Proof of symmetries in electrostatics

In summary: BTW, the OP didn't even mention there's a... @Bob SIn summary, the direction of the electric field must be radial, for a spherical charge distribution to remain invariant after applying a rotation matrix to its field.
  • #1
Trifis
167
1
I need a mathematical proof that should indicate the following: The direction of the electric field must be radial, for a spherical charge distribution to remain invariant after applying a rotation matrix to its field.

Analogously how can we prove that the electric field of a infinite cylinder is perpedicular to its surface?

Moreover, consider our augmentation at the boundary value problem of electrostatics. Dirichlet's condition was once more taken as granted, of course there not due to some symmetry but nevertheless the principle remains the same!

In other words, why is the electric field of the symmetric objects the way it is? Is it an experimental result or just the result of a superposition, which could be deducted? After all, all the great consequences of the laws of electrostatics are based on these initial "symmetry" assumptions! E.g. we could not calculate the electric field of a sphere with Gauss's law without "guessing" its direction!

I need maths in order to unravel this symmetry mystery...
 
Physics news on Phys.org
  • #2
Trifis said:
Analogously how can we prove that the electric field of a infinite cylinder is perpedicular to its surface?... E.g. we could not calculate the electric field of a sphere with Gauss's law without "guessing" its direction! I need maths in order to unravel this symmetry mystery...
The solution is simple. If the cylinder or sphere is conducting, then there can not be any electric field parallel to the surface. If there were, then there would be surface currents to redistribute the charge. Thus the electric field (if any) has to be perpendicular to the surface.
 
  • #3
Bob S said:
The solution is simple. If the cylinder or sphere is conducting, then there can not be any electric field parallel to the surface. If there were, then there would be surface currents to redistribute the charge. Thus the electric field (if any) has to be perpendicular to the surface.

I think the OP wants a mathematical proof, i.e., without understanding what electric field is and how it interacts with charges, prove that it is in the radial direction due to rotational invariance, since its source is rotationally invariant.
 
  • #4
sunjin09 said:
I think the OP wants a mathematical proof, i.e., without understanding what electric field is and how it interacts with charges, prove that it is in the radial direction due to rotational invariance, since its source is rotationally invariant.
All the OP needs to know is [itex] \overrightarrow{J}=\sigma\overrightarrow{E} [/itex]. If there is a component of electric field that is parallel to the surface, then there is a surface current. Therefore the electric field has to be perpendicular to the surface.
 
  • #5
@Bob S I don't mean to become ungrateful, but what I need to know is what I'm asking for (bold text). I'm already aware of the physical explanation.

I'm pretty sure, that a theorem of the following form can be found in the bibliography:
D[itex]\vec{E}[/itex](D[itex]\vec{r}[/itex]) = [itex]\vec{E}[/itex]([itex]\vec{r}[/itex]) [itex]\Rightarrow[/itex] [itex]\vec{E}[/itex]([itex]\vec{r}[/itex]) = E([itex]\vec{r}[/itex])[itex]\vec{e_{r}}[/itex]
That's what I'm looking for!
 
  • #6
I think first that you have to state that the surface of the sphere is conducting, and that the charge is static (no surface currents). Then using J = σE , you can show that if there are no surface currents, Eθ=Eφ=0. Thus there can be only a radial electric field Er.

Then you can use the 3 components of [itex] \left(\nabla\times \overrightarrow{E} \right) =0 \space [/itex] to show that [itex] \frac{\partial E_r}{\partial \theta} = \frac{\partial E_r}{\partial \varphi}=0 [/itex]. Thus Er is the same everywhere on the surface of the sphere: Curl E in spherical polar coordinates is


[tex] \left(\nabla\times \overrightarrow{E} \right)_\theta=\frac{1}{r\sin\theta}\frac{\partial E_r}{\partial \varphi} -\frac{1}{r}\frac{\partial\left(r E_\varphi \right)}{\partial r}=0 [/tex]
[tex] \left(\nabla \times \overrightarrow{E}\right)_\varphi=\frac{1}{r}\left(\frac{\partial\left(r E_\theta \right)}{\partial r} -\frac{\partial E_r}{\partial\theta } \right) =0[/tex]
[tex] \left(\nabla \times \overrightarrow{E}\right)_r = \frac{1}{r\sin \theta}\left(\frac{\partial\left(\sin\theta E_\varphi \right) }{\partial \theta}-\frac{\partial E_\theta}{\partial \varphi} \right)=0 [/tex]
 
  • #7
Well, you've mathematically elaborated your argument, namely the fact that the condition for an electrostatic equilibrium leads to the determination of the direction of the electric field.
That is appreciated.


BUT, you should be aware that there is also another proof of this, involing only symmetries rotation matrixes and reflections on planes...

I would have to compromise myself, that I'll never see that beautiful proof again:frown:
 
  • #8
If the charge is rotationally symmetric, so is the field. Therefore, the field cannot have tangential components. (For any tangential component at any point, a rotation along the axis connecting that point and the origin will change that component.) For cylinder, instead of considering rotational symmetry, consider reflection symmetries about the x-y plane and any plane containing z axis, similar conclusion can be made about the tangential components.

BTW, the OP didn't even mention there's a conductor.
 
Last edited:
  • #9
[itex]\nabla . E(r,\phi,\theta)=\rho(r)/\epsilon[/itex]

According to my calculation , if we define vector filed F as :

[itex]F(r,\phi,\theta)=\frac{\partial E(r,\phi,\theta)}{\partial \phi }[/itex]

then we have

[itex]\nabla . F=0[/itex]

and

[itex]\nabla × F=0[/itex]

which mean [itex]\frac{\partial E}{\partial \phi }=0[/itex]


I think you can prove the same for [itex]\frac{\partial E}{\partial \theta }[/itex].
 

1. What are symmetries in electrostatics?

Symmetries in electrostatics refer to the properties of an electric field that remain unchanged when certain transformations are applied. These transformations include rotations, reflections, and translations.

2. How do symmetries affect electric fields?

Symmetries can determine the behavior of electric fields, such as the direction and strength of the field. They can also help identify the presence of certain charges or configurations of charges in a system.

3. What is the proof of symmetries in electrostatics?

The proof of symmetries in electrostatics is based on mathematical equations and principles, such as Gauss's Law and the superposition principle. These equations show that the properties of an electric field remain unchanged under certain transformations, providing evidence for the existence of symmetries.

4. Why are symmetries important in electrostatics?

Symmetries are important in electrostatics because they provide a way to simplify and understand complex electric field configurations. They also allow us to make predictions about the behavior of electric fields in different situations.

5. How are symmetries used in practical applications of electrostatics?

Symmetries are used in practical applications of electrostatics, such as in designing electronic devices and studying the behavior of charged particles in electric fields. They also play a role in theoretical studies and research in the field of electrostatics.

Similar threads

  • Sticky
  • Electromagnetism
Replies
1
Views
2K
Replies
4
Views
2K
Replies
3
Views
3K
  • Electromagnetism
Replies
11
Views
7K
  • Electromagnetism
Replies
4
Views
812
  • Introductory Physics Homework Help
Replies
2
Views
349
  • Electromagnetism
Replies
7
Views
1K
Replies
5
Views
1K
  • Electromagnetism
Replies
9
Views
3K
Replies
14
Views
1K
Back
Top