Would it be possible for a suitably equiped drone to recharge by perching on a power line? I realise that power lines carry much more current and voltage than are to be found in drone recharging devices, but perhaps they could collect just a small amount of that power via induction.
Inductive charging requires AC. Not all powerlines are AC.
This is certainly illegal unless the drone is being used by the electric company.
You mention perching on the power line, which implies clamping. If the drone does indeed clamp to the power line, you could build a current transformer into the clamp (similar to a clamp-on ammeter) and it shouldn’t be too difficult to extract power.
Details are left as an exercise for the interested reader.
Just make sure that you clamp onto an AC power line as @gnoitall says.
Maybe a portable power station?
Example:https://www.amazon.com/Anker-Portable-Generator-Station-Outdoor/dp/B09FF46FQ9/ref=sr_1_4?crid=8MCKDCE3XMM7&keywords=portable+generators&qid=1694113871&sprefix=portable+generators%2Caps%2C421&sr=8-4
Is the goal of this to allow folks like Amazon or other businesses to recharge potential delivery drones? Or is this more of a potential military application, like giving the drones in Ukraine a way to extend their range?
In both cases you are stealing power, but if you are already at war with someone then stealing a bit of power from them is probably way down on the list of issues between your countries.
The next issue you run into is that there are power lines and there are power lines. Transmission lines are the ones that look like this:
These run at very high voltages. According to Wikipedia this is a 500 kV line which is 500,000 volts. There are plenty that are even higher in voltage. At those high voltages, you basically have created a bug zapper. If a bug (aka your drone) gets too close to the line, it could draw an arc and basically get zapped like a bug.
Transmission lines carry power from where it is generated to where it needs to go to be used, i.e. from the generating station to some town many miles away. Once the power gets to that town, the voltage gets dropped down to “distribution” levels, which are anywhere from about 3,000 volts in older systems to 15,000 volts or more in newer systems. Distribution lines “distribute” the power around the town, hence the name. Individual transformers drop the voltage down to what your house uses. A typical transformer supplies 3 or 4 homes.
Here is a nice picture I found.
It will be much easier to siphon power off of distribution lines, since the voltage isn’t so ridiculously high and isn’t so likely to zap your drone into ashes. The voltage is also likely to be in a much more manageable range. Transmission lines can vary from about 30,000 volts up to a million volts or more. Distribution voltages don’t vary quite so widely. They range from maybe 3,000 volts up to 25,000, with most in the 6,000 to 12,000 range.
What’s needed is the drone equivalent of a KC-135. 
Or they have an agreement to reimburse the electric company. One thought I had was police using drones to stakeout a suspect. Instead of having the drone come back to police HQ for recharging, it could recharge from the nearest power line.
There may be other uses for this tech, some licit and some illicit. Use your imagination.
Tesla Broadcast Power!
I thought that Edison lost that argument with Westinghouse long ago. Where and why is DC power transmission used?
Technically I can see it done. You’d need an induction coil compatible with the power line’s magnetic field, a beefy voltage regulator/rectifier to convert to DC at the proper voltage, and surge protectors and isolators. Would probably add enough mass to the drone to negate any advantage.
A better idea would probably be wireless drone charging “pads” built on top of the power poles in strategic locations, with the majority of the necessary hardware on the pad. That way a drone can land, charge, and then be on it’s merry way on a log-distance run.
AC won initially in the early days because it was far easier to step AC up to very high voltages (to avoid massive losses in long-distance transmission) with simple transformers.
Using DC for long-distance transmission has several advantages, including the fact that it requires fewer conductors (only two, vs. three for three-phase AC) and the fact that it doesn’t require grid-wide frequency synchronization (it can obviously even interconnect 50Hz and 60Hz grids). It started to take off in the 1950s as high-power rectifier/inverter technology developed.
As mentioned by @andrewm, long haul bulk power transmission is high-voltage DC. But this is also the same class of power line that @engineer_comp_geek argued a drone has no business trying to parasitize, since the voltage makes it unsafe to even approach.
Stick to AC regional distribution or neighborhood power lines and maybe you could do it.
There is no simple DC transformer. Back in the old days, they would use motor-generator sets to convert DC into different voltages or back into AC, but that’s not terribly efficient so it wasn’t done much. These days we have semi-conductor based transformers and such. But even so, DC transformers and switchgear are much more expensive than AC. It’s hard to beat some coils of wire around a hunk of iron for a simple, maintenance free, reliable, low-cost transformer. And AC naturally extinguishes arcs since AC crosses zero twice per AC cycle. DC doesn’t cross zero, so DC switchgear needs to be more complicated (and expensive) to handle arc suppression.
The main advantage of DC is that you can push significantly more power through the same wire if you use DC over AC. All of your insulation, standoffs, etc. need to be designed for the peak voltage. DC always runs at peak. With AC, you effectively only get the RMS value of the power through it, which is significantly less.
Over long distances, inductive and capacitive losses are significant with AC. With DC, you don’t get these losses. This makes DC dramatically more efficient at longer distances.
The extra cost of transformers, inverters, and switchgear mean that DC will likely never be cost-effective for electrical distribution. But for transmission over longer distances, DC wins. Over shorter distances, the cost of the switchgear/inverters/transformers/etc. at either end of the line exceed the savings in wire costs and the greater efficiency. You need to go a certain distance before the savings in wire cost and the efficiency benefits exceed the extra costs of the gear at each end of the line. Once you get beyond that distance, DC becomes more cost effective.
Undersea and underground cables also have huge capacitive losses, so DC becomes the better alternative at significantly shorter distances when underwater or underground transmission is needed.
There is also the lack of a need for synchronization that @andrewm mentioned.
But then as @gnoitall said, HVDC tends to run at scary high voltages.
The Pacific DC Intertie was one of the first major HVDC lines in the U.S., dating back to the early 1970s. The main point of the Pacific DC Intertie was to bring power from hydroelectric plants as far north as Oregon down to the Los Angeles area. Vancouver Island might have had HVDC before the Pacific DC Intertie. I don’t remember. But the main appeal for Vancouver Island was the fact that the power cables needed to go underwater for a fairly significant distance. The inductive and capacitive losses would have been horrific.
I think this map is a little outdated, but it shows existing and planned HVDC transmission lines in the U.S.:
Thanks.
Pro-level drones, like the DJI Matrice have dual batteries & hot swapping. One can land it & swap one battery out at a time, which means it doesn’t power down & then need to power back up & do things like find GPS satellites again. It’s much quicker & easier to move the drone off of the target a block or two away to the safe zone, bring it down, swap batteries & send it back up. You can now plug those used batteries back into the drone truck & start charging them while the second batteries are in use. I don’t know the charge/flight times for those batteries but with a sufficient quantity you can keep it up for quite some time.
There was a thread in which someone was looking for portable fuel cells, and there were a bunch available for drones and UAVs. Presumably quickly rechargeable in the field using hydrogen/something containing hydrogen.