Question 26.9: The magnitude J(r) of the current density in a certain cylin......
The magnitude J(r) of the current density in a certain cylindrical wire is given as a function of radial distance from the center of the wire’s cross section as J(r) = Br, where r is in meters, J is in amperes per square meter, and B = 2.00 × 10^5 A/m³.This function applies out to the wire’s radius of 2.00 mm. How much current is contained within the width of a thin ring concentric with the wire if the ring has a radial width of 10.0 μm and is at a radial distance of 1.20 mm?
We note that the radial width Δr = 10 μm is small enough (compared to r = 1.20 mm) that we can make the approximation
\int B r 2 \pi r d r \approx B r 2 \pi r \Delta r
Thus, the enclosed current is 2 \pi B r^2 \Delta r=18.1 \,\mu A . Performing the integral gives the same answer.