Control fusion plasma | microwave | live formula | algebra | function plot
The dot product of the two vectors in cartesian coordinates is defined as: Multiply corresponding components and the add the results. Thus \( \vec a \cdot \vec b=a_1b_1+a_2b_2+a_3b_3\) If there are two vectors in cylindrical coordinates, \(\vec{A} = A_r \hat{r} + A_\theta \hat{\theta} + A_z \hat{z}\) \(\vec{B} = B_r \hat{r} + B_\theta \hat{\theta} + B_z … Read more
We have to use the cylindrical coordinates when the problem is cylindrical symmetry. For example, the eigenmode in a circular waveguide. Althrough there is a general theory for the differential operators in general curvilinear coordinates, it it very difficult to be understanded. There is also the formular for the differential operators in cylindrical coordinates. However … Read more
The electrons beam is usually confined by magnetics in the vacuum tube. The plasma is usually confined by magnetics in tokomaks. The motion is govend by the Newton’s 2nd law. However the cylinderical coordinates are used in both cases as the fields are axially symmetric. The common question is how to get the equations of … Read more
Maxwell’s equation in vacuum Name Integral equations Differential equations Gauss’s law Gauss’s law for magnetism Maxwell–Faraday equation(Faraday’s law of induction) Ampère’s circuital law (with Maxwell’s addition) Next, We obtain the electromagnetic wave equation in a vacuum. It is also called Helmholtz equation. Take the Curl of the Faraday equation \(\nabla \times \left(\nabla \times \mathbf {E} \right)=\nabla \times … Read more