between the coils to obtain the largest homogeneous magnetic field volume. For a couple of coaxial coils, the magnetic field generated at point P is the sum of the field vectors of both coils. The magnetic field generated by both coils on the z-axis is [1] B₂(z) = 2112 1 (4z² + 4dz + d² + L²)1/z² + dz + 1 (4z²-4dz + d² + L²)/z²-dz + d² L² + 2 d² L² + 4 The magnetic field close to the coils' z-axis can be expanded by the Taylor formula as dB. B₂(z) = B₂(0) + dz 1 d²B. (0)z +=⋅ 2 dz² bz (0)z² + O(z³). Differentiating B(z), dB dz =(z) = - (2z+d)(12z² + 12dz + 3d² + 5L²) (4z² + 4dz + d² + L²)² (z² + dz + HOIL² πT + d² 12 3/2 + 4 2 (2z-d)(12z2 12dz + 3d² + 5L²) (4z² - 4dz + d² + L²)² z² dz + which is equal to zero in z = 0. + d² L2 3/2 + 4 2 To achieve an uniform field around the center of the Helmholtz coil, the second derivative of B₂(z) must be zero too. Differentiating (dB/dz)(z) and assuming z = 0, it is possible to state that [1] d = 0.5445L. This is the distance between the two square coils to obtain the largest uniform magnetic field around the center of the coils. An analytical model was implemented to assess the optimal distance between two coils for the Helmholtz configuration. Several numerical simulations were carried out to evaluate magnetic field, magnitude, and deviation. The mathematical model is based on the differential elements of the magnetic field expressed by the Biot-Savart law, I di× F dB = Po +3 (1) where is the vector from the differential current element to generic field point P (Fig. 1). dl is the infinitesimal length vector of the current element. Но is the vacuum permeability. I is the current flowing through the element considered. The geometrical configuration of the coil is sketched in Fig. 1, where a square coil with L dimension per side lies on the x-y plane, and with the z-axis orthogonal to the coil. The magnetic field in the generic point P is obtained integrating (1) along the coil. It is possible to determine the distance, d,

between the coils to obtain the largest homogeneous magnetic field volume. For a couple of coaxial coils, the magnetic field generated at point P is the sum of the field vectors of both coils. The magnetic field generated by both coils on the z-axis is [1] B₂(z) = 2112 1 (4z² + 4dz + d² + L²)1/z² + dz + 1 (4z²-4dz + d² + L²)/z²-dz + d² L² + 2 d² L² + 4 The magnetic field close to the coils' z-axis can be expanded by the Taylor formula as dB. B₂(z) = B₂(0) + dz 1 d²B. (0)z +=⋅ 2 dz² bz (0)z² + O(z³). Differentiating B(z), dB dz =(z) = - (2z+d)(12z² + 12dz + 3d² + 5L²) (4z² + 4dz + d² + L²)² (z² + dz + HOIL² πT + d² 12 3/2 + 4 2 (2z-d)(12z2 12dz + 3d² + 5L²) (4z² - 4dz + d² + L²)² z² dz + which is equal to zero in z = 0. + d² L2 3/2 + 4 2 To achieve an uniform field around the center of the Helmholtz coil, the second derivative of B₂(z) must be zero too. Differentiating (dB/dz)(z) and assuming z = 0, it is possible to state that [1] d = 0.5445L. This is the distance between the two square coils to obtain the largest uniform magnetic field around the center of the coils. An analytical model was implemented to assess the optimal distance between two coils for the Helmholtz configuration. Several numerical simulations were carried out to evaluate magnetic field, magnitude, and deviation. The mathematical model is based on the differential elements of the magnetic field expressed by the Biot-Savart law, I di× F dB = Po +3 (1) where is the vector from the differential current element to generic field point P (Fig. 1). dl is the infinitesimal length vector of the current element. Но is the vacuum permeability. I is the current flowing through the element considered. The geometrical configuration of the coil is sketched in Fig. 1, where a square coil with L dimension per side lies on the x-y plane, and with the z-axis orthogonal to the coil. The magnetic field in the generic point P is obtained integrating (1) along the coil. It is possible to determine the distance, d,