Question 26.6: A certain cylindrical wire carries current. We draw a circle......
A certain cylindrical wire carries current. We draw a circle of radius r around its central axis in Fig. 26-24a to determine the current i within the circle. Figure 26-24b shows current i as a function of r².The vertical scale is set by i_s = 4.0 mA, and the horizontal scale is set by r²_s = 4.0 mm². (a) Is the current density uniform? (b) If so, what is its magnitude?
(a) Circular area depends, of course, on r², so the horizontal axis of the graph in Fig. 26-24(b) is effectively the same as the area (enclosed at variable radius values), except for a factor of π. The fact that the current increases linearly in the graph means that i/A =J = constant. Thus, the answer is “yes, the current density is uniform.”
(b) We find i /\left(\pi r^2\right)=(0.005 A ) /\left(\pi \times 4 \times 10^{-6} \,m ^2\right)=398 \approx 4.0 \times 10^2 \,A / m ^2 .