Electricity and Magnetism Simulations

• Robert Ehrlich of George Mason University
• Lyle Roelofs of Haverford College
• Ronald Stoner of Bowling Green University
• Jaroslaw Tuszynski of George Mason University

• FIELDS (Analysis of Vector and Scalar Fields), written by Jarek Tuszynski, displays scalar and vector fields for any algebraic or trigonometric expression entered by the user. It also computes numerically the divergence, curl, and Laplacian for the vector fields, and the gradient and Laplacian for the scalar fields. Simultaneous displays of selected quantities are provided in user-selected planes, using vector, contour, or 3-D plots. The program also allows the user to define paths along which line integrals are computed.

• GAUSS (Gauss' Law), written by Jarek Tuszynski, treats continuous charge distributions having spherical or cylindrical symmetry, and those that vary as a function of the x-coordinate only. The program allows the user to enter an arbitrary function to define either the electric field magnitude, the potential, or the charge density. It then computes the other two functions by numerical differentiation or integration, and displays all three functions. Finally, the program allows the user to enter a ";comparison function"; which is plotted on the same graph, so as to check whether his analytic solutions are correct.

• POISSON (Poisson's Equation Solved on a Grid), written by Jarek Tuszynski, solved Poisson's equation iteratively on a two-dimensional grid using the method of simultaneous over-relaxation. The user can draw arbitrary systems consisting of line charges, and charges conducting cylinders, plates, and wires, all infinite in extent perpendicular to the grid. After iteratively solving Poisson's equation, the program displays the results for the potential, electric field, or the charge density (found from the Laplacian of the potential), in the form of contour, vector, or 3-D plots. In addition, many other program features are available, including the ability to specify dielectric surfaces, along which the potential varies according to some algebraic function specified by the user.

• ICANDME (Image Charges and Multiple Expansion), written by Lyle Roelofs and Nathan Johnson, allows users to explore two approaches to the solution of Laplace's equation, the image charge method and expansion in multipole moments. In the image charge mode (IC) the user is presented with a variety of configurations involving conducting planes and point charges and is asked to ";solve"; each by placing image charges in the appropriate locations. The program displays the electric field due to all point charges, real and image, and a solution can be regarded as successful with the field due to all charges is everywhere orthogonal to all conducting surfaces. Solutions can then be examined with a variety of included software ";tools";. The multipole expansion (ME) mode of the program also permits a ";hands-on"; exploration of standard electrostatic problems, in this case the ";exterior"; problem, i.e. the determination of the field outside a specified equipotential surface. The program presents the user with a variety of azimuthally symmetric equipotential surfaces. The user ";solves"; for the full potential by adding chosen amounts of the (first 6) multipole moments. The screen shows the contours of the summed potential and the problem is ";solved"; when the innermost contour matches the given equipotential surface as closely as possible.

• ATOMPOL (Atomic Polarization), written by Lyle Roelofs and Nathan Johnson, is an exploration of the phenomenon of atomic polarization. Up to 36 atoms of controllable polarizibility are immersed in an external electric field. The program solves for and displays the field throughout the region in which the atoms are located. A close-up window shows the polarization of selected atoms and software ";tools"; allow for further analysis of the resulting electric fields. Use of this program improves the students understanding of polarization, the interaction of polarized entities and the atomic origin of macroscopic polarization, the latter via study of closely spaced clusters of polarizable atoms.

• DIELECT (Dielectric Materials), written by Lyle Roelofs and Nathan Johnson, is a simulation of the behavior of linear dielectric materials using a cell-based approach. The user controls either the polarization or the susceptibility of each cell in a (25x25) grid (with assume uniformity in the 3rd direction). Full self-consistent solutions are obtained via an iterative relaxation method and the fields P, E or D are displayed. The student can investigate the self interactions of polarized materials and many geometrical effects. Use of this program aids the student in developing understanding of the subtle relations between and meaning of P, E and D.

• ACCELQ (Fields From an Accelerated Charge), written by Ronald Stoner, simulates the electromagnetic fields in the plane of motion generated by a point charge that is moving and accelerating in two dimensions. The user chooses from among seven predefined trajectories, and sets the values of maximum speed and viewing time. The electric field pattern is recomputed after each change of trajectory or parameter; thereafter, the use can investigate the electric field, magnetic field, retarded potentials, and Poynting-vector field by using the mouse as a field probe, by using gridded overlays, or by generating plots of the various fields along cuts through the viewing plane.

• QANIMATE (Fields from an Accelerated Charge - Animated Version), written by Ronald Stoner, is an interactive animation of the changing electric field pattern generated by a point electric charge moving in two dimensions. Charge motion can be manipulated by the user from the keyboard. The display can include electric field lines, radiation wave fronts, and their points of intersection. The motion of the charge is controlled by using the arrow keys to accelerate and steer much like the accelerator and steering wheel of a car, except that acceleration must be changed in increments, and the SPACE bar can be used to engage or disengage the steering. With steering engaged, the charge will move in a circle. Unless the acceleration is made zero, the speed will increase (or decrease) to the maximum (minimum) possible value. At constant speed and turning rate, the charge can be controlled by SPACE bar alone.

• EMWAVE (Electromagnetic Waves), written by Ronald Stoner, uses animation to illustrate the behavior of electric and magnetic field in a polarized plane electromagnetic wave. The user can choose to observe the wave in free space, or to see the effect on the wave of incidence on a material interface, or to see the effects of optical elements that change its polarization. The user can change the polarization state of the incident wave by specifying its Stokes parameters. Standing electromagnetic waves can be simulated by combining the incident traveling wave with a reflected wave of the same amplitude. The user can do that by choosing appropriate values of the physical properties of the medium on which the incident wave impinges in one of the animations.

• MAGSTAT (Magnetostatics), written by Ronald Stoner, computes and displays magnetic fields in and near magnetized materials. The materials are uniform and have 3-dimensional shapes that are solids of revolution about a vertical axis. The shape of the material can be modified or chosen from a data input screen. The user has the option of generating the fields produced by a permanently and uniformly magnetized object, or of generating the fields of a magnetizable object placed in an otherwise uniform external field. Besides choosing the shape and aspect ratio of the object, the user can vary the magnetic strength (H), and magnetization (M). Each of these fields can be displayed or explored in several different ways. The algorithm for computing the fields uses a superposition of Chebyschev polynomial approximants to the H field due to "rings" of "magnetic charge".