In figure (a), two circular loops with different currents but the same radius of
ID: 1546895 • Letter: I
Question
In figure (a), two circular loops with different currents but the same radius of 10. cm, are centered on a y axis. They are initially separated by distance L = 7.5 cm, with loop 2 positioned at the origin of the axis. The currents in the two loops produce a net magnetic field at the origin, with y component By. That component is to be measured as loop 2 is gradually moved in the positive direction of the y axis. Figure (b) gives By as a function of the position y of loop 2. The horizontal axis is marked in increments of 5 cm, the vertical axis is marked in increments of 20 T, and the curve approaches an asymptote of By = 7.6 T as y .
(a) What is the current i1 in loop 1? A
(b) What is the current i2 in loop 2? A
(a) y (cm) (b)Explanation / Answer
Using Halliday Eq., we find that the net y-component field is
By = [(uo*i1*R^2)/(2pi*(R^2 + z1^2)3/2)] - [(uo*i2*R^2)/(2pi*(R^2 + z2^2)3/2)]
where z1^2 = L^2 (see Fig. 1(a)) and z2^2 = y^2 (because the central axis here is denoted y instead of z). The fact that there is a minus sign between the two terms, above, is due to the observation that the datum in Fig. 1(b) corresponding to By = 0 would be impossible without it (physically, this means that one of the currents is clockwise and the other is counter-clockwise).
(a) As y , only the first term contributes and (with By = 7.6 × 10^6 T given in this case) we can solve for i1. We obtain i1 = 7.42 A
(b) With loop 2 at y = 0.15 m (see Fig. 1(b)) we are able to determine i2 from
[(uo*i1*R^2)/(2*(R^2 + z1^2)3/2)] - [(uo*i2*R^2)/(2*(R^2 + z2^2)3/2)]
We can obtain i2 =