PX275 - E1b - examples

a maths example

considering a field:

\vec{F} = x^{2} \hat{i} + x^{2} \hat{j} + x^{2} \hat{k}

to apply stokes' theorem:

d \vec{s} = d A \hat{n} = d x d y \hat{n} = d x d y \hat{k}

\begin{aligned} \nabla \times \vec{F} & = | \begin{array}{c} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_{x} & F_{y} & F_{z} \end{array} | \\ = \hat{i} (\frac{\partial F_{z}}{\partial y} - \frac{\partial F_{y}}{\partial z}) - \hat{j} (\frac{\partial F_{z}}{\partial x} - \frac{\partial F_{x}}{\partial z}) + \hat{k} (\frac{\partial F_{y}}{\partial x} - \frac{\partial F_{x}}{\partial y}) \end{aligned}

\begin{aligned} \iint_{R} (\vec{\nabla} \times \vec{F}) \cdot d \vec{s} & = \int_{0}^{a} \int_{0}^{a} (\frac{\partial F_{y}}{\partial x} - \frac{\partial F_{x}}{\partial y}) d x d y \\ = \int_{0}^{a} \int_{0}^{a} (2 x) d x d y \\ = a^{3} \end{aligned}

to check if it a conservative field, find $\oint \vec{F} \cdot d \vec{r}$

magnetic field around a current carrying wire

the following maxwell equation gives the static magnetic field:

\vec{\nabla} \times \vec{B} = μ_{0} \vec{J}

integrating over the surface:

\int_{S} (\vec{\nabla} \times \vec{B}) \cdot d \vec{s} = \iint_{S} μ_{0} \vec{J} \cdot d \vec{s}

using stokes' theorem:

\oint_{C} \vec{B} \cdot d \vec{r} = \iint_{S} μ_{0} \vec{J} \cdot d \vec{s} = μ_{0} I

this is ampere's law
evaluating the $L H S :$

\begin{aligned} \oint_{C} \vec{B} \cdot d \vec{r} & = \int_{ϕ = 0}^{2 π} B (r) {\hat{e}}_{ϕ} \cdot r d ϕ {\hat{e}}_{ϕ} \end{aligned}