Exploring Electric Field Boundaries at a Charge Density Boundary

In summary, the conversation discusses the boundary conditions for electrostatics, specifically the equations for the electric field components above and below a boundary and the question of whether the electric field in these equations is the sum of the external field and the field due to the charge at the boundary. The conversation also addresses the misconception of multiple electric fields and clarifies that there is only one electric field, which includes all contributions. The answer to the question is that both the external field and the contribution from the surface charge exhibit a discontinuity in the electric field across the boundary.
  • #1
Sebas4
13
2
Homework Statement
Boundary conditions electrostatics.
Relevant Equations
Meaning of the electric field variables in the boundary condition equations.
Hey, I have a really short question about electrostatics.
The boundary conditions are :
[tex] \mathbf{E}^{\perp }_{above} - \mathbf{E}^{\perp}_{below} = -\frac{\sigma}{\varepsilon_{0}}\mathbf{\hat{n}} [/tex],
[tex] \mathbf{E}^{\parallel }_{above} = \mathbf{E}^{\parallel}_{below}[/tex].

My question is what is [itex] \mathbf{E}^{\perp }_{above} [/itex], [itex] \mathbf{E}^{\perp }_{below} [/itex],
[itex] \mathbf{E}^{\parallel }_{above}[/itex] and [itex]\mathbf{E}^{\parallel}_{below}[/itex], is it the total electric field component near the boundary?
So is the electric field in this equation the sum of the external field and the electric field due to the charge at the boundary?

I will try to explain my question with an example.
Let's say we have an infinite plane with homogeneous charge density [itex]\sigma[/itex].
The electric field above the plane
[tex] \mathbf{E} = \frac{\sigma}{2\varepsilon_{0}}\mathbf{\hat{z}} [/tex].
The electric field below the plane is
[tex] \mathbf{E} = - \frac{\sigma}{2\varepsilon_{0}}\mathbf{\hat{z}} [/tex].
We have a homogeneous external field pointing in the z-direction, [itex]\mathbf{E}_{external} = \mathbf{E}_{0} \mathbf{\hat{z}}[/itex].
The electric field just below the surface of the plane is
[tex] \mathbf{E}_{total below} = \left(\mathbf{E}_{0} - \frac{\sigma}{2\varepsilon_{0}}\right) \mathbf{\hat{z}} [/tex].
The electric field just above the surface of the plane is
[tex] \mathbf{E}_{total above} = \left(\mathbf{E}_{0} + \frac{\sigma}{2\varepsilon_{0}}\right) \mathbf{\hat{z}} [/tex].
If we plug this in, in the boundary condition we get
[tex] \mathbf{E}_{total above} - \mathbf{E}_{total below} = \left(\mathbf{E}_{0} + \frac{\sigma}{2\varepsilon_{0}}\right) \mathbf{\hat{z}} - \left(\mathbf{E}_{0} - \frac{\sigma}{2\varepsilon_{0}}\right) \mathbf{\hat{z}} = \frac{\sigma}{\varepsilon_{0}} \mathbf{\hat{z}} [/tex].
This is true, according to the boundary condition.

I have also another question, this also works for non-homogeneous charge density boundaries? (I think so).
 
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  • #2
Sebas4 said:
Homework Statement:: Boundary conditions electrostatics.
Relevant Equations:: Meaning of the electric field variables in the boundary condition equations.

My question is what is Eabove⊥, Ebelow⊥,
Eabove∥ and Ebelow∥, is it the total electric field component near the boundary?
So is the electric field in this equation the sum of the external field and the electric field due to the charge at the boundary?
Yes. Total vector field
Notice the external field here does not really enter into the boundary condition because of the geometry. But the E field is always the total E field (or its components).
 
  • #3
It is a common misconception that there are several electric fields. This most likely stems from the fact that the equations governing electromagnetism (Maxwell’s equations) are linear, which means that solutions to them can be superpositioned and it is therefore easy to colloquially say things like ”the electric field of charge A” when what would be more precise would be ”the contribution to the electric field from charge A” (which is more of a mouthfull). However, there is no way to independently measure such a contribution.

The above also means that the answer to your question is ”both”. Both the electric field and the contribution from the surface charge will exhibit this discontinuity because all other contributions will be continuous across the surface and so any discontinuity must arise from the surface charge.
 
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Related to Exploring Electric Field Boundaries at a Charge Density Boundary

1. What is an electric field?

An electric field is a physical field that surrounds an electrically charged particle or object. It is a vector field, meaning it has both magnitude and direction, and is responsible for the electric force experienced by other charged particles within its boundaries.

2. How is electric field strength measured?

Electric field strength is measured in units of force per charge, such as newtons per coulomb (N/C). It can also be measured in volts per meter (V/m) or electron volts per meter (eV/m). These measurements give an indication of the strength of the electric field at a specific point in space.

3. What is a charge density boundary?

A charge density boundary is a boundary between two materials or regions with different charge densities. This can occur when there is a difference in the number of positive and negative charges in each region, resulting in an electric field at the boundary.

4. How do you explore electric field boundaries?

Exploring electric field boundaries involves using various tools and techniques, such as electric field mapping, to visualize and measure the electric field at different points along the boundary. This can help scientists understand the behavior of the electric field and how it affects charged particles in the surrounding area.

5. What are some practical applications of exploring electric field boundaries?

Exploring electric field boundaries has many practical applications, including in the design and optimization of electrical systems, such as circuits and antennas. It is also important in understanding the behavior of charged particles in materials, such as semiconductors, and in fields such as electrochemistry and plasma physics.

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