Effects of a discovery of the monopole

[Somali_Physicist]

I guess i have two questions , is the magnetic field conservative or non-conservative.As far as i can see just looking at a magnetic field we have a curved path hence it wouldn't be conservative, however many textbooks assume it is.Is there something I am not seeing here?

Furthermore would monopoles change the category such a field sits in.

 

[ZapperZ]

I guess i have two questions , is the magnetic field conservative or non-conservative.As far as i can see just looking at a magnetic field we have a curved path hence it wouldn't be conservative, however many textbooks assume it is.Is there something I am not seeing here?

This is not a question that is open for debate, or something that you can judge simply by "looking". There IS a clear and direct TEST of any field for it to be "conservative". Find the curl of this field. Then ask yourself : "What does it mean if the curl of this field is zero, and what does it mean if the curl of this field is not zero?"

Furthermore would monopoles change the category such a field sits in.

One step at a time. Do the first one first.

Zz.

 

[vanhees71]

Well, after you've read about

-path independence of line integrals over vector fields
-existence of a scalar potential of a vector field and this path independence
-local version: or in other words, what this has to do with the operation curl
-and what Poincare's Lemma tells you

go ahead with a nice mind-boggling example, the potential vortex
$$\vec{V}(\vec{x}) = \frac{1}{x_1^2+x_2^2} \begin{pmatrix} -x_2 \\ x_1 \\0 \end{pmatrix}$$
and calculate $\text{curl} \vec{V}$ and then the line integral along an arbitrary circle parallel to the $x_1$-$x_2$-plane with center on the $x_3$ axis. Hint: cylinder coordinates can in this case be both a good idea and adding even more to the confusion, but it's really a good kind of confusion.

 

[Somali_Physicist]

This is not a question that is open for debate, or something that you can judge simply by "looking". There IS a clear and direct TEST of any field for it to be "conservative". Find the curl of this field. Then ask yourself : "What does it mean if the curl of this field is zero, and what does it mean if the curl of this field is not zero?"
One step at a time. Do the first one first.
[
Zz.

Curl of a field is zero implies that there is no circulation .It also means that such a vector is a gradient of some scalar potential. I guess you could also see along a closed curve you would always have zero work done.As you have to apply same work to get to a point.Wikipedia defines it as something that only depends on position not path taken.

Ok so for a magnetic field:
∫B.dl = μ₀∫J.dA
by stokes theorem:
∫(∇xB).da = ∫μ₀J.dA
therefore:
∇xB = μ₀J
∇xB =0 only if J = 0

If the curl of a vector field isn't zero than it is path dependent and hence not a gradient of a scalar potential.That would imply its not zero, so not a conservative vector field?