I'd suspect intuitively that the bandwidth should decrease as the diameter decreases, but I don't have any reasoning to back it up.
Secondly, would the actual wavelengths that it can carry, also change as a function of the diameter?
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The short answer is no, the diameter of the wire doesn't affect the bandwidth. Bandwidth can be a tricky subject. If you are talking about injecting a very high frequency sine wave at one of the wire, and seeing if it is detectable at the other end, then wires of all diameters have a surprisingly high bandwidth. But if you are talking about the ability of the wire to carry useful information, the issue gets tougher. For example, if the wire passes near other conductive objects, that will change the local impedance of the wire and cause reflections of the signal, which limit your ability to recover the signal at the other end. Similarly, if several wires are run close to each other, data from one may be inductively or capacitively coupled to anther, also interfering with the ability to recover the signal. Finally, the primary limitation in sending high-rate information over a wire is noise: If the wire is unshielded, it may pick up interfering signals that make signal recovery impossible. For all these reasons, people who want to send high-bandwidth information over a wire use coax cable. A coax cable has a thin center wire to carry the information, surrounded by an insulator, surrounded by a conductive ground shield. This arrangement maintains a constant impedance along the wire (to minimize reflections), and shields the wire from external noise sources. There is one situation, however, where a thicker wire may help. As I said, the main limitation to a high bandwidth is signal-to-noise ratio (SNR). You can achieve that by limiting the noise (as with the shielded coax). You can also achieve that by increasing the power of the signal (assuming the noise stays constant). At some point, if you keep increasing the transmit power, you'll need a thicker wire to handle the current. But in practice, there is rarely enough power in a data signal that the current-carrying ability of the wire is the limiting factor.