Producing a Radiative Electric Field

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This page explains the relationship between measured radiative electric field and the properties of charges in a system.

Calculating Radiative Electric Field

Maintained by Charles Kilpatrick --Ck (talk) 14:18, 18 November 2015 (EST)

A Mathematical Model

The radiative electric field can be generally modeled as [math]\displaystyle{ \vec{E}_{radiative} = \frac{1}{4 \pi \epsilon_0} \frac{-q \vec{a}_\perp}{c^2r} }[/math] where q is the charge of the accelerated particle, [math]\displaystyle{ \vec{a}_\perp }[/math] is the projected acceleration, c is the speed of light and r is the distance between the charge and the observation location.

A Computational Model

The following is a vPython model of Radiative Electric Field due to an instant of acceleration (a "kick") on a charged particle.

3d Radiation vPython Model

The acceleration vector, an initial kick in the +x direction, is represented by the yellow arrow in the center. The orange arrows seen in the model represent [math]\displaystyle{ \vec{E}_{radiative} }[/math] and the cyan arrows represent the corresponding [math]\displaystyle{ \vec{B}_{radiative} }[/math] at distance r from the particle.

Examples

Simple

An electron at the origin is kicked in the -y direction. At observation location (0, 0, 1), what is the direction of the radiative electric field?

The radiative electric field at this location is known as the traverse electric field, which always has a direction opposite the direction of [math]\displaystyle{ \vec{a}_\perp }[/math]. Thus, [math]\displaystyle{ \vec{E}_{radiative} }[/math] at the observation location points in the +y direction.

Middling

If the angle represented by θ is 60 degrees and [math]\displaystyle{ |a| }[/math] is 8.1 * 10^14 m/s^2, what is [math]\displaystyle{ |a_\perp| }[/math]?

Procedure

Because the angle represented by θ is opposite the [math]\displaystyle{ a_\perp }[/math] vector, we simply multiply [math]\displaystyle{ |a| }[/math] by sin(θ) to compute our answer:

[math]\displaystyle{ 8.1 * 10^{14} m/s^2 * sin(60) = 7.05 * 10^{14} m/s^2 }[/math]

Difficult

An electron experiences a kick with initial acceleration [math]\displaystyle{ a_\perp }[/math]=3.6 * 10^17 in the -z direction at a point (3, 4, -4) away from the origin. Calculate the direction and magnitude of the initial radiative field at the origin and at (2, 3, 0).

Procedure

For [math]\displaystyle{ E_{rad} }[/math] at the origin,

[math]\displaystyle{ E_{rad}=\frac{1}{4\pi\epsilon_0}\frac{-q\vec{a}_\perp}{c^2r} }[/math]

[math]\displaystyle{ r=\sqrt{3^2 + 4^2 + 4^2} \approx 6.40 }[/math]

[math]\displaystyle{ E_{rad}=9 * 10^9\frac{e(3.6 * 10^{17})}{c^2 * 6.40} = \frac{8.1 * 10^{7}}{c^2} \approx 9.0 * 10^{-10} }[/math]

Thus the magnitude of the initial radiative field at the origin is 9.0 * 10^-10 V/m.


For [math]\displaystyle{ E_{rad} }[/math] at (2,3,0),

[math]\displaystyle{ E_{rad}=\frac{1}{4\pi\epsilon_0}\frac{-q\vec{a}_\perp}{c^2r} }[/math]

[math]\displaystyle{ r=\sqrt{1^2 + 1^2 + 4^2} \approx 4.24 }[/math]

[math]\displaystyle{ E_{rad}=9 * 10^9\frac{e(3.6 * 10^{17})}{c^2 * 4.24} = \frac{1.2 * 10^{8}}{c^2} \approx 1.36 * 10^{-9} }[/math]

Thus the magnitude of the initial radiative field at (2,3,0) is 1.36 * 10^-9 V/m.


As for the directions, because [math]\displaystyle{ -qa_{\perp} }[/math] is in the -z direction, both fields also point in the -z direction.

Connectedness

1. How is this topic connected to something that you are interested in?

This topic relates to my personal interest in using modelling to enhance understanding. In the past, data such as the changes over time in large, complex data sets (e.g. the knowledge base for intelligent agents) has been extremely cumbersome and borderline unintelligible without an effectively modeled rendition of this data. In the 60's, much research was done at the University of Utah in the field of computer-generated graphics, resulting in an outpouring of 3d-modelling applications, which in turn led to the invention of iconic modelling software such as Autodesk that researchers use worldwide today. As you can see, the model of the radiative [math]\displaystyle{ \vec{E} }[/math] and [math]\displaystyle{ \vec{B} }[/math] fields provided above greatly clarifies the nature of the change in these fields over time and offers us an interactive three-dimensional perspective on something that could only be drawn statically on whiteboards in the past.

2. How is it connected to your major?

As a computer science student in the Media thread, radiative effects are integral to modeling and animating light sources. Luminance, or the total light present in a space, is generated according to an integral of vectors over [math]\displaystyle{ k }[/math], all possible outward direction vectors from the light source. As the vectors representing the rays of light expand outward, they will interact with physical objects to create shadows, or other lights to create more intense or different-colored light. This is similar to how the [math]\displaystyle{ \vec{E} }[/math] and [math]\displaystyle{ \vec{B} }[/math] fields may change and distort upon interacting with other fields as they extend outward.

3. Is there an interesting industrial application?

Most practical accelerations of charge will occur on objects with both positive and negative charges. When dipoles are accelerated on more than one axis at once, they produce a cyclical difference in [math]\displaystyle{ \vec{E} }[/math] and [math]\displaystyle{ \vec{B} }[/math] fields, resulting in electromagnetic waves, which are used in countless places: radio, microwave ovens, tv remotes, any electronic display... The list goes on.

History

While electromagnetic radiation had been found as a phenomenon that produced visible light in the early 1800's, it was not until the 1860's that James Clerk Maxwell was the first to create formal equations to measure electromagnetic field while lecturing at King's College, London. Upon doing so, he found that the speed of propagation of an electric field was approximately the speed of light and later concluded that light itself was a change in electromagnetic field. In 1887, Heinrich Hertz accelerated charges with low frequency in order to produce radio waves while testing Maxwell's equations at the University of Karlsruhe in Germany, and Wilhelm Röntgen in 1895 attached electrodes to a Ruhmkorff coil in to generate an electrostatic charge and eventually discovered X-ray radiation when he noted a flickering produced on a barium platinocyanide screen at the University of Würzburg, Germany. Finally, in 1898, Marie Curie discovered the properties of the elements Polonium and Radium that generate alpha particle radiation through the energetically-excited nuclear state (and thus disruption in charge velocity) of the isotopes at the School of Physics and Chemistry, Paris.

See also

Further reading

  • Chabay, Ruth W.; Sherwood, Bruce A.(2014). Matter and Interactions (4th ed.). John Wiley & Sons Inc. ISBN 978-1-118-87586-5.
  • Serway, Raymond A.; Jewett, John W. (2004). Physics for Scientists and Engineers (6th ed.). Brooks Cole. ISBN 0-534-40842-7.
  • Tipler, Paul (2004). Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.). W. H. Freeman. ISBN 0-7167-0810-8.
  • Reitz, John; Milford, Frederick; Christy, Robert (1992). Foundations of Electromagnetic Theory (4th ed.). Addison Wesley. ISBN 0-201-52624-7.
  • Jackson, John David (1999). Classical Electrodynamics (3rd ed.). John Wiley & Sons. ISBN 0-471-30932-X.

External links

References

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