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- #37

tech99

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Agree, but do you know what happens about the electric induction field if the acceleration is produced mechanically?

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$$\rho(t,\vec{x})=q \delta^{(3)}[\vec{x}-\vec{y}(t)], \quad \vec{j}(t,\vec{x})=\dot{\vec{y}}(t) \rho(t,\vec{x}).$$

This is even relativistically covariant. You can make it manifestly covariant by expressing it in terms of an arbitrary parametrization of the world line of the particle ##y^{\mu}(\lambda)## as

$$j^{\mu}(t,\vec{x}) = q \int_{\mathbb{R}} \mathrm{d} \lambda \dot{y}^{\mu}(\lambda) \delta^{(4)}[x-y(\lambda)].$$

Choosing the coordinate time, ##t## as the parameter you get the previous form of the charge and current density.

The solutions of the Maxwell equations are given via the potentials in Lorenz gauge using the retarded Green's function, leading to the Lienard-Wiechert potentials, or directly the fields using the so-called Jefimenko equation, which are of course the same as derived from the Lienard-Wiechert potentials.

- #39

sophiecentaur

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Won't there will be different net fields in the region for each case?Agree, but do you know what happens about the electric induction field if the acceleration is produced mechanically?

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- #41

cmb

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This might help if you like to see equations rather than diagrams and words;-

http://www.dannex.se/theory/3.html

- #42

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The 'near' field amplitude rolls off as ## 1/r^3 ## with distance r and is due almost entirely to charges oscillating back & forth around the antenna, so is conservative. The ## 1/r^3 ## term also appears in a computation of an oscillating electrostatic dipole.

The 'far' E field on the other hand is due to the interchange between the E field and the B field by Maxwell's equations. The amplitude of this E field rolls of as 1/r and is non-conservative, forming closed loops. The 1/r rolloff checks with the requirement of constant power in the E wave: intensity goes as amplitude squared and the area of the wavefront is of course ## 4 \pi r^2 ##.

cf. the second (antenna) part of https://www.physicsforums.com/insig...lds-in-electrodynamics-capacitor-and-antenna/

- #43

tech99

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If we approach the centre of the antenna on the equatorial lane, the E field increases with 1/r until we are approx lambda/5 from the antenna, then levels off and remains constant until we almost touch. When we almost touch we see the very local field across the feed-point gap. However, if we approach the ends of the dipole from any direction we see the very fast local increase, which might be 1/r^3 but to my crude measurement looked more like 1/r^2.

When we are close to an antenna, not only do we see the type of behaviour described, but we also tend to see the "radiation near field" effect, where the radiation tends to be parallel to start with, and even with a simple dipole this seems to occur closer than about lambda/5.

The magnetic field, on the other hand, tended always to increase with 1/r, as does that from a non radiating long conductor, and continued to increase this way right up to the wire in the equatorial plane. It fell to zero at the ends of the dipole. This measurement is not in accordance with the theory I have read.

- #44

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Not surprising, at least to me. Squaring theory with empirical data for rf is I guess not always easy. Anechoic chamber? Also, I used theoretical analyses based on an arbitrarily short dipole ## (<< \lambda/2). ## The results are quite different with a ## \lambda/2 ## antenna, with phase being required in consideration in determining the magnetic vector potential etc.

If we approach the centre of the antenna on the equatorial lane, the E field increases with 1/r until we are approx lambda/5 from the antenna, then levels off and remains constant until we almost touch. When we almost touch we see the very local field across the feed-point gap. However, if we approach the ends of the dipole from any direction we see the very fast local increase, which might be 1/r^3 but to my crude measurement looked more like 1/r^2.

When we are close to an antenna, not only do we see the type of behaviour described, but we also tend to see the "radiation near field" effect, where the radiation tends to be parallel to start with, and even with a simple dipole this seems to occur closer than about lambda/5.

The magnetic field, on the other hand, tended always to increase with 1/r, as does that from a non radiating long conductor, and continued to increase this way right up to the wire in the equatorial plane. It fell to zero at the ends of the dipole. This measurement is not in accordance with the theory I have read.

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