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We have that the power $P$ is energy/time. Thus to obtain the radiation rate (which is 1/time), we have to divide by an energy. The energy is the energy stored in the oscillator $U$ as the time to radiate this energy away is $\tau = U/P$. Let us assume that the particle (with charge $e$) in the oscillator performs a motion of the form  q= q_0 ...

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The binomial expansion says that $(1+x)^n=1+{n \choose 1}x^1+{n \choose 2}x^2 + ...$. This should be familiar to you for positive, integer n just by expanding out the parenthesis. For NEGATIVE n, it still holds, provided you interpret ${n \choose k}$ correctly for negative numbers; for our purposes, we just need to know ${n\choose 1}=n$ always. For very ...

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To start with the law of increasing entropy applies to isolated systems. The system you describe is isolated if one considers the total entropy of both the paramagnetic material and the permanent magnet, including any radiation. The order introduced in the paramagnetic material is balanced by a disorder in the permanent magnet plus any radiation from ...

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Hint: Try finding $U$ for both possibilities. i.e $-\vec p_1.E_2$ and $- \vec p_2 .E_1$ and see if they're the same. Potential Energy is usually defined for a system.( For example, the $U$ you find for one charge because of its interactions with another charge represents the P.E. of the two charge system. )

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