The diagram shows three arrangements of circular loops, centered on vertical axes and carrying identical currents in the directions indicated. Rank the arrangements according to the magnitudes of the magnetic fields at the midpoints between the loops on the central axes.
A) 1, 2, 3
B) 2, 1, 3
C) 2, 3, 1
D) 3, 2, 1
E) 3, 1, 2

Question

The diagram shows three arrangements of circular loops, centered on vertical axes and carrying identical currents in the directions indicated. Rank the arrangements according to the magnitudes of the magnetic fields at the midpoints between the loops on the central axes. 
A) 1, 2, 3 
B) 2, 1, 3 
C) 2, 3, 1 
D) 3, 2, 1 
E) 3, 1, 2

Quizplus · Accepted Answer

The magnetic field at the midpoint between two identical current-carrying loops is given by: $B = \frac{\mu_0 I}{2R}$ where $I$ is the current in each loop and $R$ is the radius of the loop. For arrangements 2 and 3, the currents in the loops are in the same direction, so the magnetic fields at the midpoint between the loops will add. Thus, the magnetic field at the midpoint for arrangement 2 and 3 will be: $B_{2,3} = \frac{\mu_0 I}{2R} + \frac{\mu_0 I}{2\sqrt{2}R} = \frac{\mu_0 I}{2R}(1+\frac{1}{\sqrt{2}})$ For arrangement 1, the currents in the loops are in opposite directions, so the magnetic fields at the midpoint between the loops will subtract. Thus, the magnetic field at the midpoint for arrangement 1 will be: $B_1 = \frac{\mu_0 I}{2R} - \frac{\mu_0 I}{2\sqrt{2}R} = \frac{\mu_0 I}{2R}(1-\frac{1}{\sqrt{2}})$ Since $\frac{1}{\sqrt{2}} < 1$, we have $1-\frac{1}{\sqrt{2}} < 1+\frac{1}{\sqrt{2}}$, so the magnetic field at the midpoint for arrangement 1 is smaller than for arrangements 2 and 3. Comparing the magnetic fields for arrangements 2 and 3, we have $1+\frac{1}{\sqrt{2}} < 2$, so the magnetic field at the midpoint for arrangement 2 is larger than for arrangement 3. Therefore, the ranking of the magnetic fields from largest to smallest is: arrangement 2 > arrangement 3 > arrangement 1, so the best choice is C) 2, 3, 1.

The Diagram Shows Three Arrangements of Circular Loops, Centered on Vertical