Magnetic field generated by current in semicircular loop at a point on axis

[SOMEBODYCOOL]

Homework Statement

Determine the magnetic field strength and direction at a point 'z' on the axis of the centre of a semi-circular current loop of radius R.

Homework Equations

Biot Savart Formula
d\vec{B}=\frac{\mu_{0}Id\vec{r}\times\hat{e}}{4\pi|\vec{R}-\vec{r}|^{2}}

e being the unit vector from r to R

The Attempt at a Solution

A much simpler problem is a full current loop, because one component of the magnetic field cancels out. For this problem, you'd have to deal with the half-circle arc and the straight line base separately. I was also wondering whether its easier to calculate the z and x components of B separately as well... One component is straightforward enough... I just really don't understand where to start.

 

[gabbagabbahey]

This should be a pretty straightforward application of the Biot-Savart Law. Start by finding expressions for \textbf{r}, the position vector for a general point on the semi-circular arc, and \textbf{R} the position vector for a general point on the z-axis...what do you get for those?...What does that make \hat{\mathbf{e}}? What is d\textbf{r} for a semi-circualr arc?

To makethings easier, you will want to use cylindrical coordinates.

 

[SOMEBODYCOOL]

So, the parametric representation of a point on the semi-circle would be (0, bcos(t), bsin(t)) where b is the radius of the semi-circle.
The vector R is just [d, 0, 0] where d is the distance on the axis of the point
and then the e is the unit vector from R-r
But what's dr? And where does the switch to cylindrical coord come in?

 

[SOMEBODYCOOL]

I think I got it. Thanks

 

[gabbagabbahey]

I think I got it. Thanks

If you'd like to post your result, we''ll be able to check it for you.