Hello,
I have a wire with AC current :
## Ohm's law gives :
J = \sigma E
J : the current density
\sigma : the electric conductivity
E : the electric field
From this equation it is clear that (if \sigma is real) the current density and the electric field are in phase.
## Biot-Savart law gives
B = (\mu_O/4pi)\iiint(Jxr/|r|^3)dr^3
Which shows that B and J are in phase.
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These two laws give a 0° phase shift between B and E
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## Faraday's law (in frequency domain) :
\nablaxE = iwB
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This shows that there is 90° phase shift between B and E.
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Where did I make a mistake ?
Thank you in advance
Hi Newgato,
first, for an AC current a wire has conductivity as well as inductance. So, the current will lag behind the voltage / the E field.
Second, with time dependent fields, instead of the Biot-Savart law we have to apply one of Maxwell's laws: curl B = d D / d t + J, and additionally take the retardation into account.
Third, the fields around the wire are the sum of external fields which cause the current to flow in the first place and the fields generated by this current (and the resulting charge distribution) which act back on the current. Generally, the generated E field points in the direction opposite to the external E field.
Finally, the fields generated by a current in a wire can be split (theoretically) into parts that are 90° out of phase and parts that are in phase. While the parts out of phase are dominant in the neighborhood of the wire (reactive fields), the in phase parts are dominant further away (radiation fields).
Because calculating the scattering of radiation fields even by simple structures like dipols is quite challenging, the usual approach in introductory textbooks is to assume a given current distribution, and to calculate the generated fields only.
The Biot-Savart law applies to magnetostatic conditions.
https://en.wikipedia.org/wiki/Biot%E2%80%93Savart_law
https://en.wikipedia.org/wiki/Magnetostatics
Further to Joerg Fricke's answer, if the distributions of charge and current vary with time, Biot-Savart's law is replaced by Jefimenko's equations. These are general solutions to Maxwell's equations which take account of the finite propagation speed of electric and magnetic fields.
https://en.wikipedia.org/wiki/Jefimenko's_equations
Hi Joerg Fricke
Thank you for the detailed answer, I was missing the fields that gave birth to the current, so thank you again.
I want to know when should we take into account the parasitic capacitance of the wire ?
Hi Alan Robinson Thank you for your answer.
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