Magnetized sphere in an applied field B_{0} Induced moment: m = \frac{4π}{3} Ma^{3}

• Outside field: B_{0} − ∇ (\frac{m cos θ}{r^{2}} ) – Radial: B_{0} cos θ + \frac{2m cos θ}{r^{3}}

– Tangential: −B_{0} sin θ + \frac{m sin θ}{r^{3}}

• Inside fields: B_{in} = H_{in} + 4πM – Radial B: B_{in} cos θ

– Tangential H: −H_{in} sin θ

• Continuity conditions

– Radial: B_{in} = B_{0} + \frac{2m}{a^{3}}

– Tangential: H_{in} = B_{0} − \frac{m}{a^{3}}

\{ \begin{matrix}B_{in} = B_{0} + \frac{8πM}{3} \\ H_{in} = B_{0} − \frac{4πM}{3} \end{matrix}  ⟹  \{ \begin{matrix}B_{in} + 2H_{in} = 3B_{0} \\ B_{in} = H_{in} + 4πM \end{matrix}

Spontaneous Magnetization (B_{0} = 0)

\{ \begin{matrix}B_{in} = \frac{8π}{3} M_{spont} \\ H_{in} = −\frac{4π}{3} M_{spont} \end{matrix}  ⟹  \{ \begin{matrix}B_{in} = −2H_{in} \\ B_{in} = H_{in} + 4πM_{spont} \end{matrix}

Linear susceptibility material (B_{in} = μH_{in})

3B_{0} = μH_{in} + 2H_{in} ⟹ \{ \begin{matrix}H_{in} = 3B_{0}/(μ + 2) \\ B_{in} = 3μB_{0}/(μ + 2) \\ m = B_{0}a^{3} \frac{μ − 1}{μ + 2} \end{matrix}

Type I superconductor (4πM = −H_{in} or μ = 0)

3B_{0} = 2H_{in} ⟹ \{ \begin{matrix}H_{in} = \frac{3}{2}B_{0} \\ B_{in} = 0 \\ m = −\frac{1}{2}B_{0}a^{3} \end{matrix}

M is equivalent to a surface current K = cM × n : K_{ϕ} = cM sin θ = \frac{3cB_{0}}{8π} sin θ. Vector potential

The effective surface current cM × n generates the vector potential

A_{Mϕ}(r, θ) = \frac{4π}{3} Mr sin θ for r < a
A_{Mϕ}(r, θ) = \frac{m}{r^{2}} sin θ for r > a

In addition, there is the external vector potential \frac{1}{2} B_{0} × r, or

A_{0 ϕ}(r, θ) = \frac{1}{2} B_{0}r sin θ

The B field can be obtained from the total A_{ϕ} = A_{0ϕ} + A_{Mϕ} by using

\{ \begin{matrix}B_{r} = \frac{1}{r sin θ} \frac{∂}{∂θ} ( A_{ϕ} sin θ) \\ B_{θ} = −\frac{1}{r} \frac{∂}{∂r} (r A_{ϕ}) \end{matrix}