To reduce the possibility of breakdowns, a common goal in high-voltage engineering is to minimize peak electric field values on electrodes for a given voltage. One example is the choice of conductor and shield radii for a coaxial cable. For a given shield radius *ro*, what choice of the conductor radius *ri* gives the lowest electric? The field on the inner conductor is high when it is a thin wire (*ri* « *ro*) or when the its surface is close to the shield (*ri* ≈ *ro*). We expect a minimum at an intermediate value. The electric field on the inner conductor of a coaxial cable with applied voltage *V0* is

*Er*(*ri*) =*V0*/[*ri* ln(*ro*/*ri*)].

Setting the partial derivative of *Er* with respect to *ri* equal to 0.0, we find that the minimum field occurs when

*ri*/*ro* = 1/*e*,

where *e* = 2.718. Similarly, the condition for minimum field between spherical electrodes with radii *ri* and *ro* is

*ri*/*ro* = 1/2.

In a recent electron-gun design project, the cathode was located on the spherical tip of a cylindrical electrode of radius *ri* with a constrained outer radius *ro*. For reliable gun operation, it was important to determine the peak electric field as a function of *ri* for the combined geometry. The application was ideal for numerical methods – an analytic solution would be require complex series expansions and would not offer an accuracy advantage. The first figure shows the geometry for a calculation with the two-dimensional **EStat **code. The solution volume represented a 5.0 cm length of a coaxial transmission with outer radius *ro* = 1.0 cm and applied voltage *V0* = 1.0 V. For good accuracy, the element size was about 0.01 cm. I made calculations for several choices of *ri*. It was relatively easy to change the geometry by direct editing of the text-format **EStat** control script. A geometry change was effected with a global replacement of the *ri* values in the vectors that defined the inner conductor.

For each solution I viewed a plot of |**E**| in the **EStat **analysis menu. In the auto-normalization mode, the code determined and displayed the maximum value of electric field magnitude in the solution volume. The color-coding in the figure shows the field distribution near the optimal value of *ri*. The highest field occurred near the transition from a spherical to cylindrical surface. The circles in the second figure show that maximum value of |**E**| in the solution volume as a function of the cathode radius. As expected, the best choice of *ri* was somewhat below the value for ideal spherical electrodes.

To find an accurate value of *ri* with a finite number of calculations, I made a fifth order polynomial fit to the data points (solid line in the figure). This task took only a few minutes using the numerical utility program **PsiPlot **(http://www.polysoftware.com/plot.htm). Rather than attempt to solve the resulting equation, I simply instructed the program to calculate a large number of instances over a short interval near the minimum. The best choice of inner radius was

*ri*/*ro* = 0.490.

At this value, the peak electric field for the normalized calculation was 358.2096 V/m or 3.5821(*V0*/*ro*).

For more information on **EStat** and **HiPhi** for electrostatics, please see http://www.fieldp.com/estat.html and http://www.fieldp.com/hiphi.html.