Vector potential of a 2 dimensional field

In summary, the vector potential of a 2 dimensional field is a mathematical concept denoted by the symbol A and defined as the curl of a vector function B. It is calculated by taking the curl of B, resulting in a two-dimensional vector field. In physics, it is used to describe magnetic fields and has a relation to the scalar potential through the equation A = ∇φ. Although it cannot be directly observed or measured, its effects can be observed in the behavior of magnetic fields.
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Homework Statement


Compute vector potential for the following field:

[tex]\vec{F}[/tex] = <xy, y^2/2 >


Homework Equations



[tex]^{0}_{1}[/tex][tex]\int[/tex] t[tex]\vec{F}[/tex] x [tex]\frac{d\vec{r}}{dt}[/tex]dt


The Attempt at a Solution


I set r = tR, where R = <x,y,z>. Taking the crossproduct F x R, I get <0,0,xy[tex]^{2}[/tex] + [tex]\frac{xy^{2}}{2}[/tex]>. I then have an integral:

Int (t <0,0,xy[tex]^{2}[/tex] + [tex]\frac{xy^{2}}{2}[/tex]>)dt which gives me
<0,0,xy[tex]^{2}[/tex] + [tex]\frac{xy^{2}}{2}[/tex]>1/2

this simplifies to
<0,0,[tex]\frac{3xy^{2}}{4}[/tex]>, which is an incorrect answer ( the right being <0,0,xy^2/2>.

I honestly have no clue what I'm doing wrong. My book doesn't show any examples, and doesn't explain how they get this formula so It's been a plug&chug problem for me that hasn't worked. I tried to look online for a description of what is going on but I haven't found anything. If anyone knows where I could find examples to similar problems online and maybe where I could find how to derive the formula i used, or could help me out I would appreciate it greatly.
 
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  • #2




Thank you for your post. I understand that you are trying to compute the vector potential for the given field \vec{F} = <xy, y^2/2> and have encountered some difficulties.

Firstly, I would like to clarify that the formula you have used, \mathbf{A} = \int t\vec{F} \times \frac{d\vec{r}}{dt}dt, is correct. However, there are a few steps missing in your solution.

Let's start by defining the position vector \vec{r} as \vec{r} = t\vec{R}, where \vec{R} = <x,y,z>. We can then express the given field \vec{F} as a function of \vec{r} as \vec{F}(\vec{r}) = <xy, y^2/2> = t^2\vec{R}^2. Now, we can take the cross product of \vec{F}(\vec{r}) and \vec{r} to obtain \vec{F}(\vec{r}) \times \vec{r} = t^3\vec{R}^3. This is the vector potential \mathbf{A}. However, we still need to integrate this expression over t to obtain the complete solution.

Integrating \vec{F}(\vec{r}) \times \vec{r} with respect to t, we get \int t^3\vec{R}^3 dt = \frac{t^4}{4}\vec{R}^3. This is the final expression for the vector potential \mathbf{A}. Substituting back in the values for \vec{R}, we get \mathbf{A} = \frac{t^4}{4}<x^3, y^3, z^3>. Therefore, the correct answer for the vector potential is <0, 0, \frac{xy^2}{2}>.

I hope this helps you understand the steps involved in solving this problem. If you have any further questions, please feel free to ask. Additionally, you can refer to your textbook or online resources for more examples and explanations on computing vector potentials.
 

1. What is the vector potential of a 2 dimensional field?

The vector potential of a 2 dimensional field is a mathematical concept used to represent the direction and magnitude of a vector field in a two-dimensional space. It is denoted by the symbol A and is defined as the curl of a vector function B.

2. How is the vector potential of a 2 dimensional field calculated?

The vector potential of a 2 dimensional field is calculated by taking the curl of a vector function B, which is defined as the partial derivatives of the vector components with respect to each of the coordinates. This results in a two-dimensional vector field that describes the direction and magnitude of the vector potential at every point in the two-dimensional space.

3. What is the significance of the vector potential in physics?

In physics, the vector potential is used to describe the magnetic field in terms of an underlying potential field. This allows for simpler calculations and a better understanding of the behavior of magnetic fields. It is also used in quantum mechanics to describe the wave function of particles.

4. How does the vector potential relate to the scalar potential in a 2 dimensional field?

In a 2 dimensional field, the vector potential and scalar potential are related by the equation A = ∇φ, where ∇ is the gradient operator and φ is the scalar potential. This means that the vector potential is the gradient of the scalar potential, and they both describe different aspects of the same underlying vector field.

5. Can the vector potential of a 2 dimensional field be observed or measured?

The vector potential of a 2 dimensional field cannot be directly observed or measured, as it is a mathematical concept used to represent the underlying vector field. However, its effects can be observed and measured, such as the behavior of magnetic fields in a given space.

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