The power available from a hydropower dam is P = 9.81𝑞ℎ𝜂 where P is the power in kilowatts, q is the flow rate in cubic meters per second, h is the head height in meters, and η is the efficiency factor. This paper claims that beaver dams have a head height from 0.3m-5m with most below 1.5m and a width up to 46m but usually 10m or less and mentions that beaver dams can withstand a flow rate of 1.34m^3/s per meter width for a dam with a height of 1.4m. So if we estimate a 75% efficiency and go with 1.34m^3/s per meter width, 10m width and 1.4m head that gives us:

P = 9.81 * 13.4 m^3/s * 1.4m * 0.75

for a power of 138kW.

Can you pet and or smooch the beaver for a job well done?

beavers dam up ravines, i dont think there'd be much potential energy there to capture.

Beaver dams are where beavers keep their food and children, where would you put the turbines?

bobr kurwa

Not sure if this is what you're talking about, but small-scale hydropower is definitely a thing and can be done fairly cheaply: https://youtu.be/1KyL1-0A0Gw

You're not going to generate enough electricity from a small damn to provide power to anything for more than a very small group of people though.

Usually beaver dams are designed to disperse water more broadly into a landscape, turning a stream into a wetland. Humans often do the opposite, digging a trench to turn a marsh or swamp into a stream, and then sometimes damming that stream into a lake or reservoir.