Magnetic Field in Center of Cube

The current I flowing along the edges of one face of a cube (see Figure 1.1a) produces a magnetic field in the center of the cube of magnitude B₀. Consider another cube where the current I flows along a path shown in Figure 1.1b. What magnetic field will now exist at the center of the cube?

Figure 1.1a                                          Figure 1.1b

SOLUTION

The field can be calculated directly from the current flowing in each of the edges of the cube and then expressed in terms of B₀, but it is easier to use symmetry considerations. In Figure 1.1a, the field is clearly parallel to the y-axis (ŷ is perpendicular to the face of the cube), so

Figure 1.2

For the problem at hand, we can add several of these “current loops” in order to produce the configuration given in Figure 1.1b, noting that two overlapping but opposite current legs produce no field (see Figure 1.2). From the figure, we see that there are now three faces of the cube with a current I flowing about their edges: ABCD, AFGB, and ADEF, producing the fields

So the total field B = B₀(-1, 1, 1), where the direction is along the diagonal of the cube and the magnitude is