# Maxwell's equations & Conductors

• Swapnil
In summary: There can always be slight deviations from the ideal case in real world scenarios. In summary, The fact that there could be no E-field inside a conductor is purely experimental and cannot be proven using Maxwell's equations. This idealization is seen in superconductors, while for ordinary metals of finite conductivity, a small DC field can exist according to Ohm's Law. Time varying fields are screened by circulating eddy currents, and penetrate a small distance characterized by the skin depth. The assumption of there being no current inside the conductor is more pleasing in order to prove the absence of an E-field. The "inside" of a conductor refers to the solid region where the mobile charges are, and in general, the E-field in this region would always
Swapnil
Is the fact that there could be no E-field inside a conductor purely experimental? I don't see any way to apply Maxwell's equations to prove this fact.

If there is an E field in a region with mobile charge carriers then there will be a current. So after all currents have died away, either the E field must be canceled or the medium will have run out of charge carriers. Does that help?

Zero E field is an idealization that holds for perfect conductivity where exactly enough charges can move to the surface to screen out an ambient field. This is actually seen in superconductors, although related phenomena like the Meissner effect require quantum mechanics to explain. For ordinary metals of finite conductivity, a small DC field can exist inside according to Ohm's Law
$$\vec{J}=\sigma\vec{E}$$

Time varying fields, on the other hand, are screened by circulating eddy currents, and penetrate a small distance characterized by the skin depth. This is covered under discussions of wave propagation in conducting media, in upper class E&M books like Schwartz or Reitz and Milford. Here's a link that contains the derivations in sections 6.19-6.20
"www.sp.phy.cam.ac.uk/teaching/em/waves.pdf"[/URL]

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Dick said:
If there is an E field in a region with mobile charge carriers then there will be a current. So after all currents have died away, either the E field must be canceled or the medium will have run out of charge carriers. Does that help?
Yeah, this makes sense. Eventhough in order to prove that there is no E-field inside the conductor we are assuming that there is no current inside the conductor, the assumption of there being no current is more pleasing.

BTW, whenever we say "inside" a conductor (in the context of there being no E-field), do we always refer to the region where the conductor is solid?

If you mean inside a void in a conductor - sure, E can be non-zero in there.

Dick said:
If you mean inside a void in a conductor - sure, E can be non-zero in there.
No, I meant to say that in general when we say "There is no E-field inside a conductor," what does "inside" actually mean?

For example, consider spherical conductor shell. It hass two regions -- the region between the inner and outer surface of the conductor which is solid and the region between the center of the conductor and the inner surface which hollow. Which region would you call the "inside" of the conductor?

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I would call 'inside the conductor' inside of the conducting shell.

Dick said:
I would call 'inside the conductor' inside of the conducting shell.

I am not sure which region you are referring to... the hollow region or the solid region?

The solid region - where the mobile charges are.

So, in general, the E-field in the solid region of a conductor would always be zero (for perfect conductors of course), right?

Swapnil said:
So, in general, the E-field in the solid region of a conductor would always be zero (for perfect conductors of course), right?

Anyone? ...

Swapnil said:
So, in general, the E-field in the solid region of a conductor would always be zero (for perfect conductors of course), right?

Yes, it would always be zero. Just keep in mind that this, like much of classical physics, is an idealization.

## 1. What are Maxwell's equations?

Maxwell's equations are a set of four partial differential equations that describe the behavior of electric and magnetic fields. They were developed by physicist James Clerk Maxwell in the 19th century and are considered one of the cornerstones of classical electromagnetism.

## 2. What is the significance of Maxwell's equations?

Maxwell's equations provide a unified framework for understanding the relationship between electric and magnetic fields and how they interact with matter. They have been used to make numerous predictions and have been confirmed by experiments, solidifying their importance in the field of electromagnetism.

## 3. What is the role of conductors in Maxwell's equations?

Conductors are materials that allow the flow of electric current. In Maxwell's equations, conductors are important because they have a unique effect on the behavior of electric fields. They can redistribute electric charges and create surface currents, which have implications for the propagation of electromagnetic waves.

## 4. How do Maxwell's equations apply to conductors?

In the presence of a conductor, Maxwell's equations are modified to include boundary conditions that account for the effects of the conductor. These conditions describe how electric and magnetic fields behave at the surface of the conductor and are essential for accurately modeling electromagnetic phenomena involving conductors.

## 5. What are some practical applications of Maxwell's equations and conductors?

Maxwell's equations and the study of conductors have numerous practical applications, including the design of electrical circuits, antennas, and communication systems. They are also essential for understanding and developing technologies such as wireless charging, radar, and electromagnetic shielding.

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