Some notes on the special phi term at 22:10 in the context of E&M. While it does seemingly produce a valid electrostatic field with a divergence and curl of zero. Applying Stoke's theorem finds that the curl of the electric field isn't exactly 0 everywhere, rather there is a delta function located at the origin. This suggests a changing magnetic field which invalidates it as an electrostatic field.
You may have worked this out yourself already but the constant times phi term comes up when we don't consider the full azimuthal range. Once you do consider the full azimuthal range, the "special" phi term must be zero to keep the potential single valued.
This helped me come to grips with some of the math and ideas behind using cylindrical coordinates with separation of variables. Thanks!
Some notes on the special phi term at 22:10 in the context of E&M. While it does seemingly produce a valid electrostatic field with a divergence and curl of zero. Applying Stoke's theorem finds that the curl of the electric field isn't exactly 0 everywhere, rather there is a delta function located at the origin. This suggests a changing magnetic field which invalidates it as an electrostatic field.
You may have worked this out yourself already but the constant times phi term comes up when we don't consider the full azimuthal range. Once you do consider the full azimuthal range, the "special" phi term must be zero to keep the potential single valued.
hand writing 🤮