2.1.2 Two-dimensional systems

The field depends on the inverse of the square of the distance, as we have shown before. However, when studying many interesting academic problems, such as an infinitely long charged line, we are actually considering problems in two dimensions. The symmetry with respect to an axis, say the z-axis, makes the field an any other related quantity depend only on the transverse coordinates, x and y in a cartesian system. It is therefore a 2-D problem. In fact, by choosing any plane z=const. we get a visualization of the 2-D system, which remains invariant under a z-translation. In this way, a couple of parallel charged lines would correspond in two dimensions to a couple of point charges.

Coulomb's Law must be suitably modified to work in two dimensions, and the 1/r2 law changes to 1/r.

The following example is designed to visualize the field of a charged line:

The charge line is simulated by means of 40 charges aligned, of 1000 u each. Four unit charges are assumed to be located at 100, 200 and 400 units of distance from the line. It can be observed that the force on the unit charges decreases as the inverse of the distance from the line. To check this, select the test charges with the right button of the mouse and read position and force values on the inspector panel.

The field results to be radial to the line, as shown in the following exmple:

Wait for a few seconds until all the field lines have been plotted. Rotate the figure. You will see that the field lines differ noticeably from those expected for an infinite line, due to the finite length of the simulated line compared to the dimensions of the observation region.