Fields in the End of a Slot

The figure shows a few equipotentials and field lines of the electrostatic field in the bottom of a rectangular slot with grounded walls and base. The slot has unit width. There is a source in the middle of the slot somewhere off the top of the chart, providing electric flux which flows down the tubes bounded by the field lines and spreads out as it reaches the conducting walls or floor.

The three green equipotentials have V/Vo = 0.5, 1.0, and 1.5 while the four field lines have U/Vo = -1.5, -1.0, -0.5 and +1.0. The line x =0 (not shown) has U = 0 and the plot has reflection symmetry about this line, so other field lines could be added without further effort. U is the stream or flux function conjugate to V.

Note the orthogonal crossings of the U and V lines, and the unit aspect ratio (forming curvilinear "squares"). Also note that the field lines are normal to the V = 0 wall except at the corners where E = 0 so the boundary condition "no transverse E" is clearly true at the end of every line.

The equipotentials are given by pi.y = arcsinh((V/Vo)sec(pi.x)) while the lines of constant U are given by pi.y = arccosh((U/Vo)csc(pi.x)). These are extracted from the conformal mapping W(z) = U +jV = (Vo)sin(pi.z) where z = x + jy , j*j = -1 , and pi = 3.14159.. . This maps the real axis into a |_| shaped slot, so an easy problem - the field of a parallel plate capacitor - maps into the field inside a slot with a square end.

The use of a spreadsheet with plotting facilities allows a set of y values for each curve to be generated from a set of x values. About 20 points per curve give a fairly smooth appearance, so a hand calculator could also have been used (scanned hand plot).

For display on the Web, this 250 by 250 pixel chart was exported in Macintosh "pict" format and converted to "gif" format by a shareware program, ready for uploading and a hyperlink.