Electrodynamics: Electrostatic field potencial in Cartesian coordinates

In summary, the given problem involves finding an electrostatic field potential function in Cartesian coordinate system, given that the absolute permitivity is a coordinate function with a constant A. The equations used to solve the problem include \vec{D}=ε\vec{E}, \vec{E}= - grad\varphi, and div\vec{D}=ρ. However, without given charge distribution or boundary conditions, it is difficult to fully solve the problem.
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Homework Statement



It's given that absolute permitivity is a coordinate function: ε (x, y, z) = Asin(x)cos(y), where A=const

Homework Equations



We need to find an electrostatic field potential function [itex]\varphi[/itex] in Cartesian coordinate system.

The Attempt at a Solution



I tired to solve, but I don't know if it's ok. Check, please?

[itex]\vec{D}[/itex]=ε[itex]\vec{E}[/itex] and [itex]\vec{E}[/itex]= - grad[itex]\varphi[/itex]
div[itex]\vec{D}[/itex]=ρ
ρ=[itex]\frac{dDx}{dx}[/itex]+[itex]\frac{dDy}{dy}[/itex]+[itex]\frac{dDz}{dz}[/itex]
[itex]\vec{E}[/itex]=[itex]\vec{x}[/itex]0εEx+[itex]\vec{y}[/itex]0εEy+[itex]\vec{z}[/itex]0εEz
Dx=εEx
Dy=εEy
Dz=εEz
then
ρ=d Asin(x)cos(y)E x / dx + dAsin(x)cos(y)E y /dy + d Asin(x)cos(y)Ez / dz=Asin(x)sin(y)[itex]\frac{dE}{dx}[/itex]+Acos(x)cos(y)[itex]\frac{dE}{dy}[/itex]+Asin(x)cos(y)[itex]\frac{dE}{dz}[/itex]=Asin(x)sin(y) d [itex]\varphi[/itex]2 /dx2 +Acos(x)cos(y) d [itex]\varphi[/itex]2 / dy2 + Asin(x)cos(y) d [itex]\varphi[/itex] 2/ dz2

and now i don't know. ;D
 
Last edited:
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  • #2
You're not given any charge distribution, any boundary conditions?
 

FAQ: Electrodynamics: Electrostatic field potencial in Cartesian coordinates

1. What is an electrostatic field potential in Cartesian coordinates?

An electrostatic field potential in Cartesian coordinates is a representation of the electric potential at a particular point in space due to a distribution of electric charges. It is a scalar quantity that describes the strength and direction of the electric field at that point.

2. How is an electrostatic field potential calculated in Cartesian coordinates?

The electrostatic field potential in Cartesian coordinates is calculated using the formula V(x,y,z) = kQ/r, where V is the potential, k is the Coulomb constant, Q is the magnitude of the charge, and r is the distance from the charge to the point at which the potential is being calculated.

3. What is the relationship between electrostatic field potential and electric field strength?

The electrostatic field potential is directly related to the electric field strength. The electric field is the negative gradient of the potential, meaning that it is the rate of change of the potential with respect to distance. This relationship is described by the equation E = -∇V.

4. How does the electrostatic field potential vary with distance from a point charge?

The electrostatic field potential varies inversely with distance from a point charge. As the distance from the charge increases, the potential decreases. This relationship is described by the inverse square law, which states that the potential is proportional to the inverse of the distance squared.

5. What are some applications of electrostatic field potential in Cartesian coordinates?

Electrostatic field potential in Cartesian coordinates has many practical applications, including in designing electronic circuits, analyzing the behavior of charged particles in electric fields, and understanding the properties of materials in electrostatic environments. It is also used in electromagnetics, plasma physics, and other areas of science and engineering.

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