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The basic reason why your approach is too simple So the first problem for you to meditate on is: $\int dt~ r \ne r t,$ because $r = r(t)$ is going to be depending on time. For example, if something is purely falling up/down radially towards/away from the Earth, then the proper expression is instead:$$m \ddot r = -\frac{GM}{r^2},$$and the proper way to solve ...

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This is two body system. In such case you should make the following change of variables $\bf r=\bf r_1-\bf r_2$ and $\bf R=\frac{m_1 \bf r_1+m_2 \bf r_2}{m_1+m_2}$. It corresponds to center of mass system In such case the Lagrangian has the simple form: L=\frac{m_1\dot r_1^2}{2}+\frac{m_2\dot r_2^2}{2}+\frac{q_1 q_2}{|\bf r_1-\bf r_2|}=\frac{(m_1+m_2)\dot ...

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Lubos Motl answered this question a long time ago.

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So I take particle A and place it in space, then I place particle B 1,000,000 light years away from particle A. Alright. But, just to be sure: In astronomy, and (prehaps somewhat more recently) in cosmology and in physics in general, we understand this measure of "having been apart" as chronometric mutual separation. In your example this means ...

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While the first inclination is to drag out inverse square laws of point sources ... you have to take into account the surface area of the exposure. The radiation is fairly constant. a 1 cm sphere would receive the same level of radiation as a 1 meter sphere. what would change is the radiation per square cm. when looking axially the halogen filament could be ...

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The intensity goes as $r^{-2}$, not $r^{-3}$. One meter is 100 times further away than 1 cm, thus 1/10,000th the intensity. Side Note 1 How often are you within 10 cm (~3.94 inches) of a halogen bulb? The article states that 15 minutes under constant exposure at 10 cm can elicit erythema. However, who sits under a halogen bulb 10 cm away for 15 minutes ...

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The quoted document refers to medical radiation dose, in terms of intensity of radiation absorbed by tissue. In your case, the Sun's radiation is taken at the Earth's surface. Since intensity is inversely proportional to square of distance from source, it will be actually $0.01\%$ of the Sun's radiation intensity in the ultraviolet region at the Earth's ...

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