The law of conservation of energy states that energy cannot be created or destroyed. It can only change form. In this sense, the term 'renewable energy' is a bit misleading. The label renewable refers to the fact that the stock of this energy is constantly being renewed, driven the Sun's radiation.
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Solar radiation heats the Earth's atmosphere in some places more than in others as the Earth spins on it's axis. Those parts of the atmosphere that are heated become less dense, so expand and rise, causing cooler air to move in. The effect is the wind we feel on the surface of the Earth. Cooling causes the same effect in reverse. The wind can be used to turn turbines and generate electricity.
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The intensity of the Sun's radiation averages about 1400 W/m^2 over the surface of the Earth. This can be converted into electricity using photovoltaic cells assembled into solar panels.
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Water can be heated in tanks by the Sun's radiation. The water is then used to provide heat in people's homes.
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Biofuels are the a possible renewable energy source, capable of being used in place of fossil fuels. They grow using the Sun's energy to power photosynthesis, and can later be made into ethanol and other such fuels and maybe partly mixed with normal fossil fuel to power cars and planes.
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Hydroelectric power uses water falling from height to turn turbines. The water is raised to the height in the first place by evaporation, the energy for which is provided by the Sun. As it flows over waterfalls and along rivers, gravitational potential energy is being lost, can can be turned into electrical energy using a turbine.
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Sustainably managed trees can be considered to be a renewable energy source. A tree is planted for every one cut down, and the trees cut down can be burnt to produce electricity in power stations.
Non renewable energy sources are everything else. Nuclear, fossil fuels, geothermal (which uses heat from the Earth's interior to heat water) are all non renewable. The first two energy sources causes pollution, and occasionally a nuclear power station explodes and causes a great deal of pollution, but geothermal is generally said to be 'green' energy source, and non polluting.