Pulsed Point Dipole: Jackson's E&M Explained

In summary, the charge and current densities in the coulomb gauge demonstrate causality by representing a point dipole flashing on and off at time t=0, with the delta prime terms indicating a change in density over time.
  • #1
keniwas
59
1
In Jackson's E&M book, he talks about using the following charge and current densities to demonstrate causality in the coulomb gauge.

[tex]\rho (\vec{r},t)=\delta(x)\delta(y)\delta'(z)\delta(t)[/tex]
[tex]\vec{J}(\vec{r},t})=\delta(x)\delta(y)\delta(z)\delta'(t)\hat{z}[/tex]

since they represent a point dipole flashing on and off at time t=0. I understand the 'flashing' aspect since the densities are only non-zero only at t=0. What I don't understand is why this represents a point dipole... why is the [tex]\delta'(z)[/tex] primed in the charge density? and likewise with the [tex]\delta'(t)[/tex] in the current?

Any thoughts?
 
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  • #2
The delta primed terms in the charge and current densities represent a point dipole because they represent a source of electric field that is localized at a single point in space. The delta prime terms represent a change in the charge or current density over time, which is necessary for a point dipole to occur. This is because the electric field produced by a point dipole depends on the difference in charge or current density between two times (t0 and t1). Therefore, the delta prime terms indicate that the charge or current density changes from time t0 to time t1.
 

1. What is a pulsed point dipole?

A pulsed point dipole is a theoretical model used in electromagnetism to represent a single point of charge that is oscillating or changing over time.

2. How is Jackson's E&M used to explain pulsed point dipoles?

Jackson's E&M, or Jackson's Electrodynamics, is a textbook commonly used in physics courses to explain the fundamental principles of electromagnetism. It can be used to describe the behavior of pulsed point dipoles by applying its equations and concepts to the specific scenario.

3. What are some real-life applications of pulsed point dipoles?

Pulsed point dipoles have many practical applications, including in radar and sonar technology, where they are used to detect and locate objects based on the reflection of electromagnetic waves.

4. How does a pulsed point dipole differ from a static point dipole?

A static point dipole is a non-moving point of charge, while a pulsed point dipole is a point of charge that is changing over time. This means that the electric and magnetic fields produced by a pulsed point dipole will also vary over time, while those of a static point dipole will remain constant.

5. Are there any limitations or assumptions when using pulsed point dipoles in calculations?

Yes, there are some limitations and assumptions when using pulsed point dipoles. For example, the model assumes that the dipole is very small compared to the wavelength of the electromagnetic waves being studied. It also assumes that the dipole is isolated and not influenced by any other charges or fields in its surroundings.

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