Electrostatic Boundary Conditions

In summary, according to the book, as an infinite, uniformly charged plane (positive charge) is approached, an electric field is created that does not depend on the distance from the plane. However, as the plane is crossed, a discontinuity in the field is noted. The difference in magnitude between the field above and below the plane is determined by Gauss' law.
  • #1
Hypercube
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Hello PF community,

I am currently self-studying electrodynamics from Griffiths textbook, and I'm at a point where the book discusses electrostatic boundary conditions. If someone can please check if my reasoning is right.

So, as I am approaching an infinite, uniformly charged plane (let the charge be positive), there is electric field pointing away from the plane, which has a magnitude equal to:

$$\vec E=\frac{\sigma}{2\epsilon_0}\hat n$$

where ##\hat n## is a vector perpendicular to the plane. This can be derived from Gauss' law, and it shows that electric field of a point does not depend on the distance from the plane. Magnitude on the other side of the plane would be the same, except it would have opposite direction.

However, according to the book, as I'm approaching the boundary and eventually cross it, there is a discontinuity. The author writes:

$$E_{above}-E_{below}=\frac{\sigma}{\epsilon_0}$$

Note that the author does not have these in bold. This implies that (somehow) the magnitude of the electric field above the plane is not equal to the magnitude of the electric field below it. How is that possible? I expected RHS to be 0 in this case.

Thanks.
 
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  • #2
Hypercube said:
Note that the author does not have these in bold.
Probably an oversight.

The magnitude can be different (and for an ideal plate capacitor the difference in magnitude will be this value, because the field on the outside is zero), but the boundary condition is the change in the field vector.
 
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  • #3
Perhaps Griffiths was thinking in terms of a component of the field, e.g. ##E_z## for a sheet parallel to the xy-plane. Unlike the magnitude ##E##, this can be positive or negative.
 
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  • #4
jtbell said:
Perhaps Griffiths was thinking in terms of a component of the field, e.g. ##E_z## for a sheet parallel to the xy-plane. Unlike the magnitude ##E##, this can be positive or negative.
Yes. Griffiths was in fact referring to the component of the field perpendicular to the surface. I happen to have a copy of his book handy and checked again, to make sure. It was neither an oversight, nor an imprecisely stated result. The relation given in Griffiths 3rd edition says:
Eabove - Ebelow = σ/ε0
 
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  • #5
But I still don't quite understand what this means.

Yes, he is talking about perpendicular component. But if I have a flat, infinite plane and the charge is uniformly distributed, perpendicular component is the only component, right?

Suppose that this charged plane lies in the x-y plane. At all points below z-axis the field is constant and negative, and at all points above z-axis, field is constant and positive, and there is discontinuity as we "cross" the boundary. Is the below graph correct?
upload_2018-1-22_5-11-45.png


If the field above is uniform and constant, and the field below is uniform and constant, then the difference is a simple matter of ##E_{above}-E_{below}##. Instead, he applies Gauss' Law, states limiting condition where ##\epsilon## (distance between upper and lower Gaussian surface parallel to the plane) converges to zero, etc, etc... Intuition tells me that there must be something that I overlooked.
 

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  • #6
I’m traveling now, so I don’t have my copy of Griffiths at hand to see the exact context for myself. However, that limiting process seems to be what one would use for the more general case of a non-planar sheet of charge, in which the field is not uniform at finite distances from the sheet.

Perhaps Griffiths is actually discussing the more general case, or perhaps he is simply anticipating it.
 
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  • #7
jtbell is right. Griffiths is indeed discussing the general case of a non-flat sheet of charge. The OP is a particular case. The diagram given by OP is correct for an infinite plane sheet of charge. But the relation in Griffiths is more generally valid. For an infinite flat sheet, as hypercube says, the perpendicular component is the only component. In the general case, even if there is a parallel com[ponent, the discontinuity in the perpendicular component still follows the relation given in Griffiths.
 
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  • #8
I understand now. Thank you for your replies.
 

1. What are electrostatic boundary conditions?

Electrostatic boundary conditions are a set of rules that govern the behavior of electric fields at the interface between two different materials or regions. They describe how the electric potential and electric field must be continuous and consistent across this boundary.

2. What is the importance of electrostatic boundary conditions in scientific research?

Electrostatic boundary conditions play a crucial role in many areas of scientific research, including physics, chemistry, and engineering. They are essential for understanding the behavior of electric fields in different materials and for solving complex problems involving electric charges and fields.

3. What are the types of electrostatic boundary conditions?

There are two main types of electrostatic boundary conditions: Dirichlet and Neumann. Dirichlet boundary conditions specify the electric potential at the boundary, while Neumann boundary conditions specify the normal component of the electric field at the boundary.

4. How are electrostatic boundary conditions applied in practical situations?

In practical situations, electrostatic boundary conditions are used to determine the behavior of electric fields in various materials, such as conductors, insulators, and semiconductors. They are also used in the design and analysis of electronic devices, such as capacitors, transistors, and integrated circuits.

5. What are some common misconceptions about electrostatic boundary conditions?

One common misconception about electrostatic boundary conditions is that they only apply to static electric fields. In reality, they can also be applied to time-varying electric fields. Another misconception is that they only apply to two-dimensional situations, but they can also be applied to three-dimensional cases.

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