Magnetic Field above a Rectangular Circuit

[Contingency]

Homework Statement

On the XY there lies a conducting wire-rectangle with sides parallel to the axis.
The current is given and constant.
What is the magnitude of the magnetic field along an axis parallel to the z axis, going through the intersection of the rectangle's diagonals?

Homework Equations

$$\vec { dB } =\frac { I }{ c{ \left| \vec { r } \right| }^{ 3 } } \vec { dl } \times \vec { r }$$

The Attempt at a Solution

Due to symmetry, I can expect the field along this axis to be in the z direction. I can integrate along just one of each pair of the circuit's sides and double the result.
What I'm having trouble with is the cross product in Biot-Savart's Law - it seems to me that the angle ${ \theta }_{ \vec { dl } ,\vec { r } }$changes throughout the integral. If this is true then it seems the calculation of the field is not too easy..
I'd just like to make sure I'm correct before I get into it - does ${ \theta }_{ \vec { dl } ,\vec { r } }$change throughout the integral?

 

[VantagePoint72]

Remind yourself how the calculation goes for an infinite line of current using the Biot-Savart law. The same calculation will apply to each piece of the loop, just with finite integration bounds.

 

[Contingency]

I haven't calculated that field with Biot-Savart.. But I presume the implied answer is that the angle changes and this is not negligible.. Right?

 

[VantagePoint72]

Yes, the angle changes. It is usually one of the first calculations done in any EM text after introducing the Biot-Savart law. You can take at look at "Introduction to Electrodynamics" by Griffiths, for instance.