to mental, emotional, or motivational forces especially at the unconscious level. The mental forces involved in psychodynamics are often divided into two parts: (a) the interaction of the emotional and motivational forces that affect behavior and mental states, especially on a subconscious level; (b) inner forces affecting behavior: the study of the emotional and motivational forces that affect behavior and states of mind.

Freud proposed that psychological energy was constant (hence, emotional changes consisted only in displacements) and that it tended to rest (point attractor) through discharge (catharsis).

In mate selection psychology, psychodynamics is defined as the study of the forces, motives, and energy generated by the deepest of human needs.

In general, psychodynamics studies the transformations and exchanges of "psychic energy" within the personality. A focus in psychodynamics is the connection between the energetics of emotional states in the Id, ego and super-ego as they relate to early childhood developments and processes. At the heart of psychological processes, according to Freud, is the ego, which he envisions as battling with three forces: the id, the super-ego, and the outside world. The id is the unconscious reservoir of libido, the psychic energy that fuels instincts and psychic processes. The ego serves as the general manager of personality, making decisions regarding the pleasures that will be pursued at the id's demand, the person's safety requirements, and the moral dictates of the superego that will be followed. The superego refers to the repository of an individual's moral values, divided into the conscience – the internalization of a society's rules and regulations – and the ego-ideal – the internalization of one's goals. Hence, the basic psychodynamic model focuses on the dynamic interactions between the id, ego, and superego. Psychodynamics, subsequently, attempts to explain or interpret behaviour or mental states in terms of innate emotional forces or processes.

History

Ernst von Brücke, early developer of psychodynamics

Freud used the term psychodynamics to describe the processes of the mind as flows of psychological energy (libido) in an organically complex brain. The idea for this came from his first year adviser, Ernst von Brücke at the University of Vienna, who held the view that all living organisms, including humans, are basically energy-systems to which the principle of the conservation of energy applies. This principle states that "the total amount of energy in any given physical system is always constant, that energy quanta can be changed but not annihilated, and that consequently when energy is moved from one part of the system, it must reappear in another part." This principle is at the very root of Freud's ideas, whereby libido, which is primarily seen as sexual energy, is transformed into other behaviours. However, it is now clear that the term energy in physics means something quite different from the term energy in relation to mental functioning.

Psychodynamics was initially further developed by Carl Jung, Alfred Adler and Melanie Klein. By the mid-1940s and into the 1950s, the general application of the "psychodynamic theory" had been well established.

In his 1988 book Introduction to Psychodynamics – a New Synthesis, psychiatrist Mardi J. Horowitz states that his own interest and fascination with psychodynamics began during the 1950s, when he heard Ralph Greenson, a popular local psychoanalyst who spoke to the public on topics such as "People who Hate", speak on the radio at UCLA. In his radio discussion, according to Horowitz, he "vividly described neurotic behavior and unconscious mental processes and linked psychodynamics theory directly to everyday life."

In the 1950s, American psychiatrist Eric Berne built on Freud's psychodynamic model, particularly that of the "ego states", to develop a psychology of human interactions called transactional analysis which, according to physician James R. Allen, is a "cognitive-behavioral approach to treatment and that it is a very effective way of dealing with internal models of self and others as well as other psychodynamic issues.".

Around the 1970s, a growing number of researchers began departing from the psychodynamics model and Freudian subconscious. Many felt that the evidence was over-reliant on imaginative discourse in therapy, and on patient reports of their state-of-mind. These subjective experiences are inaccessible to others. Philosopher of science Karl Popper argued that much of Freudianism was untestable and therefore not scientific. In 1975 literary critic Frederick Crews began a decades-long campaign against the scientific credibility of Freudianism. This culminated in Freud: The Making of an Illusion which aggregated years of criticism from many quarters. Medical schools and psychology departments no longer offer much training in psychodynamics, according to a 2007 survey. An Emory University psychology professor explained, “I don’t think psychoanalysis is going to survive unless there is more of an appreciation for empirical rigor and testing.”

Freudian analysis

According to American psychologist Calvin S. Hall, from his 1954 Primer in Freudian Psychology:

Freud greatly admired Brücke and quickly became indoctrinated by this new dynamic physiology. Thanks to Freud's singular genius, he was to discover some twenty years later that the laws of dynamics could be applied to man's personality as well as to his body. When he made his discovery Freud proceeded to create a dynamic psychology. A dynamic psychology is one that studies the transformations and exchanges of energy within the personality. This was Freud’s greatest achievement, and one of the greatest achievements in modern science, It is certainly a crucial event in the history of psychology.

At the heart of psychological processes, according to Freud, is the ego, which he sees battling with three forces: the id, the super-ego, and the outside world. Hence, the basic psychodynamic model focuses on the dynamic interactions between the id, ego, and superego. Psychodynamics, subsequently, attempts to explain or interpret behavior or mental states in terms of innate emotional forces or processes. In his writings about the "engines of human behavior", Freud used the German word Trieb, a word that can be translated into English as either instinct or drive.

In the 1930s, Freud's daughter Anna Freud began to apply Freud's psychodynamic theories of the "ego" to the study of parent-child attachment and especially deprivation and in doing so developed ego psychology.

Jungian analysis

At the turn of the 20th century, during these decisive years, a young Swiss psychiatrist named Carl Jung had been following Freud's writings and had sent him copies of his articles and his first book, the 1907 Psychology of Dementia Praecox, in which he upheld the Freudian psychodynamic viewpoint, although with some reservations. That year, Freud invited Jung to visit him in Vienna. The two men, it is said, were greatly attracted to each other, and they talked continuously for thirteen hours. This led to a professional relationship in which they corresponded on a weekly basis, for a period of six years.

Carl Jung's contributions in psychodynamic psychology include:

1. The psyche tends toward wholeness.
2. The self is composed of the ego, the personal unconscious, the collective unconscious. The collective unconscious contains the archetypes which manifest in ways particular to each individual.
3. Archetypes are composed of dynamic tensions and arise spontaneously in the individual and collective psyche. Archetypes are autonomous energies common to the human species. They give the psyche its dynamic properties and help organize it. Their effects can be seen in many forms and across cultures.
4. The Transcendent Function: The emergence of the third resolves the split between dynamic polar tensions within the archetypal structure.
5. The recognition of the spiritual dimension of the human psyche.
6. The role of images which spontaneously arise in the human psyche (images include the interconnection between affect, images, and instinct) to communicate the dynamic processes taking place in the personal and collective unconscious, images which can be used to help the ego move in the direction of psychic wholeness.
7. Recognition of the multiplicity of psyche and psychic life, that there are several organizing principles within the psyche, and that they are at times in conflict.

Electromagnetic wave equation

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Electromagnetic_wave_equation 

The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field E or the magnetic field B, takes the form:

$${\displaystyle \begin{array}{rl}\left(v_{\mathrm{p}\mathrm{h}}^2{\mathrm{\nabla }}^2-\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}\right)\mathbf{E}& =\mathbf{0}\\ \left(v_{\mathrm{p}\mathrm{h}}^2{\mathrm{\nabla }}^2-\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}\right)\mathbf{B}& =\mathbf{0}\end{array}}$$

where

$${\displaystyle v_{\mathrm{p}\mathrm{h}}=\frac{1}{\sqrt{\mu \epsilon }}}$$

is the speed of light (i.e. phase velocity) in a medium with permeability μ, and permittivity ε, and ∇² is the Laplace operator. In a vacuum, v_ph = c₀ = 299792458 m/s, a fundamental physical constant. The electromagnetic wave equation derives from Maxwell's equations. In most older literature, B is called the magnetic flux density or magnetic induction. The following equations$${\displaystyle \begin{array}{rl}\mathrm{\nabla }\cdot \mathbf{E}& =0\\ \mathrm{\nabla }\cdot \mathbf{B}& =0\end{array}}$$predicate that any electromagnetic wave must be a transverse wave, where the electric field E and the magnetic field B are both perpendicular to the direction of wave propagation.

The origin of the electromagnetic wave equation

A postcard from Maxwell to Peter Tait.

In his 1865 paper titled A Dynamical Theory of the Electromagnetic Field, James Clerk Maxwell utilized the correction to Ampère's circuital law that he had made in part III of his 1861 paper On Physical Lines of Force. In Part VI of his 1864 paper titled Electromagnetic Theory of Light, Maxwell combined displacement current with some of the other equations of electromagnetism and he obtained a wave equation with a speed equal to the speed of light. He commented:

The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws.

Maxwell's derivation of the electromagnetic wave equation has been replaced in modern physics education by a much less cumbersome method involving combining the corrected version of Ampère's circuital law with Faraday's law of induction.

To obtain the electromagnetic wave equation in a vacuum using the modern method, we begin with the modern 'Heaviside' form of Maxwell's equations. In a vacuum- and charge-free space, these equations are:

$${\displaystyle \begin{array}{rl}\mathrm{\nabla }\cdot \mathbf{E}& =0\\ \mathrm{\nabla }\times \mathbf{E}& =-\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}\\ \mathrm{\nabla }\cdot \mathbf{B}& =0\\ \mathrm{\nabla }\times \mathbf{B}& ={\mu }_0{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}\end{array}}$$

These are the general Maxwell's equations specialized to the case with charge and current both set to zero. Taking the curl of the curl equations gives:

$${\displaystyle \begin{array}{rl}\mathrm{\nabla }\times \left(\mathrm{\nabla }\times \mathbf{E}\right)& =\mathrm{\nabla }\times \left(-\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}\right)=-\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(\mathrm{\nabla }\times \mathbf{B}\right)=-{\mu }_0{\epsilon }_0\frac{{\mathrm{\partial }}^2\mathbf{E}}{\mathrm{\partial }{t}^2}\\ \mathrm{\nabla }\times \left(\mathrm{\nabla }\times \mathbf{B}\right)& =\mathrm{\nabla }\times \left({\mu }_0{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}\right)={\mu }_0{\epsilon }_0\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(\mathrm{\nabla }\times \mathbf{E}\right)=-{\mu }_0{\epsilon }_0\frac{{\mathrm{\partial }}^2\mathbf{B}}{\mathrm{\partial }{t}^2}\end{array}}$$

We can use the vector identity

$${\displaystyle \mathrm{\nabla }\times \left(\mathrm{\nabla }\times \mathbf{V}\right)=\mathrm{\nabla }\left(\mathrm{\nabla }\cdot \mathbf{V}\right)-{\mathrm{\nabla }}^2\mathbf{V}}$$

where V is any vector function of space. And

$${\displaystyle {\mathrm{\nabla }}^2\mathbf{V}=\mathrm{\nabla }\cdot \left(\mathrm{\nabla }\mathbf{V}\right)}$$

where ∇V is a dyadic which when operated on by the divergence operator ∇ ⋅ yields a vector. Since

$${\displaystyle \begin{array}{rl}\mathrm{\nabla }\cdot \mathbf{E}& =0\\ \mathrm{\nabla }\cdot \mathbf{B}& =0\end{array}}$$

then the first term on the right in the identity vanishes and we obtain the wave equations:

$${\displaystyle \begin{array}{rl}\frac{1}{c_0^2}\frac{{\mathrm{\partial }}^2\mathbf{E}}{\mathrm{\partial }{t}^2}-{\mathrm{\nabla }}^2\mathbf{E}& =\mathbf{0}\\ \frac{1}{c_0^2}\frac{{\mathrm{\partial }}^2\mathbf{B}}{\mathrm{\partial }{t}^2}-{\mathrm{\nabla }}^2\mathbf{B}& =\mathbf{0}\end{array}}$$

where

$${\displaystyle c_0=\frac{1}{\sqrt{{\mu }_0{\epsilon }_0}}=2.99792458\times {10}^8\text{m/s}}$$

is the speed of light in free space.

Covariant form of the homogeneous wave equation

Time dilation in transversal motion. The requirement that the speed of light is constant in every inertial reference frame leads to the theory of Special Relativity.

These relativistic equations can be written in contravariant form as

$${\displaystyle ◻A^{\mu }=0}$$

where the electromagnetic four-potential is

$${\displaystyle A^{\mu }=\left(\frac{\varphi }{c},\mathbf{A}\right)}$$

with the Lorenz gauge condition:

$${\displaystyle {\mathrm{\partial }}_{\mu }{A}^{\mu }=0,}$$

and where

$${\displaystyle ◻={\mathrm{\nabla }}^2-\frac{1}{c^2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}}$$

is the d'Alembert operator.

Homogeneous wave equation in curved spacetime

Main article: Maxwell's equations in curved spacetime

The electromagnetic wave equation is modified in two ways, the derivative is replaced with the covariant derivative and a new term that depends on the curvature appears.

$${\displaystyle -{A^{\alpha ;\beta }}_{;\beta }+{R^{\alpha }}_{\beta }{A}^{\beta }=0}$$

where ${\displaystyle {R^{\alpha }}_{\beta }}$ is the Ricci curvature tensor and the semicolon indicates covariant differentiation.

The generalization of the Lorenz gauge condition in curved spacetime is assumed:

$${\displaystyle {A^{\mu }}_{;\mu }=0.}$$

Inhomogeneous electromagnetic wave equation

Main article: Inhomogeneous electromagnetic wave equation

Localized time-varying charge and current densities can act as sources of electromagnetic waves in a vacuum. Maxwell's equations can be written in the form of a wave equation with sources. The addition of sources to the wave equations makes the partial differential equations inhomogeneous.

Solutions to the homogeneous electromagnetic wave equation

Main article: Wave equation

The general solution to the electromagnetic wave equation is a linear superposition of waves of the form

$${\displaystyle \begin{array}{rl}\mathbf{E}(\mathbf{r},t)& =g(\varphi (\mathbf{r},t))=g(\omega t-\mathbf{k}\cdot \mathbf{r})\\ \mathbf{B}(\mathbf{r},t)& =g(\varphi (\mathbf{r},t))=g(\omega t-\mathbf{k}\cdot \mathbf{r})\end{array}}$$

for virtually any well-behaved function g of dimensionless argument φ, where ω is the angular frequency (in radians per second), and k = (k_x, k_y, k_z) is the wave vector (in radians per meter).

Although the function g can be and often is a monochromatic sine wave, it does not have to be sinusoidal, or even periodic. In practice, g cannot have infinite periodicity because any real electromagnetic wave must always have a finite extent in time and space. As a result, and based on the theory of Fourier decomposition, a real wave must consist of the superposition of an infinite set of sinusoidal frequencies.

In addition, for a valid solution, the wave vector and the angular frequency are not independent; they must adhere to the dispersion relation:

$${\displaystyle k=|\mathbf{k}|=\frac{\omega }{c}=\frac{2\pi }{\lambda }}$$

where k is the wavenumber and λ is the wavelength. The variable c can only be used in this equation when the electromagnetic wave is in a vacuum.

Monochromatic, sinusoidal steady-state

The simplest set of solutions to the wave equation result from assuming sinusoidal waveforms of a single frequency in separable form:

$${\displaystyle \mathbf{E}(\mathbf{r},t)=\mathrm{\Re }\left\{\mathbf{E}(\mathbf{r})e^{i\omega t}\right\}}$$

where

• i is the imaginary unit,
• ω = 2π f  is the angular frequency in radians per second,
•  f  is the frequency in hertz, and
• ${\displaystyle e^{i\omega t}=\cos (\omega t)+i\sin (\omega t)}$ is Euler's formula.

Plane wave solutions

Main article: Sinusoidal plane-wave solutions of the electromagnetic wave equation

Consider a plane defined by a unit normal vector

$${\displaystyle \mathbf{n}=\frac{\mathbf{k}}{k}.}$$

Then planar traveling wave solutions of the wave equations are

$${\displaystyle \begin{array}{rl}\mathbf{E}(\mathbf{r})& ={\mathbf{E}}_0{e}^{-i\mathbf{k}\cdot \mathbf{r}}\\ \mathbf{B}(\mathbf{r})& ={\mathbf{B}}_0{e}^{-i\mathbf{k}\cdot \mathbf{r}}\end{array}}$$

where r = (x, y, z) is the position vector (in meters).

These solutions represent planar waves traveling in the direction of the normal vector n. If we define the z direction as the direction of n, and the x direction as the direction of E, then by Faraday's Law the magnetic field lies in the y direction and is related to the electric field by the relation

$${\displaystyle c^2\frac{\mathrm{\partial }B}{\mathrm{\partial }z}=\frac{\mathrm{\partial }E}{\mathrm{\partial }t}.}$$

Because the divergence of the electric and magnetic fields are zero, there are no fields in the direction of propagation.

This solution is the linearly polarized solution of the wave equations. There are also circularly polarized solutions in which the fields rotate about the normal vector.

Spectral decomposition

Because of the linearity of Maxwell's equations in a vacuum, solutions can be decomposed into a superposition of sinusoids. This is the basis for the Fourier transform method for the solution of differential equations. The sinusoidal solution to the electromagnetic wave equation takes the form

$${\displaystyle \begin{array}{rl}\mathbf{E}(\mathbf{r},t)& ={\mathbf{E}}_0\cos (\omega t-\mathbf{k}\cdot \mathbf{r}+{\varphi }_0)\\ \mathbf{B}(\mathbf{r},t)& ={\mathbf{B}}_0\cos (\omega t-\mathbf{k}\cdot \mathbf{r}+{\varphi }_0)\end{array}}$$

where

• t is time (in seconds),
• ω is the angular frequency (in radians per second),
• k = (k_x, k_y, k_z) is the wave vector (in radians per meter), and
• ${\displaystyle {\varphi }_0}$ is the phase angle (in radians).

The wave vector is related to the angular frequency by

$${\displaystyle k=|\mathbf{k}|=\frac{\omega }{c}=\frac{2\pi }{\lambda }}$$

where k is the wavenumber and λ is the wavelength.

The electromagnetic spectrum is a plot of the field magnitudes (or energies) as a function of wavelength.

Multipole expansion

Assuming monochromatic fields varying in time as ${\displaystyle e^{-i\omega t}}$, if one uses Maxwell's Equations to eliminate B, the electromagnetic wave equation reduces to the Helmholtz equation for E:

$${\displaystyle ({\mathrm{\nabla }}^2+k^2)\mathbf{E}=0,\mathbf{B}=-\frac{i}{k}\mathrm{\nabla }\times \mathbf{E},}$$

with k = ω/c as given above. Alternatively, one can eliminate E in favor of B to obtain:

$${\displaystyle ({\mathrm{\nabla }}^2+k^2)\mathbf{B}=0,\mathbf{E}=-\frac{i}{k}\mathrm{\nabla }\times \mathbf{B}.}$$

A generic electromagnetic field with frequency ω can be written as a sum of solutions to these two equations. The three-dimensional solutions of the Helmholtz Equation can be expressed as expansions in spherical harmonics with coefficients proportional to the spherical Bessel functions. However, applying this expansion to each vector component of E or B will give solutions that are not generically divergence-free (∇ ⋅ E = ∇ ⋅ B = 0), and therefore require additional restrictions on the coefficients.

The multipole expansion circumvents this difficulty by expanding not E or B, but r ⋅ E or r ⋅ B into spherical harmonics. These expansions still solve the original Helmholtz equations for E and B because for a divergence-free field F, ∇² (r ⋅ F) = r ⋅ (∇² F). The resulting expressions for a generic electromagnetic field are:

$${\displaystyle \begin{array}{rl}\mathbf{E}& =e^{-i\omega t}\sum _{l,m}\sqrt{l(l+1)}\left[a_E(l,m){\mathbf{E}}_{l,m}^{(E)}+a_M(l,m){\mathbf{E}}_{l,m}^{(M)}\right]\\ \mathbf{B}& =e^{-i\omega t}\sum _{l,m}\sqrt{l(l+1)}\left[a_E(l,m){\mathbf{B}}_{l,m}^{(E)}+a_M(l,m){\mathbf{B}}_{l,m}^{(M)}\right],\end{array}}$$

where ${\displaystyle {\mathbf{E}}_{l,m}^{(E)}}$ and ${\displaystyle {\mathbf{B}}_{l,m}^{(E)}}$ are the electric multipole fields of order (l, m), and ${\displaystyle {\mathbf{E}}_{l,m}^{(M)}}$ and ${\displaystyle {\mathbf{B}}_{l,m}^{(M)}}$ are the corresponding magnetic multipole fields, and a_E(l, m) and a_M(l, m) are the coefficients of the expansion. The multipole fields are given by

$${\displaystyle \begin{array}{rl}{\mathbf{B}}_{l,m}^{(E)}& =\sqrt{l(l+1)}\left[B_l^{(1)}{h}_l^{(1)}(kr)+B_l^{(2)}{h}_l^{(2)}(kr)\right]{\mathbf{\Phi }}_{l,m}\\ {\mathbf{E}}_{l,m}^{(E)}& =\frac{i}{k}\mathrm{\nabla }\times {\mathbf{B}}_{l,m}^{(E)}\\ {\mathbf{E}}_{l,m}^{(M)}& =\sqrt{l(l+1)}\left[E_l^{(1)}{h}_l^{(1)}(kr)+E_l^{(2)}{h}_l^{(2)}(kr)\right]{\mathbf{\Phi }}_{l,m}\\ {\mathbf{B}}_{l,m}^{(M)}& =-\frac{i}{k}\mathrm{\nabla }\times {\mathbf{E}}_{l,m}^{(M)},\end{array}}$$

where h_l^(1,2)(x) are the spherical Hankel functions, E_l^(1,2) and B_l^(1,2) are determined by boundary conditions, and

$${\displaystyle {\mathbf{\Phi }}_{l,m}=\frac{1}{\sqrt{l(l+1)}}(\mathbf{r}\times \mathrm{\nabla })Y_{l,m}}$$

are vector spherical harmonics normalized so that

$${\displaystyle \int {\mathbf{\Phi }}_{l,m}^{\ast }\cdot {\mathbf{\Phi }}_{l^{\prime },m^{\prime }}d\mathrm{\Omega }={\delta }_{l,l^{\prime }}{\delta }_{m,m^{\prime }}.}$$

The multipole expansion of the electromagnetic field finds application in a number of problems involving spherical symmetry, for example antennae radiation patterns, or nuclear gamma decay. In these applications, one is often interested in the power radiated in the far-field. In this regions, the E and B fields asymptotically approach

$${\displaystyle \begin{array}{rl}\mathbf{B}& \approx \frac{e^{i(kr-\omega t)}}{kr}\sum _{l,m}(-i{)}^{l+1}\left[a_E(l,m){\mathbf{\Phi }}_{l,m}+a_M(l,m)\hat{\mathbf{r}}\times {\mathbf{\Phi }}_{l,m}\right]\\ \mathbf{E}& \approx \mathbf{B}\times \hat{\mathbf{r}}.\end{array}}$$

The angular distribution of the time-averaged radiated power is then given by

$${\displaystyle \frac{dP}{d\mathrm{\Omega }}\approx \frac{1}{2k^2}{\left|\sum _{l,m}(-i{)}^{l+1}\left[a_E(l,m){\mathbf{\Phi }}_{l,m}\times \hat{\mathbf{r}}+a_M(l,m){\mathbf{\Phi }}_{l,m}\right]\right|}^2.}$$