(Real physicists of a delicate disposition may with to skip this post, as it contains some extreme mutilation of some rather important science)
In this post, waves and radiation are synonymous
- electric fields cause magnetic fields
- magnetic fields cause electric fields
(These are based on two of Maxwell's four equations)
So if you have one, you get the other, and vice-versa.
In fact, you get a self-sustaining effect where the magnetic field creates an electric field, which creates a magnetic field with creates an electric field, etc, etc.
This is called an electromagnetic wave, and the electric/magnetic field combo travels through empty space at 300,000,000m/s or so.
This wave* can have a wide range of frequencies depending on its source:
Non-Ionising radiation:
Low frequency radiation can be caused by changing currents in electric wires. The electric field from the wire can cause a magnetic field which causes an electric field around other nearby wires, disrupting any signals they may be carrying. This is what ED was worried about
At higher frequencies, the waves are great for communication. You can create and detect them with radio antennas and at higher frequencies still, see them with your eyes or a camera.
Ionising radiation:
At very high frequencies, starting with UV light, and including X-rays, the electric field in the wave can disrupt the arrangement of electrons in the atoms in chemicals in your body, notably DNA. This means you can end up with with the wrong DNA in your cells, and that can cause cancer. However, the chance of problems arising is small, and increases with the amount of radiation you are exposed to
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Some elements, generally those with large atoms near the bottom of the periodic table, are unstable. They tend to fall apart into smaller atoms, and release high-frequency EM waves as they do so.
Being exposed to the waves emitted by these decaying atoms is unhealthy, but luckily, the high-frequency waves tend to be absorbed by air, or water, or metals, or pretty much anything else nearby. This is why nuclear waste is often stored underwater or underground. All the high-frequency radiation is absorbed by the water or rock.
However, the amount of radiation that hits you increases dramatically as you get closer, and so does the risk of vital chemicals in your body being affected.
This means that almost touching it is far worse that standing near to a piece of radioactive material.
If you get a small amount stuck on your skin, or stuck in your lungs, or eat it, then it is constantly touching you and you are exposed for a very long time. The chance of your DNA being damaged and getting cancer can be very high.
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To sum up:
Low frequency radiation --> safe as long as there isn't enough to cause heat damage. Caused by signals in wires, and for radio communication, and is light.
High frequency radiation --> pretty safe when only for a short time, very dangerous (causes cancer) when you are exposed for a long time, or at high intensities, or when you're very close to the source
Large atoms decaying into smaller ones (aka radioactive material) --> emit high-frequency radiation --> extremely dangerous when touched or stuck in your body, because you are close to the source
The hazard suits pictured above are to prevent workers getting radioactive material stuck in/on their bodies, which would result in them receiving large amounts of high-frequency radiation, which would be very likely to cause cancer by messing with their DNA.
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* You may have heard that light is made up of particles called photons. This is kinda true too. It behaves like a wave, and a bit like a particle. Take your pick.