Evaluating Scalar Field in Spherical Coordinates

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Homework Help Overview

The problem involves evaluating the scalar field defined by the expression ##f(r, \theta, \phi)= \mid 2\hat{r}+3\hat{\phi} \mid## in spherical coordinates. Participants are exploring the appropriate mathematical tools and concepts to approach this evaluation.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Some participants consider using the law of cosines to evaluate the magnitude of the vector sum, while others question whether this is the correct approach given the context of spherical coordinates.
  • There is a discussion about the angle between the vectors involved, with suggestions to clarify the notation used for angles to avoid confusion.
  • One participant notes a correction in the original expression, changing ##\hat{\phi}## to ##\hat{\theta}##, which prompts further exploration of the implications of this change.
  • Questions arise regarding the determination of the angle between the vectors in spherical coordinates and whether a conversion to Cartesian coordinates might simplify the problem.

Discussion Status

The discussion is ongoing, with various approaches being considered. Some participants have provided insights into the orthogonality of the coordinate system, suggesting that the vectors involved may be orthogonal, which could simplify the evaluation. However, there is no explicit consensus on the method to proceed.

Contextual Notes

Participants are navigating potential confusion regarding the notation for angles and the implications of using different coordinate systems. The correction of the vector notation has introduced additional considerations into the discussion.

bowlbase
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Homework Statement


Evaluate the scalar field ##f(r, \theta, \phi)= \mid 2\hat{r}+3\hat{\phi} \mid## in spherical coords.


Homework Equations


Law of Cosines?

##\mid \vec{A} + \vec{B} \mid = \sqrt{A^2+B^2+2ABCos(\theta)}##

The Attempt at a Solution



I'm not sure the law of cosines is exactly what I'm suppose to use but so far it is the only thing that I've found that seems to fit the way the problem is presented.

If this is the correct way then:
##\mid 2\hat{r}+3\hat{\phi} \mid=\sqrt{2^2+3^2+12cos(\theta)}##


Am I doing this correctly?
 
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bowlbase said:

Homework Statement


Evaluate the scalar field ##f(r, \theta, \phi)= \mid 2\hat{r}+3\hat{\phi} \mid## in spherical coords.


Homework Equations


Law of Cosines?

##\mid \vec{A} + \vec{B} \mid = \sqrt{A^2+B^2+2ABCos(\theta)}##

The [itex]\theta[/itex] which occurs in this expression is not the spherical coordinate [itex]\theta[/itex]. This is obviously going to be a source of confusion, so you need to find a different letter for the angle between [itex]\vec A[/itex] and [itex]\vec B[/itex], such as [itex]\alpha[/itex]:
[tex] \|\vec{A} + \vec{B}\| = \sqrt{A^2 + B^2 + 2AB\cos \alpha}[/tex]

However I think the intended method is to start from
[tex] \|2 \hat r + 3 \hat \phi\|^2 = (2 \hat r + 3 \hat \phi) \cdot (2 \hat r + 3 \hat \phi)[/tex]

The Attempt at a Solution



I'm not sure the law of cosines is exactly what I'm suppose to use but so far it is the only thing that I've found that seems to fit the way the problem is presented.

If this is the correct way then:
##\mid 2\hat{r}+3\hat{\phi} \mid=\sqrt{2^2+3^2+12cos(\alpha)}##


Am I doing this correctly?

Now all you need is the angle [itex]\alpha[/itex] between [itex]\hat r[/itex] and [itex]\hat \phi[/itex].
 
I made a mistake. The question should be: ##\mid 2\hat{r} +3\hat{\theta} \mid##
 
I'm not sure how I would go about finding the angle between the two vectors in spherical. I could probably switch them to Cartesian but is there a simpler way via spherical?
 
Spherical coordinates are orthogonal coordinates! Thus [itex]\hat{r}[/itex] and [itex]\hat{\phi}[/itex] are orthogonal with unit norm...
 
So just 90° then?
 

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