Week 7A: Work down by a vector field

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

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

with

r(t)=Rcos(ωt)ex+Rsin(ωt)ey.\vec{r}(t) = R\cos(\omega t)\vec{e}_x + R\sin(\omega t)\vec{e}_y.

and,

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

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

W=CFdr=Γ(dFydxdFxdy)dA=ΓdA=πR2.\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 ρ(x,y,z)\rho(x,y,z). It is said that charges act as a source of electric field (EE). The associated (Gauss’) law of nature reads,

E=ρϵ0,\nabla \cdot E = \frac{\rho}{\epsilon_0},

with ϵ0\epsilon_0 some proportionality constant of nature. If take the electric-field flux integral of a closed surface Γ\Gamma, associated with a volume VV, we get,

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

Applying Gauss’ theorem to his law of nature,

ΦE=VEdV=Vρϵ0dV=Qenclosedϵ0,\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 QenclosedQ_{\text{enclosed}} is the total charge enclosed within the volume VV. It seems that we have rewritten the aforementioned differential form Gauss’ law into an integral form,

ΓEndA=Qenclosedϵ0.\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 R1R_1, and minor radius R2R_2 can be formed by bending a cylinder of radius R2R_2 with length 2πR12\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=2πR22R1V = 2\pi R_2^2R_1 will at least serve as a good approximation of the corresponding donut’ volume, but what is it exactly?

The marvelous design of this website is taken from Suckless.org