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Using mass ($M$), length ($L$), time ($T$), and electric current ($A$) as fundamental quantities, the dimensions of magnetic permeability are:
Magnetic permeability $\mu_0$ appears in: $F = \dfrac{\mu_0}{4\pi}\dfrac{I_1 I_2 l}{r}$
$[\mu_0] = \dfrac{[F][r]}{[I]^2[l]} = \dfrac{MLT^{-2} \cdot L}{A^2 \cdot L} = MLT^{-2}A^{-2}$
Alternatively, from $B = \mu_0 H$: $[H] = AL^{-1}$, $[B] = MT^{-2}A^{-1}$, so $[\mu_0] = \frac{[B]}{[H]} = \frac{MT^{-2}A^{-1}}{AL^{-1}} = MLT^{-2}A^{-2}$
Find all $x$ satisfying $(\cot^{-1}x)^2 - 7(\cot^{-1}x)+10>0$.
Let $u=\cot^{-1}x$. Solve $u^2-7u+10>0$, i.e., $(u-2)(u-5)>0$.
This gives $u<2$ or $u>5$.
Since $\cot^{-1}x$ is a decreasing function with range $(0,\pi)$:
- $u<2 \Rightarrow \cot^{-1}x<2 \Rightarrow x>\cot2$ (decreasing function reverses inequality)
- $u>5 \Rightarrow \cot^{-1}x>5 \Rightarrow x<\cot5$
Solution: $x\in(-\infty,\cot5)\cup(\cot2,\infty)$
\(r_n = a_0 \cdot \dfrac{n^2}{Z}\) โ \(r \propto n^2\)
\(E_n = -13.6 \cdot \dfrac{Z^2}{n^2}\) eV โ \(E \propto -\dfrac{1}{n^2}\)
As orbit number increases, radius increases (electrons move farther out) while energy becomes less negative (electrons become less tightly bound). For hydrogen (Z=1): \(r_1 = 0.529\) ร (Bohr radius).
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