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sams

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^{th}Edition, page 44, Example 1.8.2, the curl of the central force field is zero.

1. Why are central force fields irrotational?

2. Why are central force fields conservative?

Any help is much appreciated...

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- Thread starter sams
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sams

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1. Why are central force fields irrotational?

2. Why are central force fields conservative?

Any help is much appreciated...

- #2

dRic2

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- #3

sams

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dRic2

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Why would you say so? If ##f## is a generic function then you can't say much about it.

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bobob

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Why would you say so? If ##f## is a generic function then you can't say much about it.

It is not a generic function. It's the gradient of a scalar. That and vector identities are all you need.

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Why can't you just take the curl of the function as @dRic2 has suggested? This is not something ambiguous. You should be able to tell

Zz.

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sams

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Thank you once again for your kind efforts...

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rotational fieldsarenon-conservative. Is there any physical meaning or physical interpretation? This leads to the questions raised above: why are central force fieldsirrotationalandconservative? In other words, why alsoirrotational fieldsareconservative?

Thank you once again for your kind efforts...

I'm a bit puzzled by this.

A field having a "rotational"component, by definition, will have a "rotational impulse" (I made up that phrase), and so, there is something "gained" along a rotational path. Something that has no rotational component, shouldn't have such an impulse and will not provide any type of gain along that path.

So it is conservative or non-conservative by definition. Your question is like asking why a straight line has no curvature.

Zz.

- #9

sams

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Can't a linear field be also non-conservative? For example, a time-varying magnetic field!Something that has no rotational component, shouldn't have such an impulse and will not provide any type of gain along that path.

So again, is it always the case of a straight line that results in a conservative field? Are there other factors that affects conservative fields?Your question is like asking why a straight line has no curvature.

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Can't a linear field be also non-conservative? For example, a time-varying magnetic field!

First of all, I didn't see any indication that we're talking about time-varying fields. Secondly, a magnetic field is already non-conservative in the static case.

So again, is it always the case of a straight line that results in a conservative field? Are there other factors that affects conservative fields?

Your question then is: is there any mathematical function, whose curl is zero, that ........ So this IS a mathematical issue, contrary to what you stated earlier.

Zz.

- #11

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If ## \nabla \times E \neq 0 ## anywhere in a region, that means by Stokes theorem, ## \\ ## (Note: Stokes theorem says## \int (\nabla \times E) \cdot \hat{n} \, dA=\oint E \cdot dl ##),## \\ ## ## \oint E \cdot dl \neq 0 ## for at least one path that this integral might take where, in integrating over the loop, the finish point is the starting point. This means that there is non-zero work over this loop. Thereby, the force is non-conservative if ## \nabla \times E \neq 0 ##.## \\ ## If ## \nabla \times E =0 ## everywhere, by similar arguments we see we have a conservative force.

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