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6

Suppose you start with two kilograms of C-14. After 5730 years you have one kilogram left. Call that piece A. Now get another kilogram of C-14, call it piece B, and put it next to piece A. You now have two identical pieces of C-14, and yet one of them (A) is supposed to half-decay in 2865 years and the other (B) is supposed to half-decay in 5730 years? Do ...

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putting it simply: activity law states that : dn/dt is proportional to n. which means the rate of reaction of any substance depends on the amount of the substance itself. because there is a greater amount of carbon initially, the probability of some of it decaying is higher , than when there is less amount of it left.

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Imagine a sample of 1000 atoms with a half-life of 1 hour. That means every hour, the sample is reduced to 50% of its size. After one hour, you are left with 500 atoms. How much time for that new sample (500 atoms) to be reduced to 50% (250 atoms) ? In your interpretation : For the new sample to be reduced to 50%, it needs to lose 250 atoms. Since it ...

2

Suppose that decay worked the way you proposed, half as many atoms take half as long to decay. It sounds sort of plausible at first, but consider this: how does any one atom know when it's allowed to decay? It can't just roll a die and decay if it rolls a 1, it has to know how big the sample it's in is, and adjust its probability of decaying accordingly. If ...

59

The right way to think about this is that, over 5,730 years, each single carbon-14 atom has a 50% chance of decaying. Since a typical sample has a huge number of atoms1, and since they decay more or less independently2, we can statistically say, with a very high accuracy, that after 5,730 years half of all the original carbon-14 atoms will have decayed, ...

0

The mass of radioactive materials follows the ordinary differential equation: $$m'(t)=-am(t),$$ where $m$ is the mass and $a$ a positive constant - i.e., constant relative rate of decay. This implies $$m(t)=m(0)\mathrm{e}^{-at}. \tag{1}$$ If $T_h$ is half life, then $$m(T_h)=m(0)\mathrm{e}^{-aT_h}=\frac{1}{2}m(0),$$ which implies that $$T_h=\frac{\log ... 17 I know exactly where you're coming from. If I can put it into my own words: If it takes a sample some amount of time to decay, shouldn't a sample of half the size take half the time to decay? I have fallen into this seemingly sensical but somehow incorrect belief more than once. Here's a graph that shows what I believe you're currently thinking. The ... 9 Logically, shouldn't it take 2,865 years for the quarter to decay, rather than 5,730? Imagine that the quantity q(n) of something decays as$$q(n) = Q\cdot 2^{-n} where $n$ is the number of half-lifes. Initially, there is quantity $q(0) = Q\cdot 2^0 = Q$ of something. After 1 half-life, there is $q(1) = Q \cdot 2^{-1} = \frac{Q}{2}$ remaining. ...

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I think you're confused simply by the language. Remember that it's a quarter of the original sample. So it's like compounding interest in the bank. You start with initial principal, once the interest is compounded, you might say that the percentage of that principal is ADDED TO the "principal", and then a percentage of THAT is calculated, and added to that ...

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The term "Tokamak" refers to a design, not a size. The planed ITER reactor has the goal of 500 megawatts output. So it would take approximately 300,000,000 such reactors to produce the same power as the solar energy reaching the Earth. https://www.iter.org/factsfigures

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My textbook uses $R$ for activity as corresponding to the decay rate.

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You also you need to take into account the radiation that is generated by the interactions within the Al (unless that is neglected). This modifies the thickness of Al required. The correct name for that term is 'Buildup Factor'. There is a chapter in Cember & Johnson that explains it very nicely. A.

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This isn't an answer because as it stands it's hard to answer your question, but it got a bit long for a comment. You ask whether a magnetic field causes time dilation, but since there are no magnetic monopoles (or at least we've never found any) I can't think of a situation where you could usefully separate out the effects of the magnetic field from all ...

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Since an electromagnetic field is associated with a non-vanishing energy-momentum tensor, which in turn is the source term of gravity in the Einstein field equations, the answer is yes - if you were able to produce a strong enough field you would eventually get a measurable gravitational time dilation as compared to a reference clock far away. It should be ...

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