![]() ![]() I may or may not discuss this in class if not I likely at least talked about how that combination of 2 cos(theta) r-hat sin(theta) theta-hat is special (approx lecture 20) field lines of a pure electric dipole.nb - Mathematica notebook to plot the field lines of a pure electric dipole, namely E = C/r 3 (2 cos(theta) r-hat sin(theta) theta-hat).worked SOV problem given sigma on a shell.pdf - worked separation of variables problem likely discussed very quickly in class (approx lecture 19).legendre polynomials.pdf - summary of information about the Legendre polynomials (approx lecture 17).separation of variables - 3D cube.nb - Mathematica notebook for separation of variables 3D cube problem (approx lecture 16).separation of variables - 2D semi-infinite.nb - Mathematica notebook for separation of variables 2D semi-infinite problem (approx lecture 16).orthogonality of sine functions.nb - Mathematica notebook demonstrating the orthogonality of sine functions (approx lecture 16).charge density from image problem.nb - Mathematica notebooks for the canonical image problem (approx lecture 15).relaxation examples.pdf - output images for some relaxation examples shown in class (approx lecture 14).field from a spherical shell.nb - Mathematica code demonstrating the "Assuming" command (approx lecture 6).What you should already know about electric field and potential.pdf (approximatelly lecture 1).Parallel Equations for the Electric and Magnetic Fields.pdf - List of most of the important equations in Phys 441, organized according to electric and magnetic field versions of similar equations.The axial symmetry is mathematically described by the independence of physical quantities, such as the magnetic field \(\varvec\) is the direction of quasisymmetry.Here are some handouts I have prepared to help clarify or add information on several selected topics. In a tokamak 1, the reactor vessel is axially symmetric (see Fig. In the approach to nuclear fusion based on magnetic confinement, charged particles (the plasma fuel) are trapped in a doughnut-shaped (toroidal) reactor with the aid of a suitably designed magnetic field. Nuclear fusion is a technology with the potential to revolutionize the way energy is harvested. Due to the vanishing rotational transform, these solutions are however not suitable for particle confinement. The obtained solutions hold in a toroidal volume, are smooth, possess nested flux surfaces, are not invariant under continuous Euclidean isometries, have a non-vanishing current, exhibit a vanishing rotational transform, and fit within the framework of anisotropic magnetohydrodynamics. This result is achieved by a tailored parametrization of both magnetic field and hosting toroidal domain, which are optimized to fulfill quasisymmetry. Here, we prove the existence of weakly quasisymmetric magnetic fields by constructing explicit examples. Nevertheless, the existence of such magnetic configurations lacks mathematical proof due to the complexity of the governing equations. ![]() Quasisymmetric magnetic fields may allow the realization of next generation fusion reactors (stellarators) with superior performance when compared with tokamak designs. A quasisymmetry is a special symmetry that enhances the ability of a magnetic field to trap charged particles. ![]()
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