Ampère's Law — Question 2

Question 2

A uniformly conducting cylinder has different field laws inside and outside

An infinite solid cylindrical wire of radius a carries total current I uniformly through its cross section toward +z. The surrounding medium is vacuum. Use vacuum permeability $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{T\,m/A}$ where numerical values are needed.

1. Use a concentric circular path to derive B(r) for r less than a and r greater than a.

2. For a=2.00 mm and I=8.00 A, find the field at r=a/2, a and 2a; identify the maximum.

Question 2 - Solution

A uniformly conducting cylinder has different field laws inside and outside

(a) Method and calculation

$$J=\frac I{\pi a^2},\quad I_{\mathrm{enc}}(r)=I\frac{r^2}{a^2},\quad 2\pi rB=\mu_0I_{\mathrm{enc}}.$$$$\boxed{B(r)=\begin{cases}\mu_0Ir/(2\pi a^2),&r\le a,\\\mu_0I/(2\pi r),&r\ge a.\end{cases}}\quad\vec B\parallel\hat\phi.$$

(b) Interpretation and further calculation

$$B(a)=\frac{\mu_0(8.00)}{2\pi(.00200)}=.800\,\mathrm{mT}.$$$$\boxed{B(a/2)=.400\,\mathrm{mT},\quad B(2a)=.400\,\mathrm{mT}};\quad B_{\max}=B(a).$$

Physical check

The field is continuous at the surface in the absence of a singular surface-current sheet, rises linearly inside and decays outside.