Difference between a conservative and nonconservative vector fields

In summary: Can you give an example of a non-conservative force-field?I'm not sure what you are asking. Are you asking for a real life example of a non-conservative field? If so, I'm not sure I can help you.
  • #1
mjordan2nd
177
1
From a mathematical standpoint I have no trouble understanding the difference between a conservative vector field and a non-conservative vector field. It's rather simple. The conservative field can be reduced to some functions gradient vector, doesn't care what path you decide to take, and always returns 0 on a closed loop. However, on my calculus test today on a problem about work I got an unexpected answer. A particle started at the origin, went around the plane z=y/2 ranging from {(x, y), -1<=x<=1, -2<=y<=2} and returned to the origin. The fact that it traversed a closed loop made bells go off in my head. I though, "what a stupid question. No-brainer." But for some reason I didn't put down 0 and worked out the line integral. I got 3, no matter how many times I checked myself. I'm absolutely convinced I did the math correctly on that problem. Our physics professor had always told us that if the displacement vector is 0 then no work was done. At the time of my test I was so convinced to what my physics professors had told me, I convinced myself I was wrong and put down 0 anyway. I was baffled by that problem, so I thought about it on my way to the computer lab once I got out of my test. And it hit me. What if the force field wasn't conservative? It would make perfect sense. That's why the answer, was in fact, 3. However I don't quite understand. What would be a real life example of a non-conservative force-field? Or any real non-conservative vector field for that matter. What are the physical differences between them? Or are non-conservative fields purely a mathematical construct? I'm still grappling with this. I can't get my head quite around it, though I'm glad I figured out why my answer was always 3.

Sorry if this should be in calculus instead...
 
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  • #2
Hint: What is the curl of a magnetic field?

Zz.
 
  • #3
Heh, I have no idea.

I'll look it up. Thanks.
 
  • #4
If the field was "conservative", then there would be no work done in moving around a closed path. Did you check to make sure the field was conservative?
 
  • #5
No, unfortunately I did not even think of the possibility that the field could have been non-conservative at the time. I have been so used to seeing work being force * displacement, I didn't check. I checked on plenty of other problems, but it eluded me on this one. Heh, guess it happens. Now that I think back on it, however, I'm almost certain that it was not a conservative field.
 

Related to Difference between a conservative and nonconservative vector fields

1. What is the difference between a conservative and nonconservative vector field?

A conservative vector field is one in which the line integral around a closed path is always zero, meaning that the work done by the vector field is independent of the path taken. A nonconservative vector field, on the other hand, may have a nonzero line integral around a closed path, meaning that the work done is dependent on the path taken.

2. How can I tell if a vector field is conservative or nonconservative?

You can determine if a vector field is conservative by checking if it satisfies the condition of being irrotational, meaning that the curl of the vector field is equal to zero. If the vector field is not irrotational, then it is nonconservative.

3. What are some real-life examples of conservative and nonconservative vector fields?

An example of a conservative vector field is the gravitational force field, where the work done by gravity is independent of the path taken. An example of a nonconservative vector field is the force of friction, where the work done is dependent on the path taken due to energy being lost as heat.

4. What is the significance of conservative and nonconservative vector fields?

The distinction between conservative and nonconservative vector fields is important in physics and engineering because it helps us understand the principles of energy conservation and work done. Conservative vector fields are associated with conservative forces, which can be described using potential energy, while nonconservative vector fields are associated with nonconservative forces that cannot be described using potential energy.

5. Can a vector field be both conservative and nonconservative?

No, a vector field cannot be both conservative and nonconservative. This is because the properties that define a conservative vector field (such as being irrotational) are contradictory to the properties of a nonconservative vector field (such as having a nonzero curl).

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