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Magnetic Simulation

Magnetic Field (from the paper)

The magnetic field components in the x and z directions are given by:

Bx=BrRπ[α+P1(k+)−α−P1(k−)]B_x = \frac{B_r R}{\pi} \left[ \alpha_+ P_1(k_+) - \alpha_- P_1(k_-) \right] Bz=BrRπ(ρ+R)[β+P2(k+)−β−P2(k−)]B_z = \frac{B_r R}{\pi (\rho + R)} \left[ \beta_+ P_2(k_+) - \beta_- P_2(k_-) \right]

Where ( P_1(k) ) and ( P_2(k) ) are elliptic integrals of the first and second kind:

P1(k)=K−21−k2(K−E)P_1(k) = \mathcal{K} - \frac{2}{1 - k^2} \left( \mathcal{K} - \mathcal{E} \right) P2(k)=−γ1−γ2(P−K)−11−γ2(γ2P−K)P_2(k) = -\frac{\gamma}{1 - \gamma^2} \left( \mathcal{P} - \mathcal{K} \right) - \frac{1}{1 - \gamma^2} \left( \gamma^2 \mathcal{P} - \mathcal{K} \right)

Additional relations:

ξ±=z±L\xi_{\pm} = z \pm L α±=1ξ±2+(ρ+R)2\alpha_{\pm} = \frac{1}{\sqrt{\xi_{\pm}^2 + (\rho + R)^2}} β±=ξ±α±\beta_{\pm} = \xi_{\pm} \alpha_{\pm} γ=ρ−Rρ+R\gamma = \frac{\rho - R}{\rho + R} k±2=ξ±2+(ρ−R)2ξ±2+(ρ+R)2k_{\pm}^2 = \frac{\xi_{\pm}^2 + (\rho - R)^2}{\xi_{\pm}^2 + (\rho + R)^2}

Elliptic Integrals

K=K(1−k2)=∫0π2dθ1−(1−k2)sin⁡2θ\mathcal{K} = \mathbb{K}\left( \sqrt{1 - k^2} \right) = \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1 - (1 - k^2) \sin^2 \theta}} E=E(1−k2)=∫0π2dθ 1−(1−k2)sin⁡2θ\mathcal{E} = \mathbb{E}\left( \sqrt{1 - k^2} \right) = \int_0^{\frac{\pi}{2}} d\theta \, \sqrt{1 - (1 - k^2) \sin^2 \theta} P=Π(1−γ2,1−k2)=∫0π2dθ(1−(1−γ2)sin⁡2θ)1−(1−k2)sin⁡2θ\mathcal{P} = \Pi(1 - \gamma^2, \sqrt{1 - k^2}) = \int_0^{\frac{\pi}{2}} \frac{d\theta}{(1 - (1 - \gamma^2) \sin^2 \theta) \sqrt{1 - (1 - k^2) \sin^2 \theta}}

Magnetic Gradient (Mathematical Derivation)

The magnetic gradient in the x and z directions is derived as:

(∇B)x=Bx∂Bx∂x+Bz∂Bz∂xB(\nabla B)_x = \frac{B_x \frac{\partial B_x}{\partial x} + B_z \frac{\partial B_z}{\partial x}}{B} (∇B)z=Bx∂Bx∂z+Bz∂Bz∂zB(\nabla B)_z = \frac{B_x \frac{\partial B_x}{\partial z} + B_z \frac{\partial B_z}{\partial z}}{B}

Magnet Force (from the paper, Unverified, use with caution)

The force ( f(B) ) as a function of magnetic field strength is given by:

f(B)={3B<Msp3MspBB≥Msp3f(B) = \begin{cases} 3 & B < \frac{M_{sp}}{3} \\ \frac{M_{sp}}{B} & B \geq \frac{M_{sp}}{3} \end{cases}

The magnetic force components in the x and z directions are:

Fx=Vpf(B)Bx(∇B)xF_x = V_p f(B) B_x (\nabla B)_x Fz=Vpf(B)Bz(∇B)zF_z = V_p f(B) B_z (\nabla B)_z

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