Prove Radial Fields are conservative and at the origin.

In summary, a conservative field is a vector field where the line integral around any closed loop is zero, indicating that the work done is independent of the path taken. A radial field is a type of vector field where the vectors point towards or away from a central point, making it commonly seen in physics and engineering problems. To prove that a radial field is conservative, its curl must equal zero, which is easier to show at the origin where all vectors are directed towards or away from. Therefore, all radial fields are conservative at the origin.
  • #1
physics_197
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Prove Radial Fields are conservative... and at the origin.

1. Homework Statement

Prove the radial field F = r / |r|^p is conservative on any region not containing the origin. For what values of p is F conservative on a region that contains the origin?

2. Homework Equations

v x F = 0 (for conservation)

3. The Attempt at a Solution

I've proved the first part, but I don't know how to solve for p's such that F is conservative on a region that CONTAINS the origin.
 
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  • #2


What about if r was a unit vector (p=1)
 

1. How do you define a conservative field?

A conservative field is a type of vector field in which the line integral of the vector field around any closed loop is zero. In simpler terms, this means that the work done by the field on a particle moving along a closed path is independent of the path taken.

2. What is a radial field?

A radial field is a type of vector field in which the vectors point directly away from or towards a central point, called the origin. This type of field is commonly seen in physics and engineering, especially in problems involving electric or gravitational forces.

3. How can you prove that a radial field is conservative?

To prove that a radial field is conservative, we need to show that its curl (a measure of the rotation of the field) is equal to zero. This can be done by taking the curl of the field's components and simplifying the resulting equations. If the curl is zero, then the field is conservative.

4. Why is the origin an important point for proving the conservativeness of a radial field?

The origin is an important point for proving the conservativeness of a radial field because it is the center of the field and all vectors in the field are directed towards or away from it. This makes it easier to show that the field's curl is equal to zero, as all the vectors can be simplified to their radial components.

5. Are all radial fields conservative at the origin?

Yes, all radial fields are conservative at the origin. This is because the origin is the center of the field and all vectors in the field are directed towards or away from it. This makes it easy to show that the field's curl is equal to zero, proving its conservativeness.

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