Magnetic field due to the element of current, idl at the center of the loop is expressed as:

$\overline{){\mathbf{d}}\stackrel{\mathbf{\rightharpoonup}}{\mathbf{B}}{\mathbf{=}}\frac{{\mathbf{\mu}}_{\mathbf{0}}}{\mathbf{4}\mathbf{\pi}}{\mathbf{\left(}}\frac{\mathbf{i}\mathbf{d}\stackrel{\mathbf{\rightharpoonup}}{\mathbf{l}}\mathbf{\times}\stackrel{\mathbf{\rightharpoonup}}{\mathbf{R}}}{{\mathbf{R}}^{\mathbf{3}}}{\mathbf{\right)}}}$

The vector R projects the element to center of the loop.

The magnitude of dB can be written as:

$\begin{array}{rcl}{\mathbf{d}}\stackrel{\mathbf{\rightharpoonup}}{\mathbf{B}}& \mathbf{=}& \frac{{\mathbf{\mu}}_{\mathbf{0}}}{\mathbf{4}\mathbf{\pi}}{\mathbf{\left(}}\frac{\mathbf{id}\stackrel{\mathbf{\rightharpoonup}}{\mathbf{l}}\mathbf{\xb7}\mathbf{R}\mathbf{}\mathbf{sin}\mathbf{90}\mathbf{\xb0}}{{\mathbf{R}}^{\mathbf{3}}}{\mathbf{\right)}}\\ & \mathbf{=}& \frac{{\mathbf{\mu}}_{\mathbf{0}}\mathbf{id}\stackrel{\mathbf{\rightharpoonup}}{\mathbf{l}}}{\mathbf{4}{\mathbf{\pi R}}^{\mathbf{2}}}\end{array}$

A piece of wire is bent to form a circle with radius *r*. It has a steady current *I* flowing through it in a counterclockwise direction as seen from the top (looking in the negative *z *direction). (Figure 1)

**Part A**

What is *B** _{z}*(0), the

Express your answer in terms of *I*, *r*, and constants like *μ*_{0} and *π*.

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