What is the relationship between the amount of radioactive substance and the harm it does to the body? This is probably a very simple question, but I wonder how the amount of radioactive substance is related to the radiation it emits and therefore the harm it does to the body.
If I held 0.1g of barium-137 in my left hand and 10g of it in my right hand, for the same amount of time, would my right hand have been damaged 100 times as much as my left hand?
(I used barium-137 as an example because it's a gamma emitter. Alpha and beta particles would be absorbed internally more by the 10g mass because it's bigger.)
 A: The search term to use in order to learn about this is "linear no-threshold," or LNT. This is the hypothesis that the biological harm is proportional to the radiation dose. It's hard to get reliable data, but LNT appears to be a poor approximation at low doses.
At low doses, there may even be an effect called radiation hormesis. A variety of experiments seem to show cases in which low levels of
radiation activate cellular damage control mechanisms, increasing the
health of the organism. For example, there is evidence that exposure
to radiation up to a certain level makes mice grow faster; makes
guinea pigs' immune systems function better against diphtheria;
increases fertility in female humans, trout, and mice; improves fetal
mice's resistance to disease; reduces genetic abnormalities in
humans; increases the life-spans of flour beetles and mice; and
reduces mortality from cancer in mice and humans.
I've had a hard time evaluating the evidence about radiation hormesis and LNT. A couple of authors who have worked on this sort of thing are Tubiana and Little, and they reach different conclusions.
There are obviously important public health and public policy implications for all this, so it would be nice to have a definitive answer, but I just don't think we're ever likely to get one. Ideally you'd raise a gazillion lab rats to adulthood with different amounts of ionizing radiation, and compare their health. In reality, the kinds of population doses we talk about in cases like Hiroshima, Chernobyl, and Fukushima are so low that if LNT were true, the number of lab rats required in order to do a statistically conclusive controlled study would be prohibitively expensive. (Even in the cases of Hiroshima and Nagasaki, most of the surviving population was exposed to relatively low doses. In studies of survivors that look for cancer, for example, it's hard to disentangle the effects of nuclear radiation from the effects of things like burning buildings, which emit nasty carcinogenic smoke.)
A: There are different aspects to this, as you may have gathered from the other answers.
One aspect is the amount of ionization that is produced in your tissue. This is a simple linear relationship. Each unit of a particular kind of radiation produces the same amount of ionization. It produces the same amount of energy deposited in affected tissue. It produces the same amount of radicals, dislocation of molecules, etc. Twice as much radiation produces twice the result, etc.
The other aspect is, how much harm is produced by different amounts of radiation. On this aspect, there are some subtle points, and some controversy.
First, it is complicated by the fact that there is a statistical component. Some of the effects show up as probabilities of certain harm. Thus, radiation effects are often quoted in terms of probability of producing excess cancers, and similar statements. 
The next complication is the idea of a minimum value of radiation that might not be harmful, or might even be beneficial. This is controversial. The buzz word is "hormesis." 
Above some threshold, it seems fairly clear that twice as much radiation is twice as bad. This holds up to some other threshold where death is likely to occur. Above that it does not make sense to talk about additional harm. You are dead.
So, that much isn't controversial. There is some discussion about fixing those limits, and the exact degree of harm from a given unit of radiation. 
What is controversial is that lower threshold. There is some evidence that, below some minimum, living things can recover from small exposure to radiation. That it won't produce adverse effects. There is even some evidence, controversial evidence, that small doses of radiation give resistance to larger doses. It's kind of like you get a base tan before going to the tropical beach.
The problem here is, it is very difficult to be accurate enough to be confident of any such claims.
The competing idea to this threshold is the no-threshold idea. That idea says that radiation is bad down to arbitrarily small dose. So if dose  produces 1 additional cancer in a population of 1000 people so exposed, then the no-threshold idea says a dose of half  will, on average, produce 1 additional cancer in a population of 2000 people so exposed. And so on to smaller and smaller doses.
Consider what would be required. If, for example, you wanted to test a dose that would be expected to produce 1 extra cancer in 10,000 people. And you want to do this in a way that lets you have a standard deviation in your answer of 1%. You would need to expose many 100's of thousands of people and observe the excess cancers. But look. You would then have given, possibly, 100 people cancer. This is generally considered monstrous, and so medical researchers are not allowed.
But we do have x-ray data. And some other exposures. And researchers spend a lot of time arguing about how to interpret these. We also have exposures of lab rats and such. And again, there is a lot of controversy over these.
A: Mass (kg) is proportional to activity (Bq) which in turn is proportional to dose (Sv). @AndreiGeanta mentioned effective dose which is statistically proportional to the adverse effects of radiation for the whole body averaged.
