## Week 7A: Work down by a vector field

Reconsider the previous example for the computation of work along a closed loop $C$,

$W = \oint_C \vec{F} \cdot \mathrm{d}\vec{r}.$

with

$\vec{r}(t) = R\cos(\omega t)\vec{e}_x + R\sin(\omega t)\vec{e}_y.$

and,

$\vec{F}(x,y) = -y\vec{e}_x.$

Introducing $\Gamma$ as the surface enclosed by the loop $C$, and using Green’s theorem we may write,

$\begin{align} W &= \oint_C \vec{F} \cdot \mathrm{d}\vec{r} \\ &= \iint_{\Gamma} \left(\frac{\mathrm{d}F_y}{\mathrm{d}x} - \frac{\mathrm{d}F_x}{\mathrm{d}y}\right) \mathrm{d}A\\ &= \iint_{\Gamma} \mathrm{d}A = \pi R^2. \end{align}$

Thank you Green’s theorem! for this simple elaboration.

## Week 7A: Electric field of a charge distribution

Consider a material with a net electric charge density distribution $\rho(x,y,z)$. It is said that charges act as a source of electric field ($E$). The associated (Gauss’) law of nature reads,

$\nabla \cdot E = \frac{\rho}{\epsilon_0},$

with $\epsilon_0$ some proportionality constant of nature. If take the electric-field flux integral of a closed surface $\Gamma$, associated with a volume $V$, we get,

$\Phi_E = \oiint_{\Gamma} \vec{E}\cdot \vec{n}\mathrm{d}A,$

Applying Gauss’ theorem to his law of nature,

$\Phi_E = \iiint_V \nabla \cdot \vec{E} \mathrm{d}V = \iiint_V \frac{\rho}{\epsilon_0} \mathrm{d}V = \frac{Q_{\text{enclosed}}}{\epsilon_0},$

Where $Q_{\text{enclosed}}$ is the total charge enclosed within the volume $V$. It seems that we have rewritten the aforementioned differential form Gauss’ law into an integral form,

$\oiint_{\Gamma} \vec{E}\cdot \vec{n}\mathrm{d}A = \frac{Q_{\text{enclosed}}}{\epsilon_0}.$

Thank you Gauss! for your contributions to the theory of Electromagnetism.

## Week 7B: Tokamak design

The term Tokamak is adopted from the Russian abbreviation for a toroidal chamber with magnetic coils and it refers to an engineering design for creating and maintaining a donut-shaped (torus) plasma. Such a toroidal plasma can be engineered to have near zero thermal losses, and is a candidate design to achieve the million of degrees Kelvin needed for obtaining the energy of nuclear fusion. Conceptually, a donut of major radius $R_1$, and minor radius $R_2$ can be formed by bending a cylinder of radius $R_2$ with length $2\pi R_1$ such that the ends become connected to each other. The question is: Will the volume change as a result of the bending process? Intuitively, the volume of the cylinder $V = 2\pi R_2^2R_1$ will at least serve as a good approximation of the corresponding donut’ volume, but what is it exactly?