Expressing a Field in Spherical Coordinates as Cartesian Vectors

In summary: I think what you have is correct. However, you can simplify it a bit by using the identities you listed for ar and aθ, and also by using the fact that r = √(x^2 + y^2 + z^2). So you should end up with something like:In summary, F = (xsin(θ)cos(∅) + ysin(θ)sin(∅) + zcos(θ)) / √(x^2 + y^2 + z^2)^3.
  • #1
yoamocuy
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Homework Statement


A field is given in spherical coordinates as F=[cos(θ)/r2]∙ar+[sin(θ)/r]∙aθ. Express F in terms of x, y, z, ax, ay, az

Homework Equations



ar∙ax=sin(θ)cos(∅)
ar∙ay=sin(θ)sin(∅)
ar∙az=cos(θ)
aθ∙ax=cos(θ)cos(∅)
aθ∙ay=cos(θ)sin(∅)
aθ∙az=-sin(θ)
x=r*sin(θ)*cos(∅)
y=r*sin(θ)*sin(∅)
z=r*cos(θ)
r=√(x2+y2+z2 )
cos(θ)=z/r
∅=tan-1(y/x)

The Attempt at a Solution


cos(θ)/r2*[sin(θ)cos(∅)ax+sin(θ)sin(∅)ay+cos(θ)az]+sin(θ)/r*[cos(θ)cos(∅)ax+cos(θ)cos(∅)ay-sin(θ)az]

z/r3*[sin(θ)cos(∅)ax+sin(θ)sin(∅)ay+cos(θ)az]+sin(θ)/r*[cos(θ)cos(∅)ax+cos(θ)cos(∅)ay-sin(θ)az]

(z*r)/r4*[sin(θ)cos(∅)ax+sin(θ)sin(∅)ay+cos(θ)az]+sin(θ)/r*[cos(θ)cos(∅)ax+cos(θ)cos(∅)ay-sin(θ)az]

z/r4*[xax+yay+zaz]+sin(θ)/r*[cos(θ)cos(∅)ax+cos(θ)cos(∅)ay-sin(θ)az]

z/(x2+y2+z2)3*[xax+yay+zaz]+sin(θ)/r*[cos(θ)cos(∅)ax+cos(θ)cos(∅)ay-sin(θ)az]

That's about as far as I've gotten. I'm not even sure if what I've done so far is on the right track or not :/ I'm not sure what to do with the 2nd half of this equation?
 
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  • #2
Use the fact that [itex]\vec{F} = (\vec{F}\cdot\hat{a}_x)\hat{a}_x + (\vec{F}\cdot\hat{a}_y)\hat{a}_y + (\vec{F}\cdot\hat{a}_z)\hat{a}_z[/itex].

Calculate [itex]\vec{F}\cdot \hat{a}_x[/itex] using the various dot products you listed above. Then convert from the spherical variables to the Cartesian variables.
 
  • #3
vela said:
Use the fact that [itex]\vec{F} = (\vec{F}\cdot\hat{a}_x)\hat{a}_x + (\vec{F}\cdot\hat{a}_y)\hat{a}_y + (\vec{F}\cdot\hat{a}_z)\hat{a}_z[/itex].

Calculate [itex]\vec{F}\cdot \hat{a}_x[/itex] using the various dot products you listed above. Then convert from the spherical variables to the Cartesian variables.

Isn't that what I did above? I didn't originally show the ax, ay, az in my work but I just added them in there for clarity.
 
  • #4
Oh, OK. I didn't see the unit vectors in your original attempt, so I figured you were doing it all wrong and didn't bother to look too closely.
 

1. What is the difference between spherical and cartesian coordinates?

Spherical coordinates are a system of representing points in 3-dimensional space using a distance from the origin, an inclination angle from the positive z-axis, and an azimuth angle from the positive x-axis. Cartesian coordinates, on the other hand, use three perpendicular axes (x, y, and z) to represent points in 3-dimensional space.

2. How do you convert from spherical to cartesian coordinates?

To convert from spherical to cartesian coordinates, you can use the following formulas:
x = r * sin(θ) * cos(ϕ)
y = r * sin(θ) * sin(ϕ)
z = r * cos(θ)
where r is the distance from the origin, θ is the inclination angle, and ϕ is the azimuth angle.

3. What are the advantages of using spherical coordinates?

Spherical coordinates are particularly useful for describing points on a sphere or other curved surfaces, as they use angles rather than distances to represent points. They are also helpful in situations where the distance from the origin is more important than the direction of the point.

4. How are spherical coordinates used in real-world applications?

Spherical coordinates have many practical applications, such as in astronomy for describing the position of stars, in navigation for representing locations on the Earth's surface, and in physics for describing the position of particles in a spherical potential. They are also commonly used in computer graphics and 3D modeling.

5. Are there any limitations to using spherical coordinates?

One limitation of spherical coordinates is that they can only represent points in 3-dimensional space. They also become more complex to use in higher dimensions, so cartesian coordinates are often preferred in those cases. Additionally, some calculations, such as finding the distance between two points, may be more difficult to perform in spherical coordinates compared to cartesian coordinates.

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