Why a E field cannot hold a charge in space?

[Buffu]

From the book,

You can't construct an electrostatic field that will hold a charged particle in vacuum.

Justification :
Suppose we have electric field in which there exist a point P ar which a +ve charged particle would be in stable equilibrium. That means that any small displacement of the particle from P must bring it to a place where an electric field acts to push it back toward P. But that means that a little sphere around P must have E pointing inward everywhere on its surface. That contradicts Gauss' law , for there is no -ve charge in the sphere. In other words you can't have have an empty region where electric field points all outward or inward which is needed for stable equilibrium.

In terms or electric potential, a stable position for a charged particle must be where the potential is either lower than that at all neighbouring points or higher than that at all neighbouring points. Clearly neither is possible as potential is a function whose average value over a sphere is always equal to its value at the centre.

I did not understand,
1. Why does fields coming inwards the sphere contradicts the Gauss law ?
2. I am unable to connect the potential picture and electric field picture. Why does the point P must have lowest or highest potential than neighbouring particles ?

3. I understand the fact that "potential is a function whose average value over a sphere is always equal to its value at the centre". It is because of Laplace's equation but I don't understand how this is connected here. In potential picture there is no sphere after all right ?

For the first question, I think it contracdicts the Gauss law because there is a +ve inside the sphere or the field lines should be outward not inward, which results in -ve flux instead of +ve flux. Am I correct ?

 

[Nugatory]

From the book,

Which book?

 

[Buffu]

Which book?

Purcell 2nd edition.

 

[Nugatory]

1. Why does fields coming inwards the sphere contradicts the Gauss law ?
2. I am unable to connect the potential picture and electric field picture. Why does the point P must have lowest or highest potential than neighbouring particles ?
...
For the first question, I think it contracdicts the Gauss law because there is a +ve inside the sphere or the field lines should be outward not inward, which results in -ve flux instead of +ve flux. Am I correct ?

You are not quite correct. Remember, an electric field is defined by asking what would happen to a test particle placed in the field, but the field is there whether there's a test particle or not. Thus, the text isn't saying that there is a positive charge, it is considering whether there could be an electric field that would push a positively charged test particle to the center if we were to place one there. Such a field would clearly violate Gauss's law because there isn't any charge at all inside the sphere, so no way for there to be a non-zero flex across the surface of the sphere.

For the second question: Take a moment to review the relationship between the electric field and the potential, and the definition of the gradient.

If the particle is stable at the point P, that means that there is no net force on a particle at point P, which means that the electric field there is zero. But the electric field is the gradient of the potential, so that means that the gradient of potential must be zero at that point... And the gradient can only be zero at a local maximum or a local minimum.

 

[Buffu]

You are not quite correct. Remember, an electric field is defined by asking what would happen to a test particle placed in the field, but the field is there whether there's a test particle or not. Thus, the text isn't saying that there is a positive charge, it is considering whether there could be an electric field that would push a positively charged test particle to the center if we were to place one there. Such a field would clearly violate Gauss's law because there isn't any charge at all inside the sphere, so no way for there to be a non-zero flex across the surface of the sphere.

For the second question: Take a moment to review the relationship between the electric field and the potential, and the definition of the gradient.

If the particle is stable at the point P, that means that there is no net force on a particle at point P, which means that the electric field there is zero. But the electric field is the gradient of the potential, so that means that the gradient of potential must be zero at that point... And the gradient can only be zero at a local maximum or a local minimum.

So the sphere is not covering our charge at P instead it is close to it. right ?

 

[Nugatory]

So the sphere is not covering our charge at P instead it is close to it. right ?

Not right. The sphere completely surrounds the point P. But there is no charge at P in the problem; this is a question about the electrical field inside a region of space and whether that field violates Gauss's law.

 

[Buffu]

Yes I understand now. Thanks.