Will all of the drilling and digging to use the earth's natural heat as geothermal energy be affecting Earth's core, causing it to cool down?
If so, would it result in an ice age?
If not, how does it retain its heat?
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Part 1, see Neos answer. Earth will lose its heat no matter what we do, and our extraction of geothermal energy is insignificant (Wikipedia quotes a BP figure of 11.4 GW electrical, 28 GW heating).
To answer part 2 of your question: if the Earth's core loses its heat, this will not have a major direct impact on climate. Internal heat generation is estimated by Davies and Davies (2010) to be roughly 47 TW. With a surface area of 5.1 × 1014 m2, this translates to roughly 0.1 W/m2. This can be compared to the other flows in the climate system, illustrated by Trenberth and Fasullo, 2012:
So, from a climate perspective, internal heat generation is not important. See also this post on skepticalscience.
However, we might lose our atmosphere, which would have inconvenient consequences. An ice age would be the least of our worries. A subsequent question would be: (How long) would Earth's atmosphere last without a global magnetic field? That is a different question and I'm not sure if we really know the answer.
This question is relevant, Why is the inside of the Earth so hot?
The short answer is the core is losing heat no matter what we do. You see, heat is transported from the core to the surface, but its important to think of heat in terms of energy. Since there is a finite amount of energy within the earth, we are actually transferring energy from the inside to the out. It works similar to a internal combustion engine found inside your car. You are converting a potential difference (High and Low temperature) into mechanical energy. In Earth's case, this mechanical energy is represented as as convection cells, which drive plate tectonics. Eventually, an engine runs out of gas, or in Earth's case, energy, and will begin to cool off.
When the core cools down, I don't think an ice age would be the right think to say. Mars would be a good example of what might happen to Earth when it loses most of its heat. There will no longer be tectonic events like volcanism, Earth will be a cold ball of mass. There will be a lot of ice, but eventually cosmic radiation and solar winds will destroy the atmosphere without the protection of the Earth's magnetic field leaving a barren planet with a largely homogenous surface. So sure, it will cause an ice age, but the ultimate destiny of the planet is barren, with a solid non-convecting mantle and core.
I want to add this this is largely speculation, that we really don't know what will happen. I just assume that Earth will share a similar fate to Mars. Mars once had a magnetic field protecting its atmosphere, but as the planet cooled off the field disappeared. Mars's historic magnetic field is an area of contentious research.
As gerrit pointed out, Venus has an atmosphere without a magnetic field, so this is clearly postulation. Perhaps an expert will shed light on this question (How long) would Earth's atmosphere last without a global magnetic field?
Put a frying pan on a stove burner and make the pan hot. Measure its temperature every minute over half an hour or so to get an idea of how rapidly it naturally cools.
Then start the experiment over again. This time, take a needle and touch and hold its tip to the frying pan so that it acts as a heat sink. The relative sizes of frying pan and needle will approximate "Earth and current methods of geothermal energy transfer". Measure the cooling over the next half hour.
If you run the two experiments a few times and compare results, you should find that it's practically impossible to detect any difference. The results can be extrapolated to estimate effects on the Earth, and a plausible conclusion is that it won't make a meaningful difference.
The simple reason is that our current geothermal efforts (as well as any currently projected future efforts) are so vanishingly small when compared to the size of the Earth that it has less effect than we can measure.
Now, that doesn't mean that some radical future change in technology won't change things. But no one can answer on those terms except to assert that we could make a significant difference if we could advance technology far enough.
Yes. But by how much? Let's do some rough math. We'll just be concerned with orders of magnitude here.
Suppose we have a uniform sphere the size of the Earth. Call it 1021 cubic meters.
Suppose this sphere is made of rock that is four times more dense than water. Water weighs 1000 kg per cubic meter.
Of course the Earth is not uniform; it is made up of rocks that are less dense and metals which are more dense. We're doing some rough math here.
And let's suppose that the interior of our planet is of uniform temperature, say, 5000 Kelvin.
Again, of course the Earth is not uniformly hot throughout. Again, we're doing rough math here, just to get an idea of the order of magnitude involved.
Let's suppose that our ball of rock is not producing new heat. Of course the Earth is producing new heat inside it, for instance, from radioactive elements in the core. But let's suppose that it is not.
And let's suppose that our ball of rock has a specific heat capacity of 0.8 joules per kilogram * kelvin. The specific heat capacity roughly speaking tells us how much energy is in some amount of a substance at a particular temperature. So multiply that out.
(1021 cubic meters) x
(4000 kg / cubic meter) x
(5000 kelvin) x
(0.8 Joules per kilogram * kelvin) =
1.6 x 1028 joules
We're just looking for an order of magnitude here. Our ball of rock has roughly 1028 joules of thermal energy.
Now let's suppose that we extract some amount of those joules. Total energy consumption of humanity from all sources -- nuclear, gas, etc -- is about 1018 joules per year. If we got 100% of that from our ball of hot rock, it would cool it off by one-ten-billionth of its total heat every year.
That's making the worst possible assumptions; of course we do not get anywhere even close to all our power from geothermal, the energy we do get was just going to be wasted into the atmosphere eventually anyway, the earth does make its own heat, and so on. We could get our total power needs met by geothermal energy for trillions of years without worrying about cooling the core.
The same way anything other ball of rock retains its heat. Heat, like all forms of energy, is retained indefinitely until something acts to remove it. I'm not clear on what question you're actually asking here.
This answer to the question 'Why has Earth's core not become solid?' over on Physics seems to claim the answer is no.
The core isn't hot just because of remnant heat left over from formation, the heat energy in the core is continually renewed by radioactive processes.
This energy from the core must already be continually dissipated up through the mantle, through the crust, into the atmosphere and eventually into space (or else the planet would be heating up).
All we could possibly do is speed the dissipation of this energy through the crust, any energy we extract would get to the surface anyway.
As others have pointed out geothermal energy is a tiny fraction of what heats our atmosphere, most of that comes from the sun.
Even if this were not the case, for us to cause an ice age, would require us to have near complete control over geothermal release through artificial means. We would have to extract enough energy over a long enough period from deep enough in the earth that there was no longer significant natural heat dispersion through the crust. Then we would have to stop and bottle up our manual extraction so that the heat had no other means to escape but rising through the crust in the natural way. The heat present in the atmosphere would dissipate into space far quicker than new heat would rise through the crust.
I imagine both the process of our intervening to the point of control, and our sudden relinquishing control would both have significant effects aside from climate change: earthquakes, volcanic eruptions, disrupting continental drift...
I hope it is clear that it doesn't.
You have to start with the cause of the heat: radioactive isotopes are distributed throughout the earth and, since thermal radiation occurs at the surface, the deeper you go the hotter it gets. Radioactive isotopes decay at a fixed rate and some have very long half lives so this heat is released over the lifespan of the earth. The earth's heat is not (mostly?) due to kinetic energy left over from its formation or tidal squeezing by the moon. Because of the scale nothing we do will make much difference to this situation.