DC current flows in a wire when two loops are brought close by?

[elgen]

Hi all,

I have a question on the DC current flows, when two loops are close by.

Consider a battery driving a loop with a finite resistance. There are charges on the surface of the loop, which exerts forces to direct the flow of the DC current. The changes also produce an electrostatic field outside the loop. Together with the static magnetic field due to the current loop, the Poynting vector shows that power flows from the battery into the resistive wire via the fields. From Ampere's law, V=R_1 I, where R_1 denotes the resistance of the loop.

Consider that, I bring a second DC current loop to the vicinity of the first loop.

Is the electrostatic field from the second loop going to affect the current flow of the first loop?

My guess is, yes. As the two loops get closer, the charges on the two wires will re-arrange themselves. These charges will affect the DC current flow in both wires. As a result, R_1 will be different when being measured without the second loop, and with the second loop nearby.

Thank you for the attention.

Elgen

 

[Dale]

No, this shouldn’t cause any change in current. Charges will redistribute, but only to cancel out the external electrostatic field.

 

[vanhees71]

Why should the external electrostatic field cancel out? Just look at the complete solution of the DC through a coaxial cable!

 

[Dale]

Why should the external electrostatic field cancel out? Just look at the complete solution of the DC through a coaxial cable!

It is a pretty well known result:

https://pressbooks-dev.oer.hawaii.e...rs-and-electric-fields-in-static-equilibrium/

 

[vanhees71]

I was talking about a DC circuit and the one example you can evaluate completely analytically. There you see that there is a radial electric field between the conductors, which together with the magnetic fields leads to the energy transport from the "battery" to the resistor. You find this complete treatment, e.g., in Sommerfeld, Lectures on Theoretical Physics vol. 3 (electrodynamics).

 

[Dale]

I was talking about a DC circuit and the one example you can evaluate completely analytically. There you see that there is a radial electric field between the conductors, which together with the magnetic fields leads to the energy transport from the "battery" to the resistor. You find this complete treatment, e.g., in Sommerfeld, Lectures on Theoretical Physics vol. 3 (electrodynamics).

That isn't what the OP is asking about. The OP is asking about two nearby loops. Using a coaxial cable for the loops should be fine, but the point is that there are two nearby loops. Each one has its own battery and therefore its own current.

The OP wants to know what the additional current is in one loop due to the presence of the other loop nearby.

 

[vanhees71]

Ok, then I misunderstood the question. Of course the em. field of one loop has an influence on the moving charges in the other. The dominating effect is the magnetic interaction between the currents.

 

[Dale]

That will not change the current in the other loop.

 

[elgen]

Thank you both for the comment.

When a wire with a finite conductivity is connected to a battery, current flows around and there are charges on the the surface.

When another wire is brought close to the first wire, the charges will re-arrange themselves to reach an equilibrium state.

Since the charges exert Lorentz force on the current, the path of the current flow will change slightly leading to a change in resistance.

In many references, when people discuss conductors, due to the high conductivity, charges are considered moving freely to the surface to have E=0 inside the conductor. However, in my opinion, E cannot be 0, as it leads to the current flow by J=\sigma E. This non-zero E is determined by the charges, which are affected by the changes from the neighboring loop.

 

[Dale]

Since the charges exert Lorentz force on the current, the path of the current flow will change slightly leading to a change in resistance.

I am pretty skeptical of this. Do you have a professional scientific reference that confirms this?

 

[elgen]

This is the relevant article. In section III, it discusses the electron movement inside a conductor, and shows how electrostatic force and lorentz force work together to reach equilibrium inside.

This, is referred as the Hall-effect electric field, and it shows is 10^(-6) smaller compared to the longitudinal electric field.

https://www.researchgate.net/public...long_straight_strip_carrying_a_steady_current

I have not identified an article discussing two neighboring conducting wires. Extending the argument in the paper, I think, the hall-effect from the second loop would be the cause of the current change in the primary loop.

 

[hutchphd]

I think, the hall-effect from the second loop would be the cause of the current change in the primary loop.

I think any misunderstanding here is semantic. One must carefully carefully define "current in a wire" If the wire is perfectly conducting then there will be no change in net current. If the sensing wire is a good Hall effect material, then obviously DC Hall effect sensors work. If I understand their method of operation, they use a "magnetic circuit" to direct and concentrate the DC magnetic field onto the sensor.
Semantics and size.

 

[vanhees71]

If you mean "perfectly conducting" as "superconducting" than you have to argue differently anyway. The usual linear constitutive relations for classical electromagnetics in medium don't apply here. One should note that superconductivity involves "Higgsing" electromagnetic gauge invariance and thus leading to different equations in the superconducting medium (something like the London equations).

Of course it's true that for a complete treatment of the wire you have to take into account the Hall effect. This is already important for relativistic consistency. However, since the drift speeds of the electrons in a usual household wire is in the order of only 1mm/s (millimeters per second!) the Hall effect is completely negligible for all practical purposes.

 

[hutchphd]

Yes. By "perfect conductors" I was referring to the mythical wires that appear on circuit diagrams. These are super-duper superconductors!

 

[vanhees71]

There's no meaning to the geometrical view of circuit diagrams. theoretically they are all living in a single point since at various places retardation is neglected (by neglecting the "displacement current" in the Ampere Maxwell Law).