Electrostatics and symmetry

In summary, the question asks to find the direction and variables that the electrostatic field depends on in a uniformly charged plane. The solution involves studying the invariances of symmetry, specifically rotational and translational symmetries in the cylindrical coordinate system. While one attempt concludes that the field only depends on r, the correct solution shows that it actually depends on z due to the placement of charges in the xOy plane.
  • #1
ShaddollDa9u
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Homework Statement



"Find the direction and the variables which the electrostatic field depends on at all points of the plane (xOy) uniformly charged with the density of positive charges ϱ"

The Attempt at a Solution



So first of all, I have to study the Invariances of symmetry. I tried to switch to the cylindrical coordinate system (Ur, Uθ, Uz) and I observe that there is an Rotational symmetry following Uθ and a Translational symmetry following Uz, which made me conclude that the electrostatic field only depends on r.

2. Homework Equations

However, my teacher found that there is a Rotational symmetry following Uθ and a Translational symmetry following Ur, which means that the field only depends on z.
Shouldn't it be the opposite ?? How did he find so ??

Thank you
 
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  • #2
All the charges are in the xOy plane. How does that lead to a translation symmetry in z?
 

1. What is electrostatics and how does it relate to symmetry?

Electrostatics is the branch of physics that studies the behavior of electric charges at rest. It deals with the forces between these charges and the electric fields they produce. Symmetry, on the other hand, refers to the balance and proportion of elements in a system. In the context of electrostatics, symmetry plays a crucial role in determining the electric field and potential of a system, as well as the distribution of charges.

2. What is Coulomb's law and how does it apply to electrostatics?

Coulomb's law states that the force between two point charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them. This law is important in electrostatics as it allows us to calculate the force between two charges and understand how they interact with each other.

3. How does the concept of symmetry help in solving electrostatic problems?

The concept of symmetry helps in solving electrostatic problems by simplifying the calculations and reducing the complexity of the problem. Symmetry allows us to make assumptions about the distribution of charges and the electric field, making it easier to solve for the desired quantities.

4. What is Gauss's law and how is it related to symmetry?

Gauss's law states that the electric flux through a closed surface is equal to the total charge enclosed by that surface. This law is closely related to symmetry, as it allows us to use the symmetry of a charge distribution to simplify the calculation of the electric flux and solve for the electric field.

5. How does symmetry impact the behavior of electric fields in different materials?

Symmetry plays a significant role in determining the behavior of electric fields in different materials. The symmetry of a material's crystal structure can affect its dielectric properties, which in turn affects the electric field and potential inside the material. Additionally, the symmetry of a material can also determine its piezoelectric and pyroelectric properties, which involve the generation of electric fields in response to mechanical stress and temperature changes, respectively.

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