Question

In a particular region there is a uniform current density of 18 a/m2 in the positive z direction. what is the value of line integralvector b·dvector s when that line integral is calculated along the three straight-line segments from (x, y, z) coordinates (4d, 0, 0) to (4d, 3d, 0) to (0, 0,0) to (4d, 0, 0), where d = 30 cm?

Answer 1

By Stokes' theorem, the integral is 0. If $R$ is the triangular region bounded by the given line segments composing the curve $C$, then

$\displaystyle\int_{\partial R}\mathbf b\cdot\mathrm d\mathbf s=\iint_R\nabla\times\mathrm d\mathbf r$

where $\nabla\times F=\mathrm{curl}(0,0,18)=0$.

Just to verify this, we can parameterize the path by

$C=C_1\cup C_2\cup C_3$
$\begin{cases}C_1:=\{(4d,3dt,0)~|~0\le t\le1\}\\C_2:=\{(4d(1-t),3d(1-t),0)~|~0\le t\le1\}\\C_3:=\{(4dt,0,0)~|~0\le t\le1\}\end{cases}$

$\displaystyle\int_C\mathbf b\cdot\mathrm d\mathbf s$
$=\displaystyle\int_0^1(0,0,18)\cdot(0,3d,0)\,\mathrm dt+\int_0^1(0,0,18)\cdot(-4d,-3d,0)\,\mathrm dt+\int_0^1(0,0,18)\cdot(4d,0,0)\,\mathrm dt$
$=0+0+0=0$