∇ x E in Cylindrical Cordinates

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In summary, the conversation is about rewriting the equation for the curl of the electric field in cylindrical coordinates. The equations for the del operator and the electric field in cylindrical coordinates are given, and the goal is to find the expressions for the partial derivatives of the electric field in each coordinate direction.
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Homework Statement


I would like to re-write equation according to polar coordinates. ∇ x E = determinant of Polar coordinates. My question is how can i write determinant of polar coordinates?

Homework Equations



Maxwell Equations. (Faraday)

The Attempt at a Solution



E(x,y,z)= e(x,y) e^jβz
first e is a vector. ^ means exponantial

x=r.cos(φ)
y=r.sin(φ)

polar coordinates including e(r),e(φ),e(z)
 
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  • #2
is it true?

1. row i j k
2. row d/dr d/dφ d/dz
3. row e(r) e(φ) e(z)

so where can i use
x=r.cos(φ)
y=r.sin(φ)
 
  • #3
No. That is not the appropriate form for the curl in cylinder coordinates. You should be able to find the appropriate expression by a simple obline search for ”curl in cylinder coordinates”
 
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  • #4
İf possible, could you look a book named by Fiber Lasers, Basics Tech and App's. In this book page 12, you will see the equations set in polar coordinates for Maxwell equations.

Maxwell Equations set in polar coordinates:
r1lYaV.png


My homework is by using Maxwell equations in polar coordinates, to reach this Picture. But i did, what you said. I wrote the determinant. But i could'nt find. Could you use maxwell equations ∇ x E and ∇ x H till this Picture, step by step.

Thanks!
 

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  • #5
can someone help?
 
  • #6
This entire thread is VERY confusing.

The title of the thread asked about the curl of E in cylindrical coordinate, which should be trivial to write (https://www.chegg.com/homework-help/definitions/divergence-gradient-and-curl-in-cylindrical-coordinates-2). But then, the body of the thread is asking about polar coordinates and writing its determinant?! It is cylindrical, or is it polar?

And are you asking how for cylindrical coordinates, or spherical polar coordinates? Make up your mind.

And forget about taking the curl of anything. Do you know how to use the determinant method to find the cross product of two vectors in the first place?

BTW, just for reference, this is all math, not physics.

Zz.
 
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  • #7
can someone help?

in cylindrical coordinates not polar. just book says its in polar coordinates
 
  • #10
In my judgment, the easiest way to do this is to start out with the expressions for the del operator and the electric field strength vector in cylindrical coordinates:
$$\boldsymbol{\nabla}=\mathbf{i_r}\frac{\partial }{\partial r}+\mathbf{i_{\theta}}\frac{1}{r}\frac{\partial }{\partial \theta}+\mathbf{i_z}\frac{\partial }{\partial z}$$and
$$\mathbf{E}=E_r\mathbf{i_r}+E_{\theta}\mathbf{i_{\theta}}+E_z\mathbf{i_z}$$
So, $$\boldsymbol{\nabla}\boldsymbol{\times}\mathbf{E}=\mathbf{i_r}\boldsymbol{\times}\frac{\partial \mathbf{E}}{\partial r}+\mathbf{i_{\theta}}\boldsymbol{\times}\frac{1}{r}\frac{\partial \mathbf{E}}{\partial \theta}+\mathbf{i_z}\boldsymbol{\times}\frac{\partial \mathbf{E}}{\partial z}$$So, what do you get for ##\frac{\partial \mathbf{E}}{\partial r}##, ##\frac{\partial \mathbf{E}}{\partial \theta}##, and ##\frac{\partial \mathbf{E}}{\partial z}##?
 
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1. What is the meaning of "∇ x E" in cylindrical coordinates?

The symbol "∇ x E" represents the curl of the electric field vector in cylindrical coordinates. It describes how the electric field changes in direction and magnitude at a given point in space.

2. How is "∇ x E" calculated in cylindrical coordinates?

In cylindrical coordinates, the curl of a vector field can be calculated by taking the determinant of a matrix made up of the unit vectors in the cylindrical coordinate system and the partial derivatives of the vector field with respect to each coordinate.

3. What are the applications of "∇ x E" in cylindrical coordinates?

The curl of the electric field in cylindrical coordinates is used in various areas of physics and engineering, such as electromagnetics, fluid dynamics, and heat transfer. It helps to analyze and understand the behavior of electric fields in cylindrical systems, which are commonly found in cylindrical structures and devices.

4. How does "∇ x E" relate to Gauss's Law in cylindrical coordinates?

In cylindrical coordinates, Gauss's Law can be written as a differential equation involving the curl of the electric field. This allows for the determination of the electric field at a given point in space, given the charge distribution around it.

5. Can "∇ x E" in cylindrical coordinates be simplified?

In some cases, the curl of the electric field in cylindrical coordinates can be simplified using symmetries or special conditions in the problem. However, in general, it is a complex and fundamental quantity that cannot be simplified further.

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