- #1

- 199

- 21

- Homework Statement
- Do surface integral using spherical coordinate system over

$$A = (x, y, z)/(x^2 + y^2 + z^2)^{3/2}$$

Surface is a sphere at origin with radius R.

- Relevant Equations
- Not gauss

I'm supposed to do the surface integral on A by using spherical coordinates.

$$A = (rsin\theta cos\phi, rsin\theta sin\phi, rcos\theta)/r^{3/2}$$

$$dS = h_{\theta}h_{\phi} d_{\theta}d_{\phi} = r^2sin\theta d_{\theta}d_{\phi}$$

Now I'm trying to do

$$\iint A dS = (rsin\theta cos\phi, rsin\theta sin\phi, rcos\theta)/r^{3/2} * r^2sin\theta d_{\theta}d_{\phi}$$

Which obviously doesn't work because I'm trying to integrate over a vector.

What I suspect I'm supposed to do is

$$\iint A \cdot n dS = \iint A \cdot e_r dS = \iint A_r dS$$

However I'm not sure what e_r is..

If expressed in spherical coordinates it's (1, 0, 0)

But I cant just do

$$(rsin\theta cos\phi, rsin\theta sin\phi, rcos\theta) \cdot (1, 0 ,0)$$

and get any resonable answer.

So I'm stuck because I can't extract A_r from A.

$$A = (rsin\theta cos\phi, rsin\theta sin\phi, rcos\theta)/r^{3/2}$$

$$dS = h_{\theta}h_{\phi} d_{\theta}d_{\phi} = r^2sin\theta d_{\theta}d_{\phi}$$

Now I'm trying to do

$$\iint A dS = (rsin\theta cos\phi, rsin\theta sin\phi, rcos\theta)/r^{3/2} * r^2sin\theta d_{\theta}d_{\phi}$$

Which obviously doesn't work because I'm trying to integrate over a vector.

What I suspect I'm supposed to do is

$$\iint A \cdot n dS = \iint A \cdot e_r dS = \iint A_r dS$$

However I'm not sure what e_r is..

If expressed in spherical coordinates it's (1, 0, 0)

But I cant just do

$$(rsin\theta cos\phi, rsin\theta sin\phi, rcos\theta) \cdot (1, 0 ,0)$$

and get any resonable answer.

So I'm stuck because I can't extract A_r from A.