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\\abla*\mathrm d\mathbf r`

where
`\\abla* 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)~\\C_2:=\~0\le t\le1\\\C_3:=\(4dt,0,0)~\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`