Magnetic Field (from the paper)
The magnetic field components in the x and z directions are given by:
$$
B_x = \frac{B_r R}{\pi} \left[ \alpha_+ P_1(k_+) - \alpha_- P_1(k_-) \right]
$$
$$
B_z = \frac{B_r R}{\pi (\rho + R)} \left[ \beta_+ P_2(k_+) - \beta_- P_2(k_-) \right]
$$
Where ( P_1(k) ) and ( P_2(k) ) are elliptic integrals of the first and second kind:
$$
P_1(k) = \mathcal{K} - \frac{2}{1 - k^2} \left( \mathcal{K} - \mathcal{E} \right)
$$
$$
P_2(k) = -\frac{\gamma}{1 - \gamma^2} \left( \mathcal{P} - \mathcal{K} \right) - \frac{1}{1 - \gamma^2} \left( \gamma^2 \mathcal{P} - \mathcal{K} \right)
$$
Additional relations:
$$
\xi_{\pm} = z \pm L
$$
$$
\alpha_{\pm} = \frac{1}{\sqrt{\xi_{\pm}^2 + (\rho + R)^2}}
$$
$$
\beta_{\pm} = \xi_{\pm} \alpha_{\pm}
$$
$$
\gamma = \frac{\rho - R}{\rho + R}
$$
$$
k_{\pm}^2 = \frac{\xi_{\pm}^2 + (\rho - R)^2}{\xi_{\pm}^2 + (\rho + R)^2}
$$
Elliptic Integrals
$$
\mathcal{K} = \mathbb{K}\left( \sqrt{1 - k^2} \right) = \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1 - (1 - k^2) \sin^2 \theta}}
$$
$$
\mathcal{E} = \mathbb{E}\left( \sqrt{1 - k^2} \right) = \int_0^{\frac{\pi}{2}} d\theta , \sqrt{1 - (1 - k^2) \sin^2 \theta}
$$
$$
\mathcal{P} = \Pi(1 - \gamma^2, \sqrt{1 - k^2}) = \int_0^{\frac{\pi}{2}} \frac{d\theta}{(1 - (1 - \gamma^2) \sin^2 \theta) \sqrt{1 - (1 - k^2) \sin^2 \theta}}
$$
Magnetic Gradient (Mathematical Derivation)
The magnetic gradient in the x and z directions is derived as:
$$
(\nabla B)_x = \frac{B_x \frac{\partial B_x}{\partial x} + B_z \frac{\partial B_z}{\partial x}}{B}
$$
$$
(\nabla B)_z = \frac{B_x \frac{\partial B_x}{\partial z} + B_z \frac{\partial B_z}{\partial z}}{B}
$$
Magnet Force (from the paper, Unverified, use with caution)
The force ( f(B) ) as a function of magnetic field strength is given by:
$$
f(B) =
\begin{cases}
3 & B < \frac{M_{sp}}{3} \
\frac{M_{sp}}{B} & B \geq \frac{M_{sp}}{3}
\end{cases}
$$
The magnetic force components in the x and z directions are:
$$
F_x = V_p f(B) B_x (\nabla B)_x
$$
$$
F_z = V_p f(B) B_z (\nabla B)_z
$$
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