∇E = 0 in an Ohmic, 2d Hall conductor with constant B?

In summary, ∇E = 0 is a mathematical expression that represents a constant electric field and no spatial variation in an Ohmic, 2d Hall conductor with a constant magnetic field (B). This type of conductor follows Ohm's Law and has a linear relationship between current and voltage. The term "2d Hall conductor" refers to a two-dimensional material with a thickness much smaller than its length and width, often used in Hall effect experiments. A constant magnetic field is necessary to observe the Hall effect, which is used to measure charge carrier density and mobility. ∇E = 0 causes electrons to take a curved path due to the Lorentz force, resulting in the Hall effect and the accumulation of electrons on one side
  • #1
ManDay
159
1
I've been trying to figure this out for long now but unfortunally, I'm not able to prove that ∇E = 0 in an Ohmic, 2d Hall conductor E = Rj + v×B with B = const (and orthogonal to j).

There is quite a bit of subtlety involved in how to interpret v in that sort of ad-hoc generalization of Ohm's law, and I'm not 100% sure I got that right. If we define ρ by j := ρv and assume a static background-charge ρ' so that ∇E = (ρ' + ρ)/ε I end up at

∇E = -1/ρ²(B×j + B²/(ρR)j)·∇ρ

from Ohm's relation. However, I don't see how that would prove that ∇E = 0 given that this PDE appears to have nontrivial solutions for j's which satisfy ∇j = 0.

Any ideas would be greatly appreciated.

Edit: I just thought of something: Perhaps this depends on assumptions concerning the carriers of charge. In particular that the mobile charges are strictly of one sign and thus ρ > 0! Then it could possibly be shown that the PDE cannot be satisfied. Investigating...
 
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  • #2

Thank you for your question. It is indeed a complex issue to prove that ∇E = 0 in an Ohmic, 2d Hall conductor with a constant magnetic field and orthogonal current. As you have noted, there are some subtleties involved in interpreting the velocity term in the generalized Ohm's law. However, I would like to offer some insights and suggestions that may help you in your investigation.

Firstly, it is important to note that the equation you have derived for ∇E is not a PDE, but rather a vector equation. This means that it must hold for each component of ∇E to be equal to zero. So, it is not sufficient to show that one component is zero, but rather all components must be zero for the equation to hold.

Secondly, you are correct in thinking that the assumption of ρ > 0 is crucial in proving that ∇E = 0. This is because the charge carriers in an Ohmic conductor are assumed to be of one sign, and the current density is proportional to the velocity of these carriers. If we have a mixture of positive and negative carriers, then the current density will not be proportional to the velocity and the equation for ∇E will not hold.

Lastly, I would suggest looking into the concept of continuity equation, which relates the divergence of current density to the change in charge density. This may provide some insights into the problem and help you in your investigation.

I hope this helps and good luck with your research.
 

1. What does ∇E = 0 mean?

∇E = 0 is a mathematical expression that represents the fact that the electric field (E) is constant and has no spatial variation in an Ohmic, 2d Hall conductor with a constant magnetic field (B). This means that the electric field lines are parallel and evenly spaced throughout the conductor.

2. What is an Ohmic conductor?

An Ohmic conductor is a material that follows Ohm's Law, which states that the current through a conductor is directly proportional to the voltage across it. This means that an Ohmic conductor has a linear relationship between current and voltage, and its resistance remains constant regardless of the applied voltage.

3. What is a 2d Hall conductor?

A 2d Hall conductor refers to a two-dimensional material that has a thickness much smaller than its length and width. This type of conductor is often used in Hall effect experiments, where a magnetic field is applied perpendicular to the direction of current flow, resulting in a measurable voltage perpendicular to both the current and the magnetic field.

4. What is the significance of a constant magnetic field (B) in this scenario?

In this scenario, a constant magnetic field (B) is necessary to observe the Hall effect, which is the voltage generated perpendicular to the applied current and magnetic field. This effect is only present in materials with charge carriers that can move freely, such as in metals, and is used to measure the charge carrier density and mobility.

5. How does ∇E = 0 affect the behavior of electrons in the conductor?

∇E = 0 means that the electric field is constant and has no spatial variation in the conductor. This has the effect of causing the electrons to take a curved path due to the Lorentz force, which is the force experienced by a moving charge in a magnetic field. This results in the Hall effect, where the electrons accumulate on one side of the conductor, creating a measurable voltage perpendicular to the applied current and magnetic field.

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