Vectors in different coordinate systems

[Ahmad Kishki]

how do i write vectors in polar coordinate? And what will the azimuth coordinate represent?

I was trying to figure out the vector connecting a ring to its center using polar coordinates, so that i would perform an integration over d(phi) (finding the electric field due to a semicircle at the origin). I wrote the vector as (-rho, phi,0) but then i ran into trouble with the units since the units of phi are radians. I worked this out in cartesian pretty easily but i am eager to know how this will work in polar, since polar here seems to me more natural.

Thanks in advance :)

 

[jedishrfu]

Welcome to PF!

I think you need to include the radius as one component. In your integration you need the Jacobian too
as in r dr dtheta which provides the missing units of measure.

Here's some info on polar coordinates and circles:

http://en.wikipedia.org/wiki/Polar_coordinate_system#circle

Show us your work and we can help you even better.

 

[Ahmad Kishki]

I know the jacobian, and infact already found it in this case, but the issue is not here. Here the vector is included in the integration, but the phi component has units radians, while the rho has meters. The idea here is that maybe my idea about vectors in cylindrical / polar coordinates is wrong. This is so since the length of a vector can not be (rho + phi)^1/2 since the rho and phi have different units.

 

[Ahmad Kishki]

Rho^2 +phi^2 (first post sorry)

 

[Ahmad Kishki]

 

[Ahmad Kishki]

View attachment 74747

Welcome to PF!

I think you need to include the radius as one component. In your integration you need the Jacobian too
as in r dr dtheta which provides the missing units of measure.

Here's some info on polar coordinates and circles:

http://en.wikipedia.org/wiki/Polar_coordinate_system#circle

Show us your work and we can help you even better.

Here is the problem.. I am stuck at finding the vector that goes from the ring to the origin. And assuming i know that vector, which phi should i use to obtain the value of the electric field at the origin? I mean phi changes with position. I realize there is an easier way, but i want to know this way as well. (Sorry for multiple posts, this is my first time using a forum, soi am not sure i get the hang of it)

 

[jedishrfu]

As I look at the wolfram examples like arc length, they don't treat the vector in a component fashion but as a vector.

http://mathworld.wolfram.com/PolarCoordinates.html

And that may be why you hung up here, you're trying to work with it in its native form.

I think they would define your vector as being normal to the arc ie the second derivative of the arc line.