Current density in flat electrical conductors
This program can be used to study how currents flow through metallic conductors. Any two-dimensional arrangement of electrical conductors can be drawn and placed at a fixed potential at certain points. For example, you can imagine conductor paths on a circuit board. Flat conductors made of different materials, i.e. with different electrical conductivity and different thicknesses, can be used. The arrangements can be quite complex, including electrical circuits with several resistors. The program calculates the potential, the electric field strength, the current density and the generated power density (thermal power) at each location in the metal. Field lines can be displayed in the electric field strength image, and current lines, i.e. lines along which the current flows, can be displayed in the current density image. The program can be used to display inhomogeneous current density distributions and to calculate electrical resistances for irregularly shaped objects. The resistance can be determined by reading the potential differences (voltage) between two locations and reading the total current between these locations. Kirchhoff's node rule and the voltage drop on the wire can also be demonstrated well.
Operation
Start the program and enter the number of grid cells nx and ny for the two spatial directions at the top left. Under "Size of a cell", enter the size of a grid cell in meters. Then specify the potential on which the points you are drawing are to be placed. Selecting "Fixed potential" or "Variable potential" determines whether the potential entered for the points drawn is fixed (battery terminal, electrical connection, etc.) or whether it is an internal point of the conductor at which the potential is to be calculated by the program. Points with a fixed potential are marked with a white border. You can select the material with the specified thickness in the selection box. You can add new materials to the list yourself using the "Add material to list" button. This opens an extra window in which you must enter the name, the conductivity of the material and the thickness in millimetres. The conductivity is the reciprocal of the specific resistance, Siemens is the reciprocal of Ohm. If you then press "Add material", you will be asked for the color with which it should be displayed. The points with the previously selected parameters are drawn by moving the mouse with the left mouse button pressed. Use the right mouse button to delete points. Completely outlined areas can be filled in by pressing the Ctrl key and simultaneously clicking in the area with the left mouse button. Points can only be drawn in the color representations of potential, power density and material. Please note that a current flow between two grid elements that only touch at the corner is not possible. By ticking the box at the bottom left, the location(x,y) is displayed at the cursor. Once the arrangement has been drawn, press the "Start" button. In the case of very thin ladders that connect large areas of ladders, it may be necessary to press the start button several times, as convergence is slow in such cases.
The four buttons at the bottom left are used to select which quantity is to be displayed (electric field strength, current density, power density or the materials). In each image, the corresponding values can be read off at the mouse position. If the "Display as 3D grid graphic" checkbox is selected, the respective physical quantities are displayed as a multidimensional grid graphic. In this display, the image can be rotated by moving the mouse in the image while holding down the left mouse button. By moving the mouse while holding down the right mouse button, the light source can be moved to achieve optimum illumination of the display.
In the color representation of the electric field strength, a field line can be drawn into the image by pressing the left mouse button on the conductor. A field line image can be created by clicking on it repeatedly. The same is possible in the color image of the current density, where current lines are displayed, i.e. lines that follow the direction of the current density. As the electrons drift through the metal during current flow without absorbing any significant kinetic energy, the streamlines run in the same way as the field lines.
Numerical realization
In electrical conductors composed of different materials, the potential cannot be calculated simply by solving Laplace's equation. Instead, the equation div j = 0 is assumed here (continuity equation for the current density). It follows that the Laplace operator applied to conductivity*potential is equal to zero. This equation is solved iteratively, keeping the potential fixed at locations with fixed potential (Dirichlet boundary conditions) and using Neumann boundary conditions at boundaries bordering an insulator. Special care is taken in the numerical treatment of interfaces of different materials. The field strength is calculated from the current density (field strength=current density/conductivity). Note that surface charge densities are present at interfaces of different materials. To calculate the field lines, the direction of the electric field is followed from the cursor until a point with a fixed potential is reached. The same applies to the current lines where the direction of the current density is followed.