Force field in spherical polar coordinates

[MatinSAR]

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:

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 :

First experssion is 0 because $\theta = \dfrac {\pi} {2}$. I don't know how to integrate over $\theta$ when it is a constant.

 

[TSny]

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.

 

[TSny]

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.

 

[MatinSAR]

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 ...

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.

 

[TSny]

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?

 

[MatinSAR]

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.

 

[TSny]

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 $\vec F \cdot d\vec \lambda$

 

[MatinSAR]

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.

 

[hutchphd]

Sorry for taking your time ... Should $dr$ be 0 for the circle?

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.

 

[MatinSAR]

Is this from a published source?? If so, please identify.

Yes.
This book is Arfken mathematical methods for physicists.

 

[Orodruin]

Should dr be 0 for the circle?

You tell us. What is the definition of dr?

 

[MatinSAR]

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.

 

[Orodruin]

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?

 

[MatinSAR]

So how does radius change along the circle if r=1?

It doesn't change since radius is constant for a circle.

 

[PhDeezNutz]

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.

 

[MatinSAR]

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.

 

[PhDeezNutz]

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.

 

[Orodruin]

It doesn't change since radius is constant for a circle.

And therefore dr is …

 

[MatinSAR]

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!

And therefore dr is …

0 i think.

 

[Orodruin]

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 $dr/dt = 0$.

 

[MatinSAR]

@Orodruin @TSny @PhDeezNutz

Thank you for your help.