Along the dipole axis ( )

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Electrostatic fields and potential · Magnetostatics

The Boundary-Value Problem, Ch The magnetic multipole expansion and the magnetic scalar potential

Problem

Along the dipole axis (θ=0\theta=0), away from the origin, the electric dipole field is Eax(r)=2p4πε0r3,E_{\rm ax}(r) = \frac{2p}{4\pi\varepsilon_0 r^3}, and the full field also carries the point term −p3ε0δ3(r)-\dfrac{p}{3\varepsilon_0}\delta^3(\mathbf{r}). Along the axis of a magnetic dipole, away from the origin, Bax(r)=μ0m2πr3,B_{\rm ax}(r) = \frac{\mu_0 m}{2\pi r^3}, and the full field carries the point term +2μ0m3δ3(r)+\dfrac{2\mu_0 m}{3}\delta^3(\mathbf{r}). ((a)) For each case, form the ratio of the coefficient multiplying δ3(r)\delta^3(\mathbf{r}) to the coefficient multiplying 1/r31/r^3 in the axial term, keeping the signs. Each ratio is a pure number, because δ3(r)\delta^3(\mathbf{r}) and 1/r31/r^3 both carry units of m−3\mathrm{m^{-3}}; call the two numbers kEk_E and kBk_B. ((b)) Find kB/kEk_B/k_E, and state what it says about the size and the sign of the magnetic point term relative to the electric one.

Browse the problem library