# File:Displacement current in capacitor.svg

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## 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 ${\displaystyle I\,}$ flowing through a wire, which creates a magnetic field ${\displaystyle \mathbf {B} \,}$ around it. The magnetic field is found from Ampere's law: ${\displaystyle \oint _{\partial S}\mathbf {B} \cdot d\mathbf {l} =\mu _{0}\int _{S}(\mathbf {J} +\epsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}})\cdot d\mathbf {S} \,}$ The equation says that the integral of the magnetic field ${\displaystyle \mathbf {B} \,}$ around a loop ${\displaystyle \partial S\,}$ is equal to the current ${\displaystyle \mathbf {J} \,}$ through any surface spanning the loop, plus a term depending on the rate of change of the electric field ${\displaystyle \mathbf {E} \,}$ 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 ${\displaystyle S_{1}\,}$, has current ${\displaystyle I\,}$ passing through it so Ampere's law gives the correct magnetic field: ${\displaystyle B={\frac {\mu _{0}I}{2\pi r}}\,}$ But surface ${\displaystyle S_{2}\,}$ 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: ${\displaystyle B=0\,}$ 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 ${\displaystyle \mathbf {E} \,}$. Between the capacitor's plates, the electric field is increasing, so the rate of change of electric field through the surface ${\displaystyle S_{2}\,}$ is positive, and its magnitude gives the correct value for the field field ${\displaystyle \mathbf {B} \,}$ 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 Source Own work Author Chetvorno Permission(Reusing this file) I Chris Burks (Chetvorno) the author release this work into the public domain for any use whatever.

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current14:28, 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 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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