Gauss' Law for Magnetic Fields — Question 8

Question 8

A magnetic flux tube expands as its field weakens

A thin flux tube has negligible flux through its side wall. Its two end cuts are perpendicular to the local field. At the first cut B1=.800 T and A1=1.50 square centimeters; at the second B2=.300 T.

1. Find the second cross-sectional area.

2. If the second cut instead has its normal at 60 degrees to the field, find its geometric area for the same flux.

Question 8 - Solution

A magnetic flux tube expands as its field weakens

(a) Method and calculation

$$-B_1A_1+B_2A_2=0\Rightarrow A_2=B_1A_1/B_2.$$$$\boxed{A_2=4.00\,cm^2}.$$

(b) Interpretation and further calculation

$$\Phi=B_2A_{\mathrm{tilted}}\cos60^\circ=B_2A_2.$$$$\boxed{A_{\mathrm{tilted}}=8.00\,cm^2}.$$

Physical check

A flux tube wall follows field lines; an arbitrary geometric tube would generally have side flux.