### Pset 9, Abubakar Abid (It may take a second for the LaTeX-javascript library to load...)

1. Diamagnetism can be estimated as $$\chi_m = \frac{M}{H} = -\mu_0 \frac{q^2 Z r^2}{4 m_e V}$$ Using numbers from literature, and assuming that the solid is (predominantly) carbon, we find that susceptibility is: $$\chi_m = -\mu_0 \frac{q^2 6 (70 \text{pm})^2}{4 m_e 4/3 \pi (70 \text{pm})^3}$$ $$\chi_m = 0.0001812$$ The force is $$F = V \mu_0 \chi_m H \frac{dH}{dz} = mg$$ For a magnetic field gradient that goes to zero over the frog $$F = V \mu_0 \chi_m \frac{H^2}{\Delta z} = mg$$ $$H^2 = \frac{\Delta z \rho g}{\mu_0 \chi_m}$$ Assuming $$\Delta z$$ is 5 cm, then $$H = 1.467 MA/m$$ This would correspond to a magnetic field strength of approximately $$B = \mu_0 H = 1.8 T$$

2. The field strength of a magnetic dipole is

6. The coercivity of this material is roughly 300 Oersted, which corresponds to 0.03 Teslas. The magnetic field from a current source is $$B = \mu_0 I / (2\pi r)$$ $$0.03 T = \mu I / (2\pi 1 cm)$$ $$I = 1500 A$$