Each part of the lab reproduces a result from Module 2 numerically. Lesson 18 gave the field of point charges by superposition and of a finite line by direct integration. Lessons 20 and 21 gave Gauss’s law in its integral and point forms. Lesson 23 derived Laplace’s equation and solved it for the coax, and Lesson 25 turned that solution into the capacitance per meter, the \(96.8\ \text{pF/m}\) that Lesson 6 quoted for RG-58.

The code evaluates Lesson 18’s sum at every point of a \(200 \times 200\) grid at once. NumPy applies each arithmetic operation to a whole array, so the only loop is over the charges, and the grid of forty thousand points costs about as much as a single point would in a loop. Every charge sits in the \(z = 0\) plane, and the field is evaluated in that plane, where \(E_z = 0\) by symmetry. To model a line of charge, the code splits it into short pieces and puts a point charge \(\rho_\ell\ \Delta x\) at the center of each, which is the sum that Lesson 18’s integral is the limit of.

Every computation in Part A has a hand answer to compare with, and the notebook asks for it. The first is one line of Coulomb’s law:

The dipole is \(+1\ \text{nC}\) at \((-0.1, 0)\) and \(-1\ \text{nC}\) at \((0.1, 0)\ \text{m}\); work its field on the bisector by superposition, as in Lesson 18. The line carries \(\rho_\ell = 12.5\ \text{nC/m}\); use Lesson 18’s finite-line result and its infinite-line limit, and note how far apart they are.

The point form of Gauss’s law says \(\nabla\cdot\mathbf{E} = 0\) wherever there is no charge. The tempting test is to take the divergence of a point charge’s field on the grid and look for zero. It does not come out zero, and the reason is the third term. The grid lives in one plane, so it can take derivatives along \(x\) and \(y\) but not across the plane. \(E_z\) vanishes in the plane, yet it grows as you step above it, so its derivative along \(z\) does not vanish, and leaving it out makes the two-term sum report charge where there is none. In the notebook you find the missing term and show that it cancels the rest exactly. Earlier versions of this lab fell into the trap without noticing.

An infinite line charge along \(z\) has the field Lesson 20 found with a cylindrical Gaussian surface, and that field is the same in every plane of constant \(z\). Nothing changes along \(z\), so the two-term divergence is the whole divergence, and the grid can test the point form. The integral form becomes a statement per meter of length: the flux of \(\epsilon_0\mathbf{E}\) out through any closed contour in the plane equals the line charge inside it. The notebook integrates around squares that enclose one line, two lines of opposite charge, and none, and you explain each result with Gauss’s law.

Lesson 23 showed that in a charge-free region Laplace’s equation makes each point’s potential the average of its neighbors, and that the solution is unique once the potential on the boundaries is fixed. Part B uses both facts. The inner conductor of RG-58, of radius \(a = 0.405\ \text{mm}\), is held at \(1\ \text{V}\), and the outer, of radius \(b = 1.475\ \text{mm}\), at \(0\ \text{V}\). The code replaces every point between them with its neighbors’ average, half the grid at a time like the two colors of a checkerboard, and slightly over-corrects each step so the guess settles in a few hundred sweeps. You then compare the result with Lesson 23’s \(V(\rho)\).

The relaxed potential gives the field, \(\mathbf{E} = -\nabla V\), by finite differences. Lesson 24’s boundary condition puts the charge on the inner conductor’s surface, and Gauss’s law collects it: the flux of \(\epsilon\mathbf{E}\) out through any contour around the inner conductor is the charge per meter, \(\rho_\ell\). Dividing by the \(1\ \text{V}\) between the conductors gives \(C'\), which you compare with Lesson 25’s formula for polyethylene, \(\epsilon_r = 2.25\). The two will not agree exactly, and the notebook asks you to find out why by changing the grid.

The habits this lab practices carry into every simulation you will run. A computed field earns trust only after it reproduces a result you know, which is why each part starts with one. A grid answer carries a grid error, and the only way to see its size is to refine the grid and watch how the answer changes, which also tells you what the error comes from. The divergence trap is the third habit: the code ran, the plot looked plausible, and it was answering a question about a different field.

If your instructor assigns Part C, you measure \(C'\) with the NanoVNA of Lessons 15 and 16. Lesson 7 gave the input impedance of an open-circuited line; when the line is a small fraction of a wavelength, \(\cot\beta\ell \approx 1/\beta\ell\) and the line looks like its total capacitance \(C'\ell\). At 2 MHz a 2 m cable is \(0.02\lambda\) long, close enough that the error is under 1%. You read the reactance of two open-ended cables of different length, and the difference removes the adapter’s few picofarads, which add equally to both.

Module 2 kept every charge still. Module 3 lets charge move. Lesson 27 starts with steady current in conductors, and the rest of the module finds the magnetic field that current makes. That field completes the coax: Lesson 25 derived \(C'\), and Ampere’s law will derive the \(L'\) that Lesson 6 also quoted, so that \(Z_0 = \sqrt{L'/C'}\) is derived from start to finish.