Calculating the direction of magnetic field at a point above a loop of current

[CentreShifter]

I've uploaded my attempted solution here. The problem I'm having is highlighted at the bottom in red.

The issue I'm having is expressing the direction of H. I realize the cancellation that occurs at point (0,0,z), where the only the z-component of the H-field remains. I also realize that my final expression for $\bar{H}$ will be $\hat{z}Hcos\phi$, where $cos\phi=\frac{r}{\sqrt{r^2+z^2}}$. I'm really just having a hard time resolving the geometry of these angles to where I can actually equate the two red phi's in the image.

 

[Nate810]

\begin{figure}[H] \centering \includegraphics[width=0.3\textwidth]{attempted_solution.png} \caption{Attempted Solution}\end{figure}The solution to this problem is as follows. We want to calculate the H-field at point P due to the current element shown in the diagram. We can do this by using the Biot-Savart law as follows:\bar{H}(\vec{P}) = \frac{\mu_0I}{4\pi}\int_C \frac{d\vec{l}\times \hat{r}}{r^2} Where \vec{P} is the position of point P, I is the current flowing through the wire, \mu_0 is the permeability of free space, d\vec{l} is an infinitesimal element of the wire, and \hat{r} is a unit vector pointing from the wire element to the point P. For the current element shown in the diagram, we can set up the following integral: \bar{H}(\vec{P}) = \frac{\mu_0I}{4\pi}\int_C \frac{d\vec{l}\times \hat{r}}{r^2} Where C is the path of integration along the wire element, which can be written in parametric form as: \vec{r}(t) = \langle a\cos t, a\sin t, z\rangle Where a is the radius of the wire, and t is the parameter. The corresponding differential line element is then given by: d\vec{l} = \langle -a\sin t, a\cos t, 0\rangle dt And the unit vector \hat{r} pointing from the wire element to the point P is given by: \hat{r} = \frac{\vec{r}-\vec{P}}{|\vec{r