- #1
lowlyEEugrad
- 8
- 0
Hello friends, please help me understand this tricky aspect of Ampere's and Faraday's Laws as they relate to the static, circular magnetic field enclosing a constant current carrying wire.
(image courtesy of Wikipedia)
The existence of this magnetic field is given by the I or J terms of Ampere's equation:
[itex]\oint\textbf{B}\bullet d\textbf{l} = \mu I_{encl} + \mu\epsilon\frac{d\phi_{E}}{dt}[/itex]
[itex]\nabla[/itex] x B = μJ + [itex]\mu\epsilon[/itex][itex]\frac{d\textbf{E}}{dt}[/itex]
Before this current I is turned on, there is no current and thus no B field.
At the instant current I is turned on, B appears very close to the wire. B does not appear at far away points instantaneously. Rather it propagates out at the speed of light c.
My question concerns this propagation. Let's consider a point very close to the wire. Before current I was turned on, there was no magnetic field, or we could say B = O T. After current I is turned on, there is some magnetic field, say B = B1 with direction given by the right hand rule. This constitutes a change in magnetic field in time: [itex]\frac{d\textbf{B}}{dt} \neq 0[/itex]
Then this must generate an electric field by Faraday's law:
[itex]\nabla[/itex] x E = -[itex]\frac{d\textbf{B}}{dt}[/itex]
I believe this induced E would be located very near the wire and B = B1 which induced it. This new electric field arising means [itex]\frac{d\textbf{E}}{dt} \neq 0[/itex] which then feeds back into Ampere's law, and now we have an outwardly propagating EM wave.
But we never talk about an electric field when referring to Ampere's law in the constant current situation as shown in the picture. This leads me to believe my understanding of the propagation mechanism for the static magnetic field is mistaken.
Yet this same argument, as far as I can see, is employed by Feynman in his Lectures Vol 2 Chapter 18.4, A Travelling Wave. He uses an infinitely large sheet of constant current moving in one direction, and it results in outwardly propagating E and B.
Please help me understand how this situation is different from our normal Oersted/Ampere law where we only look at the static magnetic field. Thanks very much.
(image courtesy of Wikipedia)
The existence of this magnetic field is given by the I or J terms of Ampere's equation:
[itex]\oint\textbf{B}\bullet d\textbf{l} = \mu I_{encl} + \mu\epsilon\frac{d\phi_{E}}{dt}[/itex]
[itex]\nabla[/itex] x B = μJ + [itex]\mu\epsilon[/itex][itex]\frac{d\textbf{E}}{dt}[/itex]
Before this current I is turned on, there is no current and thus no B field.
At the instant current I is turned on, B appears very close to the wire. B does not appear at far away points instantaneously. Rather it propagates out at the speed of light c.
My question concerns this propagation. Let's consider a point very close to the wire. Before current I was turned on, there was no magnetic field, or we could say B = O T. After current I is turned on, there is some magnetic field, say B = B1 with direction given by the right hand rule. This constitutes a change in magnetic field in time: [itex]\frac{d\textbf{B}}{dt} \neq 0[/itex]
Then this must generate an electric field by Faraday's law:
[itex]\nabla[/itex] x E = -[itex]\frac{d\textbf{B}}{dt}[/itex]
I believe this induced E would be located very near the wire and B = B1 which induced it. This new electric field arising means [itex]\frac{d\textbf{E}}{dt} \neq 0[/itex] which then feeds back into Ampere's law, and now we have an outwardly propagating EM wave.
But we never talk about an electric field when referring to Ampere's law in the constant current situation as shown in the picture. This leads me to believe my understanding of the propagation mechanism for the static magnetic field is mistaken.
Yet this same argument, as far as I can see, is employed by Feynman in his Lectures Vol 2 Chapter 18.4, A Travelling Wave. He uses an infinitely large sheet of constant current moving in one direction, and it results in outwardly propagating E and B.
Please help me understand how this situation is different from our normal Oersted/Ampere law where we only look at the static magnetic field. Thanks very much.
Last edited: