diff --git a/_psets/10.md b/_psets/10.md index 3d1cbbe745bae02591e28431baeb0e0eae69652d..2377759c71d9e3920003cec0533c942aa4116d40 100644 --- a/_psets/10.md +++ b/_psets/10.md @@ -65,7 +65,34 @@ field of a magnetic dipole $$\vec{m}$$ is $$ \vec{B} = \frac{\mu_0}{4 \pi} - \left[ \frac{3 \hat{x} (\hat{x} \cdot \vec{m}) - \vec{m}}{|\vec{x}|^3} \right] + \left[ \frac{3 \hat{x} (\vec{m} \cdot \hat{x}) - \vec{m}}{|\vec{x}|^3} \right] +$$ + +The force between two magnetic dipoles $$m_1$$ and $$m_2$$ (with associated fields $$B_1$$ and +$$B_2$$) is $$F = -\nabla (m_1 \cdot B_2)$$, and the characteristic interaction energy is $$m_1 +\cdot B_2$$. This will be maximized when $$m_1$$ and $$B_2$$ are parallel. From the equation above, +we can see that the field strength will be maximized when $$\hat{x}$$ is antiparallel to $$\vec{m}$$. +Assuming a separation of 1 angstrom, the total energy is thus + +$$ +\begin{align*} +E_m &= m \cdot \frac{\mu_0}{4 \pi} \left( \frac{4 m}{r^3} \right) \\ +&= \frac{\mu_0 m^2}{\pi r^3} \\ +&= \frac{\num{1.26e-6} \si{N/A^2} (\num{-9.28e-24} \si{J/T})^2}{\pi (10^{-10} \si{m})^3} \\ +&= \num{3.5e-23} \si{J} +\end{align*} +$$ + +Meanwhile their electrostatic potential is + +$$ +\begin{align*} +E_e &= q E \\ +&= q \frac{q}{4 \pi \epsilon_0 r} \\ +&= \frac{q^2}{4 \pi \epsilon_0 r} \\ +&= \frac{(\num{1.6e-19} \si{C})^2}{4 \pi \cdot \num{8.85e-12} \si{F/m} \cdot 10^{-10} \si{m}} \\ +&= \num{2.3e-18} \si{J} +\end{align*} $$