Electric field of a neutral conductor just at the surface

[f3sicA_A]

Homework Statement

What is the electric field of a net neutral conductor just at the surface of the conductor in the presence of an electric field?

Relevant Equations

$\oint\boldsymbol{E}\cdot\text{d}\boldsymbol{a}=\frac{Q}{\epsilon_0}$

I am reading Electrodynamics by Griffiths, and in section 2.5, he mentions that one of the properties of a conductor is that the electric field just outside the conductor is perpendicular to the surface of the conductor, and gives the following image:

However, I don't understand why the electric field points outward everywhere. Here is my understanding:

If we have a net neutrally charged conductor in an external electric field, the electric field outside is only equal to the external electric field, the electric field inside the conductor is equal to 0 (due to the movement of electrons giving rise to an internal electric field balancing out the external electric field), the electric field AT the surface can be determined using Gauss' law:

$\oint\boldsymbol{E}\cdot\text{d}\boldsymbol{a}=\frac{Q}{\epsilon_0}$

$\implies E_{1}\cdot A+E_2\cdot A=\frac{\sigma\cdot A}{\epsilon_0}$

where the gaussian surface is a really tiny surface such that the electric field lines can be considered as effectively perpendicular to the gaussian surface, then #E_1# is the electric field on the outer side of the conductor and #E_2# is electric field on the inner side of the conductor. Though I say "outer" and "inner" sides of the conductor, I really mean AT the surface of the conductor itself, just two different directions that we are looking at, the outwards direction and the inwards direction.

Now then #E_2=0# because the electric field balancing out with the external electric field, giving us:

$E_1\cdot A=\frac{\sigma A}{\epsilon_0}$

$\implies\boldsymbol{E}=\frac{\sigma}{\epsilon_0}\boldsymbol{\hat{n}}$

where #\boldsymbol{\hat{n}}# is the direction perpendicular to the surface of the conductor (a tangential component cannot exist since otherwise the charges would to achieve electrostatic equilibrium). This is all seems fine, except my problem is, then it would seem to me that the electric field for half the surface of the conductor should point outwards (due to positive charge density) and the electric field on the other half of the surface of the conductor should point inwards (due to negative charge density); however, as seen in the image from Griffiths, everything is pointing outwards. Why is this?

 

[TSny]

Properties (i) through (v) listed in section 2.5 do not require the conductor to be neutral (net charge zero).

When discussing property (i), Griffiths does consider an example (shown in Fig. 2.42) where the conductor is neutral and is placed in an external field. But property (i) is valid even if the conductor has a net charge. When discussing property (v), Griffiths considers the example shown in Fig. 2.43 where the conductor is not neutral. It carries a net positive charge, and no external field is indicated.

 

[f3sicA_A]

When discussing property (v), Griffiths considers the example shown in Fig. 2.43 where the conductor is not neutral. It carries a net positive charge, and no external field is indicated.

I see, thank you for the clarification! I do realize that there is no need for the properties to hold only if the conductor is neutral; however, since it wasn't explicitly mentioned, I figured that property (v) was given by extending the example of a neutral conductor in an external electric field. However, I suppose if it was neutral, we wouldn't talk about the flow of charges around the surface (since only excess charge settles at the surface of the conductor).

 

[TSny]

...I figured that property (v) was given by extending the example of a neutral conductor in an external electric field. However, I suppose if it was neutral, we wouldn't talk about the flow of charges around the surface (since only excess charge settles at the surface of the conductor).

If the conductor is neutral and isolated (no externally applied field), then naturally there will be no charge accumulation at any point inside the conductor or on the surface of the conductor.

However, if the neutral conductor is placed in an externally applied field, free charge will move to produce some surface charge. The net surface charge will be zero since the conductor remains neutral. When equilibrium is obtained, there will be no electric field at points within the conductor, and the field just outside the conductor will be perpendicular to the surface.

A rough sketch for a neutral spherical conductor placed in an external electric field is shown here in figure 1.29. This is analogous to Griffiths' figure 2.42 for the conducting slab.

 

[f3sicA_A]

A rough sketch for a neutral spherical conductor placed in an external electric field is shown here in figure 1.29. This is analogous to Griffiths' figure 2.42 for the conducting slab.

Yes, this is exactly how I figured it should be, which is why I was confused looking at Figure 2.43 from Griffiths. Thank you for the reference!