Understanding the Relationship Between Electric Fields and Conductors

[vig]

There are some very basic questions about conductors that are bugging me:
1. I know that E=0 for a conductor is only if the fields remain static, If there was a time varying field, the electrons would be in constant motion across the conductor, meaning that at any point of time, a net tangential field does exist causing the electrons to move. So why is it that we keep applying boundary conditions that the tangential component of E field on a conductor MUST be 0?..

2. What kind of a field does a DC voltage source provide in a wire that causes a constant flow of current? I ask this because if the potential difference remains constant, the electric field must be zero (E=-delV) and if E is indeed zero, no current can flow...

I guess the answers to both are interlinked.
Would be grateful for any help.
Thanks in advance :)

 

[Dale]

1. I know that E=0 for a conductor is only if the fields remain static, If there was a time varying field, the electrons would be in constant motion across the conductor, meaning that at any point of time, a net tangential field does exist causing the electrons to move. So why is it that we keep applying boundary conditions that the tangential component of E field on a conductor MUST be 0?..

Actually, the electrostatic condition is a little stronger than just the fields remain static. Not only do the fields need to remain static, but the currents must also be 0. You can have scenarios where the fields are static, but there is a non-zero current. Such scenarios are called "magnetostatic" so as to distinguish them from electrostatic ones.

In any scenario where there is a tangential E-field on the surface of a conductor then you get a current along the surface of the conductor. Since the electrostatic condition is defined as having no currents it implies that there is no tangential E-field, essentially by definition. I.e. you certainly can have tangential E-fields on the surface of a conductor, but then it is not electrostatic.

2. What kind of a field does a DC voltage source provide in a wire that causes a constant flow of current? I ask this because if the potential difference remains constant, the electric field must be zero (E=-delV) and if E is indeed zero, no current can flow...

It is an E field. Your comment about the potential difference is incorrect. The gradient consists of spatial derivatives, so the time derivative of the potential can be zero and still have non-zero spatial derivatives, and therefore a non-zero E-field.

 

[vig]

Actually, the electrostatic condition is a little stronger than just the fields remain static. Not only do the fields need to remain static, but the currents must also be 0. You can have scenarios where the fields are static, but there is a non-zero current. Such scenarios are called "magnetostatic" so as to distinguish them from electrostatic ones.

In any scenario where there is a tangential E-field on the surface of a conductor then you get a current along the surface of the conductor. Since the electrostatic condition is defined as having no currents it implies that there is no tangential E-field, essentially by definition. I.e. you certainly can have tangential E-fields on the surface of a conductor, but then it is not electrostatic.

forgive my ignorance, but how can a STATIC field produce a current in the first place?..All that i can decipher is that if the E field changes at a constant rate, the current through the conductor must be constant.
If the tangential field is 0 only in the electrostatic case, why do we apply the condition on, say, reflection of an EM wave from a dielectric? There doesn't seem to be anything static here..
One theory i could think of is that in an ideal conductor, maybe we can assume that the moment the field is applied, the electrons rearrange themselves without any delay, implying that at any instant of time, the tangential field is 0..i don't know about the validity of this theory though..

It is an E field. Your comment about the potential difference is incorrect. The gradient consists of spatial derivatives, so the time derivative of the potential can be zero and still have non-zero spatial derivatives, and therefore a non-zero E-field.

could you pls enumerate on how exactly a spatial distribution of voltage is set up?

 

[Dale]

forgive my ignorance, but how can a STATIC field produce a current in the first place?

According to the Lorentz force law, a static E-field means that there is a steady force on charges. In a conductor, the charges are not stuck firmly to their atoms, so they will move in response to that force. That is the defining characteristic of a conductor.

All that i can decipher is that if the E field changes at a constant rate, the current through the conductor must be constant.

No, if the E-field changes at a constant rate then the current will also change at a constant rate. The current in a conductor is proportional to the E-field, not the time derivative of the E-field. This is called Ohm's law.

If the tangential field is 0 only in the electrostatic case, why do we apply the condition on, say, reflection of an EM wave from a dielectric? There doesn't seem to be anything static here..

I don't know for sure, do you have a reference?

could you pls enumerate on how exactly a spatial distribution of voltage is set up?

Well, the simplest example is a battery. The + terminal has a higher voltage than the - terminal, and they are located at different locations in space, so there is a spatial distribution of voltage.

 

[vig]

According to the Lorentz force law, a static E-field means that there is a steady force on charges. In a conductor, the charges are not stuck firmly to their atoms, so they will move in response to that force. That is the defining characteristic of a conductor.

No, if the E-field changes at a constant rate then the current will also change at a constant rate. The current in a conductor is proportional to the E-field, not the time derivative of the E-field. This is called Ohm's law.

Yea..so a static field will cause the electrons to move until a point is reached where it completely cancels out the external fields, so the net E field becomes zero.
However, if the field were time varying, the electrons would have to keep moving in an attempt to cancel out the electric field (but it never happens), causing a current.
Now if the field were to vary at a constant rate, the rate at which the electrons would move would also be constant, which is a constant current.
So in a non ideal situation, there would always be a net tangential E field on the conductor at any point.
The only thing that i can make out is, that if it were an ideal conductor (σ=∞), then there would be no time taken for the electrons to drift and cancel, so that at any point of time, the E(tan) must be zero. Is this correct?

I don't know for sure, do you have a reference?

That's the reason, also, why we carry put analysis using the methods of images for a currentcarrying element above a ground plane.

 

[Dale]

Yea..so a static field will cause the electrons to move until a point is reached where it completely cancels out the external fields, so the net E field becomes zero.

The E field is what you are calling the "net" E field. So this is already a time varying E field and therefore a time varying current. Again, the current is proportional to the E field, not the change (time derivative) of the E field.