File:Displacement current in capacitor.svg

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Displacement_current_in_capacitor.svg‎ (SVG file, nominally 744 × 800 pixels, file size: 57 KB)

[edit] Summary

Description
English: Diagram of a widely used example demonstrating need for the displacement current term in Maxwell's equations.
The diagram shows a capacitor being charged by current I\, flowing through a wire, which creates a magnetic field \mathbf{B}\, around it. The magnetic field is found from Ampere's law:
\oint_{\partial S} \mathbf{B} \cdot d\mathbf{l} = \mu_0 \int_S (\mathbf{J} + \epsilon_0 \frac {\partial \mathbf{E}}{\partial t}) \cdot d\mathbf{S} \,

The equation says that the integral of the magnetic field \mathbf{B}\, around a loop \partial S\, is equal to the current \mathbf{J}\, through any surface spanning the loop, plus a term depending on the rate of change of the electric field \mathbf{E}\, through the surface. This term, the second term on the right, is the displacement current. For applications with no time varying electric fields (unchanging charge density) it is zero and is ignored. However in applications with time varying fields, such as circuits with capacitors, it is needed, as shown below. Any surface intersecting the wire, such as S_1\,, has current I\, passing through it so Ampere's law gives the correct magnetic field:

B = \frac {\mu_0 I}{2 \pi r}\,

But surface S_2\, spanning the same loop that passes between the capacitor's plates has no current flowing through it, so without the displacement current term Ampere's law gives:

B = 0\,

So without the displacement current term Ampere's law fails; it gives different results depending on which surface is used, which is inconsistent. The 'displacement current' term provides a second source for the magnetic field besides current; the rate of change of the electric field \mathbf{E}\,. Between the capacitor's plates, the electric field is increasing, so the rate of change of electric field through the surface S_2\, is positive, and its magnitude gives the correct value for the field field \mathbf{B}\, found above.

James Clerk Maxwell added the displacement current term to Ampere's law around 1861.

Example taken from Feynman, Richard; Robert Leighton; Matthew Sands (1964) The Feynman Lectures on Physics, Vol.2, Addison-Wesley, USA, p.18-4, using slightly different terminology.
Date

10 November 2008(2008-11-10)
Source

Own work
Author

Chetvorno
Permission
(Reusing this image)

I Chris Burks (Chetvorno) the author release this work into the public domain for any use whatever.

[edit] Licensing

Public domain I, the copyright holder of this work, hereby release it into the public domain. This applies worldwide.

In case this is not legally possible:
I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

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current14:28, 20 November 2008Thumbnail for version as of 14:28, 20 November 2008744×800 (57 KB)Chetvorno (talk | contribs) (== Summary == {{Information |Description={{en|Diagram of a widely used example demonstrating need for the displacement current term in Maxwell's equations.}} The diagram shows a [[Wikip)
04:56, 11 November 2008Thumbnail for version as of 04:56, 11 November 2008744×800 (58 KB)Chetvorno (talk | contribs) ({{Information |Description=Diagram demonstrating need for displacement current term in Ampere's Law. |Source=Own work by uploader |Date=2008-11-10 |Author=Chetvorno |Permission=I Chris Burks (Chetvorno) the author rel)

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