Problem 1

 
1) Consider the circuits shown in Fig.1a and Fig.1b. The curved segments are semi-circular arcs with radii indicated in the figures. Find the magnitude and direction of the magnetic field at point P for Fig.1a (12.5 pts) and for Fig.1b (12.5 pts).  

Answer:
a)
In the first case, the magnetic fields from the two circular parts of the loop add destructively since the magnetic field from the R₁ part is into the page and the magnetic field from the R₂ part is out of the page. We have then
$\vec{B} = \frac{mu_{0}I}{4} (\frac{1}{R_{1}} - \frac{1}{R_{2}})$ into the page

b)
In the second case, the contributions to the magnetic field from both circular arcs is into the page, so therefore the net magnetic field is given by
$\vec{B} = \frac{mu_{0}I}{4} (\frac{1}{b} + \frac{1}{a})$ into the page