By 32. the right-hand rule (which is built-into Eq. 29-3) the field caused by wire 1s current, evaluated at the coordinate origin, is along the +y axis. Its magnitude B1 is given by Eq. 29-4. The field caused by wire 2s current will generally have both an x and a y component which are related to its magnitude B2 (given by Eq. 29-4) and sines and cosines of some angle. A little trig (and the use of the right-hand rule) leads us to conclude that when wire 2 is at angle 2 (shown in Fig. 29-60) then its components are B2 x = B2 sin 2 , B2 y = B2 cos 2 . The of magnitude-squared their net field is then (by Pythagoras theorem) the sum of the square of their net x-component and the square of their net y-component:
2 B 2 = ( B2 sin 2 ) 2 + ( B1 B2 cos 2 ) 2 = B12 + B2 2 B1 B2 cos 2 .
(since sin2 + cos2 =1), which we could also have gotten directly by using the law of cosines. We have i i B1 = 0 1 = 60 nT, B2 = 0 2 = 40 nT. 2R 2R With the requirement that the net field have magnitude B = 80 nT, we find
B12 + B22 B 2 1 2 = cos = cos (1/ 4) = 104, 2 B1 B2
1
where the positive value has been chosen.