Why does the capacitance of a coaxial cable depend on its length but not its impedance?
Capacitence comes from the area, proximity and intermediary material of the two conductors. The longer the conductors the more area to create higher capacitence. The cable also have inductance which depends on the thickness and length of the conductor, so the longer the cable the higher the inductance.
Impedence comes from the ratio of capacitence and inductance. So as the length of the cable increase the capacitence and inductance increase at the same rate so the ratio is going to stay the same.
Another way to look at it is that the impedance is the apparant resistance of the signal at the wavefront. Making the cable longer does not change the wavefront, it just make it go longer. So the impedance does not change.
Capacitance affects impedance. You have it the wrong way around.
Length will impact capacitance, which will change impedence, but inductance will affect impedence without changing capacitance.
Capacitance is based on the word “capacity”. A guy named Faraday discovered that two pieces of metal, separated by a “dielectric” material, can hold a charge for a certain length of time. This is called “capacitance”.
Thus, capacitance is based on how much metal there is, the space between the metals, and what type of dielectric is used. So, then, a coaxial cable will have a slightly different capacitance at 100 feet length then 10 feet. This value for capacitance is “constant”, it does not change, regardless of the signal going thru the cable (DC versus 100MHz, for example)
Impedance is related to how fast/efficiently the capacitor can charge/discharge. If you put a DC signal into a capacitor – it will charge once and stay charged until you remove the DC. If you put a variable signal (like 1MHz) into a capacitor, the circuit will act as though there is a resistor in the circuit, because the capacitor is trying to charge/discharge at that rate. Thus the impedance is dependent on the frequency of the signal, in conjunction with the capacitance of the circuit. Coaxial cables are designed to provide a known impedance at a specific frequency range.
Capacitance of the cable comes into play at low enough frequencies that the whole cable can be considered to be at the same time. In such a case, it’s trivial that capacitance scales with length, since larger capacitor plates -> more capacitance.
At high frequencies, due to the velocity factor of the cable (in coax signals travel at approx 0.6 times the speed of light in vacuum) signals don’t have time to travel through the whole cable before the start of the cable is already at a different voltage, so signals [propagate as waves](https://i.stack.imgur.com/gOPmO.jpg) down the length of the cable. The waves don’t “see” anything other than the part of the cable immediately near them – or signals reflecting and coming back from the end and meeting them – (because of that 0.6c limit) so the impedance becomes independent from the cable as a whole.