Force field in spherical polar coordinates

In summary, the concept of a force field in spherical polar coordinates involves describing the force acting on a particle in a three-dimensional space defined by radial distance, polar angle, and azimuthal angle. The force field can be expressed in terms of these coordinates, allowing for a clearer understanding of the forces affecting the particle's motion. Key aspects include the representation of vector fields, the use of gradient, divergence, and curl operations in spherical coordinates, and applications in physics, such as gravitational and electric fields. This framework is essential for analyzing systems with spherical symmetry.
  • #1
MatinSAR
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Homework Statement
A certain force field is given to me and I should do the following tasks to find out is it a conservative field or not.
Relevant Equations
pls see below.
Picture of question:
1702943544674.png

Part (a) : ##\nabla \times \vec F = 0## so a Potensial exists. I don't have problem with this part.
Part (b) : what I've done :
1702943893692.png

First experssion is 0 because ##\theta = \dfrac {\pi} {2}##. I don't know how to integrate over ##\theta ## when it is a constant.
 
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  • #2
Go back to where you wrote correctly $$d\vec{\lambda} = \hat r dr + r \hat \theta d \theta + r \sin \theta \hat \phi d \phi$$
Simplify this for integrating along the given unit circle.
 
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  • #3
The problem doesn't state whether or not the unit circle is centered at the origin of the coordinate system. So, I don't know if you are meant to assume that it is.
 
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  • #4
TSny said:
Go back to where you wrote correctly $$d\vec{\lambda} = \hat r dr + r \hat \theta d \theta + r \sin \theta \hat \phi d \phi$$
Simplify this for integrating along the given unit circle.
Is it the only way? I haven't done this before and I cannot understand what you mean ...
TSny said:
The problem doesn't state whether or not the unit circle is centered at the origin of the coordinate system. So, I don't know if you are meant to assume that it is.
I guess I don't want to assume that. According to the book final answer should be 0.

1702945890697.png
 
  • #5
MatinSAR said:
Is it the only way? I haven't done this before and I cannot understand what you mean ...

I guess I don't want to assume that. According to the book final answer should be 0.
Let's assume the unit circle is centered at the origin. For ##d\vec{\lambda}## along this circle, what can you say about the values of ##r##, ##dr##, ##d\theta## and ##\sin \theta##?

Thus, what does ##d\vec{\lambda}## simplify to?
 
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  • #6
TSny said:
Let's assume the unit circle is centered at the origin. For ##d\vec{\lambda}## along this circle, what can you say about the values of ##r##, ##dr##, ##d\theta## and ##\sin \theta##?

Thus, what does ##d\vec{\lambda}## simplify to?
I have checked my book again yet I could not find sth similar to this. I guess:
##r=1##
##\sin \theta=1##
##dr=dr##
##d \theta= 0##

If I'm right the second expression should be 0 to and I will get 0 as final answer.
 
  • #7
MatinSAR said:
I have checked my book again yet I could not find sth similar to this. I guess:
##r=1##
##\sin \theta=1##
##dr=dr##
##d \theta= 0##
OK, these look right. But you should be able to say more about ##dr##. Then you should be able to simplify the expression for ##d \vec \lambda## to a very simple result (which should make sense intuitively).

Then you can go on to think about the expression ##\vec F \cdot d\vec \lambda##
 
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  • #8
TSny said:
OK, these look right. But you should be able to say more about ##dr##. Then you should be able to simplify the expression for ##d \vec \lambda## to a very simple result (which should make sense intuitively).

Then you can go on to think about the expression ##\vec F \cdot d\vec \lambda##
Sorry for taking your time ... Should ##dr## be 0 for the circle?

Can't I answer without ##dr##? The first expression was zero and ##dr## doesn't change anything.
 
  • #9
MatinSAR said:
Sorry for taking your time ... Should ##dr## be 0 for the circle?
MatinSAR said:
Homework Statement: A certain force field is given to me and I should do the following tasks to find out is it a conservative field or not.
Relevant Equations: pls see below.

Picture of question:
View attachment 337396
Part (a) : ##\nabla \times \vec F = 0## so a Potensial exists. I don't have problem with this part.
Part (b) : what I've done :
View attachment 337397
First experssion is 0 because ##\theta = \dfrac {\pi} {2}##. I don't know how to integrate over ##\theta ## when it is a constant.
Is this from a published source?? If so, please identify.
 
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  • #10
hutchphd said:
Is this from a published source?? If so, please identify.
Yes.
This book is Arfken mathematical methods for physicists.
 
  • #11
MatinSAR said:
Should dr be 0 for the circle?
You tell us. What is the definition of dr?
 
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  • #12
Orodruin said:
You tell us. What is the definition of dr?
Radial spacing element If I have translated correctly.
Actually translating it to english is harder than its explanation for me.
 
  • #13
MatinSAR said:
Radial spacing element If I have translated correctly.
Actually translating it to english is harder than its explanation for me.
So how does radius change along the circle if r=1?
 
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  • #14
Orodruin said:
So how does radius change along the circle if r=1?
It doesn't change since radius is constant for a circle.
 
  • #15
Do you have access to Griffiths? Chapter 3.4.4. This field looks a lot like a dipole pointing in the z-direction. Just sayin’

That should answer part C.

Edit: A lot of dipole questions this week.
 
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  • #16
PhDeezNutz said:
Do you have access to Griffiths? Chapter 3.4.4. This field looks a lot like a dipole pointing in the z-direction. Just sayin’

That should answer part C.
Yes. I will check. Thanks.
 
  • #17
For part B can’t you use part A……via Stokes Theorem. Although I do think it is instructive to do the line integral directly.
 
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  • #18
MatinSAR said:
It doesn't change since radius is constant for a circle.
And therefore dr is …
 
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  • #19
PhDeezNutz said:
For part B can’t you use part A……via Stokes Theorem. Although I do think it is instructive to do the line integral directly.
Good idea!
Orodruin said:
And therefore dr is …
0 i think.
 
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  • #20
MatinSAR said:
0 i think.
Indeed.

You can also just use the parametrization ##\varphi = t## along with ##r=1## and ##\theta =\pi/2##. By definition
$$
dr = \frac{dr}{dt} dt
$$
and it should be pretty clear that ##dr/dt = 0##.
 
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  • #21
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FAQ: Force field in spherical polar coordinates

What is a force field in spherical polar coordinates?

A force field in spherical polar coordinates is a vector field that describes the distribution of forces in a system using spherical coordinates (r, θ, φ). Here, r represents the radial distance from a reference point, θ is the polar angle measured from the positive z-axis, and φ is the azimuthal angle measured from the positive x-axis in the xy-plane.

How do you express a force field in spherical polar coordinates?

A force field in spherical polar coordinates is expressed in terms of its components along the radial, polar, and azimuthal directions. It is typically written as F(r, θ, φ) = F_r(r, θ, φ) e_r + F_θ(r, θ, φ) e_θ + F_φ(r, θ, φ) e_φ, where F_r, F_θ, and F_φ are the magnitudes of the force components in the radial, polar, and azimuthal directions, respectively, and e_r, e_θ, and e_φ are the unit vectors in these directions.

How do you convert a force field from Cartesian to spherical polar coordinates?

To convert a force field from Cartesian coordinates (x, y, z) to spherical polar coordinates (r, θ, φ), you need to use the transformation equations: r = sqrt(x² + y² + z²), θ = arccos(z / r), and φ = arctan(y / x). The force components in spherical polar coordinates can then be found by projecting the Cartesian components onto the spherical unit vectors e_r, e_θ, and e_φ.

What are the applications of force fields in spherical polar coordinates?

Force fields in spherical polar coordinates are widely used in various fields of physics and engineering, including electromagnetism, gravitational studies, fluid dynamics, and quantum mechanics. They are particularly useful in problems involving spherical symmetry, such as the gravitational field around a planet or the electric field around a charged sphere.

What are the challenges in working with force fields in spherical polar coordinates?

One of the main challenges in working with force fields in spherical polar coordinates is the complexity of the mathematics involved, especially when dealing with differential equations and vector calculus. The non-linear nature of the coordinate transformations and the need to account for the varying unit vectors can make the analysis and computation more difficult compared to Cartesian coordinates.

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