# Why are central force fields irrotational and conservative?

• sams
So the ## \nabla \times E =0 ## condition is a mathematical way of saying "this force is conservative" and, more physically, "the force is conservative because the work done by it on a closed loop is 0".In summary, the curl of a central force field is zero because it is a function of only the position vector, making it irrotational. This also means that the central force field is conservative, as there is no rotational impulse and the work done on a closed loop is zero. This is a mathematical way of saying that the force is conservative and is related to the physical concept of a straight line having no curvature.

#### sams

Gold Member
In Mathematical Methods for Physicists, 6th 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...

Because the central force field is a function of only ## \mathbf r## (position vector). Try to evaluate the curl of ## f(\phi, \theta, \mathbf r) = f(\mathbf r)##

sams
In case f is in function of ϕ, θ, and r, do we consider f to be solenoidal and thus not a conservative force field?

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

dRic2 said:
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.

sams said:
In case f is in function of ϕ, θ, and r, do we consider f to be solenoidal and thus not a conservative force field?

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 mathematically if a function is conservative or not.

Zz.

sams
Thank you all for your replies and I am sorry for this confusion. In fact, I am not asking from a mathematical perspective. The mathematics is straightforward. In my questions above, I just would like to know why rotational fields are non-conservative. Is there any physical meaning or physical interpretation? This leads to the questions raised above: why are central force fields irrotational and conservative? In other words, why also irrotational fields are conservative?

Thank you once again for your kind efforts...

sams said:
Thank you all for your replies and I am sorry for this confusion. In fact, I am not asking from a mathematical perspective. The mathematics is straightforward. In my questions above, I just would like to know why rotational fields are non-conservative. Is there any physical meaning or physical interpretation? This leads to the questions raised above: why are central force fields irrotational and conservative? In other words, why also irrotational fields are conservative?

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.

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

ZapperZ said:
Your question is like asking why a straight line has no curvature.
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?

sams said:
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.

sams
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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sams and dRic2

## 1. Why are central force fields irrotational?

Central force fields are irrotational because they have a curl of zero. This means that the vector field does not have any rotational motion around a given point, and the magnitude and direction of the force at any point is solely dependent on the distance from the origin.

## 2. Why are central force fields conservative?

Central force fields are conservative because the work done by the force in moving an object from one point to another is independent of the path taken. This means that the total energy of the system is conserved, and only the initial and final positions of the object matter in calculating the work done.

## 3. How do central force fields relate to Newton's laws of motion?

Central force fields are in accordance with Newton's laws of motion, specifically the second law, which states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. In central force fields, the force acting on an object is directly proportional to its mass through the equation F = ma.

## 4. Can central force fields exist in three dimensions?

Yes, central force fields can exist in three dimensions. In fact, most natural force fields, such as gravity and electrostatic forces, are examples of central force fields that act in three dimensions. In three dimensions, the force is dependent on the distance from the origin in all three axes.

## 5. What are some real-life examples of central force fields?

Some real-life examples of central force fields include the gravitational force between the Earth and an orbiting satellite, the electrostatic force between two charged particles, and the force of attraction between the nucleus and electrons in an atom. These force fields are central because they are dependent only on the distance between the objects and the center of force.