### Challenge Problems

The accompanying figure shows a flat, infinitely long sheet of width *a* that carries a current *I* uniformly distributed across it. Find the magnetic field at the point P, which is in the plane of the sheet and at a distance *x* from one edge. Test your result for the limit $a\to 0.$

A hypothetical current flowing in the *z*-direction creates the field $\overrightarrow{B}=C\left[\left(x\text{/}{y}^{2}\right)\widehat{i}+\left(1\text{/}y\right)\widehat{j}\right]$ in the rectangular region of the *xy*-plane shown in the accompanying figure. Use Ampère’s law to find the current through the rectangle.

A nonconducting hard rubber circular disk of radius *R* is painted with a uniform surface charge density $\sigma .$ It is rotated about its axis with angular speed $\omega .$ (a) Find the magnetic field produced at a point on the axis a distance *h* meters from the center of the disk. (b) Find the numerical value of magnitude of the magnetic field when $\sigma =1{\text{C/m}}^{2},$ $R=\text{20 cm},\phantom{\rule{0.2em}{0ex}}h=\text{2 cm},$ and $\omega =400\phantom{\rule{0.2em}{0ex}}\text{rad/sec},$ and compare it with the magnitude of magnetic field of Earth, which is about 1/2 Gauss.