Homework Help: Evaluating Scalar Field

1. Oct 29, 2013

bowlbase

1. The problem statement, all variables and given/known data
Evaluate the scalar field $f(r, \theta, \phi)= \mid 2\hat{r}+3\hat{\phi} \mid$ in spherical coords.

2. Relevant equations
Law of Cosines?

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

3. 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?

2. Oct 29, 2013

pasmith

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

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

Now all you need is the angle $\alpha$ between $\hat r$ and $\hat \phi$.

3. Oct 29, 2013

bowlbase

I made a mistake. The question should be: $\mid 2\hat{r} +3\hat{\theta} \mid$

4. Oct 29, 2013

bowlbase

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?

5. Oct 29, 2013

vanhees71

Spherical coordinates are orthogonal coordinates! Thus $\hat{r}$ and $\hat{\phi}$ are orthogonal with unit norm...

6. Oct 29, 2013

bowlbase

So just 90° then?