If you find the scalar potential of a conservative vector field

In summary, the conversation is discussing whether or not the answer should include the constant of integration when finding a scalar potential. The general consensus is that the constant should technically be included, but it may be fixed to a specific value depending on boundary conditions. However, from a classical physical perspective, the difference in potential is what is relevant. It is also mentioned that scalar potentials for a given force field are not unique, so any potential will suffice.
  • #1
schattenjaeger
178
0
Should your answer include the constant of integration? I think it should but my book's answers don't, so I dunno.

Example, <2xy^3, 3y^2x^2>

answer is x^2y^3, but should I include the + C? (and yes I went through and made sure h(y) was in fact a constant
 
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  • #2
Technically, yes, you should include it.
Of course, a boundary condition might fix it to a specific constant.
From a (classical) physical point of view, it's the "difference in potential" that is physically relevant.
 
  • #3
I would say yes. But it's OK to leave it out as long as you know that adding an arbitrary constant to the potential will produce the same vector field.
 
  • #4
schattenjaeger said:
Should your answer include the constant of integration?

Did the question ask for "the" scalar potential? If so, then go to your teacher and toss a hot cup of coffee in his lap. Once you've got his attention, explain to him that scalar potentials for a given force field aren't unique.

Or did the question ask for "a" scalar potential? If so, then anyone will do.
 
  • #5
Haha, hey, I like my teacher! It's actually outta the book, and it does say >A< scalar potential, had I been thinking it would've been clear, thanks though!
 

1. What is a scalar potential?

A scalar potential is a mathematical function that describes the energy of a conservative vector field at any given point. It is used to represent the potential energy of a system that can be derived from a conservative force field.

2. How is the scalar potential of a conservative vector field found?

The scalar potential of a conservative vector field can be found by performing a line integral along any path from a reference point to the point of interest. This process is known as gradient integration and involves taking the partial derivatives of the vector field with respect to each coordinate and integrating them with respect to their respective coordinates.

3. What is the significance of a conservative vector field having a scalar potential?

If a vector field is conservative, it means that the work done by the field on any closed path is equal to zero. Having a scalar potential allows for the simplification of calculations and provides a more intuitive understanding of the vector field's behavior.

4. Can a non-conservative vector field have a scalar potential?

No, a non-conservative vector field does not have a scalar potential. This is because a non-conservative vector field has a non-zero curl, which means the work done on a closed path is not equal to zero. As a result, a scalar potential cannot be defined for a non-conservative vector field.

5. What applications does understanding the scalar potential of a conservative vector field have?

Understanding the scalar potential of a conservative vector field has many practical applications, including in electrostatics, fluid mechanics, and thermodynamics. It can also be used in the study of motion and forces in physics, as well as in the analysis of behavior of electric and magnetic fields.

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