Tuy nhiên, đây là phương trình một vòng tròn Ví dụ, dùng k = 2 cho trung tâm vòng tròn (1, 0) bán kính 1. Nói chung, tất cả những vòng tròn này đi qua nguồn gốc (trong đó f là không được định nghĩa), kể từ khi phương trình trên, khi cross-nhân, được thỏa mãn (0, 0) Tương tự như vậy, đường ngang cũng tương ứng với vòng tròn, nhưng lần này tập trung trên trục-y. | Question What is the significance of the conjugate potential Answer. Since and are conjugate d d d d dx dy and dy - dx In other words V and v are orthogonal. However since these gradients are themselves orthogonal to the lines const and const we see that the lines constant are at right angle to the equipotentials. Put another way The lines const are parallel to v and are therefore lines of force showing the direction of the force Looking at the example we just did the lines of force are given by z A Arg z-c - Arg z c Const In the homework you will see that these too are circles except for the one degenerate case when the constant is zero looking something like magnetic lines of force In fact the are the same thing. Using the Complex Potential to get the Electrostatic Field We know that we can recover the electrostatic field by just taking the gradient of E v However v d d . d dx dy dx d _ . d dx 1 dx d . dx In complex notation F z where F is the complex potential. Conclusion Conservative Vector Fields and Complex Potentials Not in Kreyzsig If E is a conservative field independent of the -coordinate or in the complex plane then E F z where F z is the associated complex potential. Example Find the electric field corresponding to Example D . 31 Notes 1. The Third Dimension In all of the above we take the third coordinate to be the z-coordinate which we cannot call z for obvious reasons So I suppose we can call it and write x y g The same formula for we used in the above examples since it is independent of . 2. Haven t we done this before Earlier we solved Dirichlet s problem using conformal mappings but had to first solve it on a simpler region usually H. Here we are doing it again from first principles and interpreting it as electrostatic force. Also it is good to do things several ways. Exercise Set 9 1. Find the potential complex potential equipotential lines and lines of force of between two parallel plates at x -5 and x 10 having potentials 200 and 500 volts .
