This study presents a fast-computed analytical model to compute the magnetic field generated by a cone with arbitrarily uniform magnetization. Based on the charge model, algebraic and geometrical analyses, the analytical expressions of the axial, azimuthal, and radial components of the magnetic field produced by the cone at any given point in space are derived. The model was in excellent agreement with the Finite Element Model (FEM), as is proved in this research. It is demonstrated that it took an average of less than 1 ms to execute each expression at a single point on the modern personal computer, whereas the FEM simulation consumed 779 s. Moreover, using the developed model, the magnetic field distribution of a cone used in magnetic resonance imaging with varying magnetization is analyzed. As a result, the distributions of the axial and radial components are a cosine-like wave with opposite directions, except for the case where the slope angle of the magnetization is equal to π/2; or, in other words, the magnetization is axial. On the other hand, the distribution of the azimuthal component is a sine-like wave, except for the noted case where the magnetization is axial.

## I. INTRODUCTION

In order to facilitate the design and optimization of permanent magnet devices,^{1–5} a fast-computed (semi-)analytical model of the magnetic field distribution of basic-shaped (such as ellipsoid, sphere, tile, ring, cone, circular and elliptical cylinders, and parallelepiped) permanent magnets have been developed.^{6–16} Moreover, due to the superposition principle,^{6,15} the magnetic field distribution of a complex-shaped permanent magnet, which is formed by the summation or substitution of basic-shaped ones, can be computed using the models of the field distribution created by these permanent magnets of elemental shapes.^{11,14} This implies that it is of great importance to obtain a fast-computed expression of the magnetic field generated by any given permanent magnets in their basic shapes. In the current literature, the magnetic field distribution of some permanent magnets with basic shapes has been developed (semi-)analytically. For instance, Ravaud *et al.*^{8} derived an analytical model to compute the magnetic field produced by a parallelepipedic magnet of various and uniform polarizations; McClurg^{10} studied the magnetic field distribution of a sphere and an ellipsoid analytically. The analytical expressions of the magnetic field distribution of elliptical cylinder permanent magnets with diametrical^{11} and axial magnetizations^{14} have also been developed. However, the derivation of mathematical models of the magnetic fields produced by conical permanent magnets is limited. Most recently, Nguyen^{15} has derived analytical expressions of three components of the magnetic field created by a conical magnet with axial magnetization. These expressions are more fast-computed than can be achieved by the Finite Element Analysis (FEA), and they are useful for the analysis of the magnetic field distribution of a conical magnet used in magnetic resonance imaging. Nonetheless, the magnetic field distribution of a conical permanent magnet with arbitrarily uniform magnetization has not yet been analytically modeled and analyzed. Although the magnetic field distribution of permanent magnets of any geometries can be solved using the FEA, this method could be time-consuming as being demonstrated in the current literature.^{11,14,15} Moreover, having in hand fast-computed expressions and understanding the magnetic field distribution could facilitate the design and optimization of devices where this type of permanent magnet could be applied, such as in magnetic resonance imaging.^{3,4,15}

To address this gap in the current literature, this research aims to derive analytical expressions of the magnetic field distribution of a permanent magnet in a cone shape with arbitrarily uniform magnetization. The analytical model is developed based on the charge model,^{6} algebraic and geometrical analyses. Based on this model, the analysis of the magnetic field distribution is also carried out to provide readers with an insight into the distribution when the magnetization varies.

The rest of this paper is organized as follows: Section II presents the step derivation of the magnetic field distribution. The results obtained from the derived analytical model are validated by those of the Finite Element Model (FEM) and described in more detail in Sec. III. Section IV analyzes the magnetic field distribution in different magnetization directions for an application in magnetic resonance imaging. Section V draws a conclusion.

## II. MATHEMATICAL FORMULATION OF THE MAGNETIC FIELD DISTRIBUTION

Due to the superposition principle, the magnetic field $HJ\u2192\u2192$ generated by a conical permanent magnet with arbitrarily uniform magnetization $J\u2192$ can be represented as the summation of the magnetic field produced by two cones with the same geometrical parameters as those of the original cone [the cone height is h (m) and the vertex angle is 2φ (rad) (Fig. 1)]; one is with the axial magnetization $JZ\u20d7$, which creates the magnetic field $HJZ\u20d7\u2192$, and the another one is with the diametrical magnetization $JY\u20d7$, which creates the magnetic field $HJY\u20d7\u2192$ (it is noted that diametrical magnetization means that the direction of the magnetization is perpendicular to the symmetry axis OZ of the cone).

Denoting the slope angle as Ω (Fig. 1) between $J\u2192$ and the horizontal plane (OXY), we have

In the author’s previous article,^{15} the magnetic field distribution $HJZ\u20d7\u2192$ of a conical permanent magnet with the axial magnetization $JZ\u20d7$ has been expressed analytically. The author recommends readers to refer to that publication for more details on the analytical expressions. For the sake of brevity, but without any loss of generality, they are not presented repeatedly in this research. This also means that, in order to compute the magnetic field distribution of a cone with arbitrarily uniform magnetization analytically, it is necessary to develop an analytical model to express the magnetic field generated by a cone with diametrical magnetization. This task is addressed in the rest of this section.

The geometrical parameters of the cone used in the derivation are depicted in Fig. 2. The height of the cone is denoted as h (m); the vertex angle is 2φ (rad); the radius of the base circle is R (m); $vb\u20d7$ and $v\u2192$ are the unit vectors normal to the base and lateral surfaces, respectively; $vh\u20d7$ and $vv\u20d7$ are projections of vector $v\u2192$ on the horizontal and vertical directions; and S and N represent the south and north poles. Without the loss of generality, it is assumed that the magnetization $JY\u20d7$ is along the OY axis.

Based on the charge model,^{6} the magnetic field produced by a cone at point N in free space (Fig. 3) can be computed using Eq. (3),

where $MN\u2192$ is the vector linking the source point M belonging to the cone at the computed point N; σ_{s} and σ_{v} are the surface and volume charge densities, respectively; and $\mu 0\u22451.256\u2009637\u20090\xd710\u22126\u2009H/m$ is the vacuum permeability.

Since the magnetic field is uniformly distributed, the volume charge density σ_{v} is equal to zero.^{12,15} Thus, only the surface charge density contributes to the magnetic field. As a result, Eq. (3) can be simplified as follows:

Now, let us consider the sectional plane O′X′Y′ parallel to plane OXY, where the source point M lies. The projection of point N on this plane is denoted as G (Fig. 3).

Figure 4 depicts the section of the cone on plane O′X′Y′ and vectors $MM\u2032\u2192,M\u2032G\u2192$ used for the following geometrical analysis.

The cylindrical coordinate system with three unit vectors $ir\u20d7$, the radial direction, $i\alpha \u20d7$, the azimuthal direction, and $iz\u20d7$, the axial direction is applied in the further derivation. In this coordinate system, source point M has coordinates (r_{M}, α, and z_{M}) and computed point N has coordinates (r, β, and z). It is worth noting that r_{M} and r are the radial distances; α and β are the azimuthal angles; and z_{M} and z are the coordinates of the axial axis.

From Figs. 3 and 4, vector $MN\u2192$ can be presented in the form of other vector components as follows:

The surface charge density, which can be calculated as the dot product of the unit normal vector and the magnetization vector^{6} generated by the base surface, is equal to zero due to the perpendicularity between the unit normal vector $vb\u20d7$ of this surface and the magnetization vector $JY\u20d7$ ($JY\u20d7\u22c5vb\u20d7=0$). Therefore, only the surface charge density of the lateral surface of the cone contributes to the magnetic field intensity. Based on Figs. 2 and 4, this density can be computed using the following equation:

Based on Fig. 5, some geometrical parameters, such as r_{M} and z_{M}, and the infinitesimal area of the lateral surface ds can be further derived using the cone’s slant height ξ as

and

After integrating Eq. (10) based on ξ, the three components of the magnetic field can be analytically expressed as follows:

### A. Expression of the axial component $HNAxial$

The axial component $HNAxial$ is obtained as follows:

where $\Xi =h2+R2$; Ψ = 2r sin φ cos(β − α) + 2z cos φ; Θ = r^{2} + z^{2}; without much difficulty, it can be algebraically proved that 4Θ > Ψ^{2}, except for the singular point where r = z = 0.

For points on the axial axis (OZ; r = 0, δ, z), the magnetic field intensity is equal to zero as a result of the integration of the odd function of cos(α) over the range [0, 2π] in Eq. (11).

The axial component of the magnetic field intensity $HNAxial(Abitr)$ is produced by a cone with arbitrarily uniform magnetization, and can be computed using the following equation:

where $HNAxial[15]$ is the axial component of a magnetic field created by a cone with the same geometrical parameters, but the magnetization has an axial orientation^{15} with the magnitude J_{Z} = J sin Ω (Fig. 1). $HNAxial[15]$ can be calculated using the analytical expression of the axial component, which was developed in the author’s previous article.^{15} For the purpose of brevity, this expression is not repeated here.

### B. Expression of the azimuthal component $HNAzim$

The azimuthal component $HNAzim$ is derived as follows:

For points on the axial axis (OZ; r = 0, δ, z), the magnetic intensity $HNAzim(OZ)$ can be expressed as follows:

where $\Xi =h2+R2$; Ψ_{0} = 2z cos φ; $\Theta 0=z2$; without much effort, it can be proved that $4\Theta 0>\Psi 02$, except for the singular point where z = 0.

The azimuthal component of magnetic field intensity $HNAzimuthal(Abitr)$ produced by a cone with arbitrarily uniform magnetization can be computed using the following equation:

where $HNAzimuthal[15]$ is the azimuthal component of the magnetic field created by a cone with the same geometrical parameters, but the magnetization has the axial orientation^{15} with a magnitude of J_{Z} = J sin Ω (Fig. 1). $HNAzimuthal[15]$ can be calculated using the analytical expression of the azimuthal component, which was developed in Ref. 15. For the purpose of brevity, this expression is not repeated here.

### C. Expression of the radial component $HNRadi$

The radial component $HNRadi$ is represented as follows:

where $\Xi =h2+R2$; Ψ = 2r sin φ cos(β − α) + 2z cos φ; Θ = r^{2} + z^{2}; and Λ = sin φ cos(β − α), and it is mentioned again that 4Θ > Ψ^{2}, except for the singular point where r = z = 0.

For the points on the axial axis (OZ; r = 0, δ, z), the radial component $HNRadi(OZ)$ can be expressed as follows:

where $\Xi =h2+R2$; Ψ_{0} = 2z cos φ; Θ_{0} = z^{2}; and it can be realized that $4\Theta 0>\Psi 02$, except for the singular point where z = 0.

The radial component of magnetic field intensity $HNRadial(Abitr)$ produced by a cone with arbitrarily uniform magnetization can be computed using the following equation:

where $HNRadial[15]$ is the radial component of the magnetic field created by a cone with the same geometrical parameters, but the magnetization has the axial orientation^{15} with the magnitude J_{Z} = J sin Ω (Fig. 1). $HNRadial[15]$ can be calculated using the analytical expression of the radial component, which was developed in Ref. 15. For the purpose of brevity, this expression is also not repeated here.

From the magnetic field intensity $H\u2192N$, the magnetic flux density $B\u2192N$ can be computed with the following equations:^{11,15}

## III. COMPARISON WITH THE RESULTS OF FINITE ELEMENT ANALYSIS

### A. Verification of the analytical model of a magnetic field produced by a cone with diametrical magnetization

In order to validate the analytical model developed in this study to compute the magnetic field generated by a diametrically magnetized cone, a Finite Element Model (FEM) was built in the Electromagnetic Simulation Software® (EMS) (EMWorks, Inc., Montreal, QC, Canada), integrated with 3D CAD INVENTOR® (AUTODESK, Inc., CA). The geometrical parameters of the cone used in this simulation are height h = 0.010 m; vertex angle 2φ = π/3 (rad); and the magnetization, which is along the axis OY, as depicted in Fig. 2, with magnitude 1 (T).

The analytical results were obtained by coding the developed expressions of the three components of the magnetic field, which were created by a cone with the same geometrical parameters and magnetization used in the FEM in MATLAB R2016b (MathWorks, Natick, MA).

The comparison results between the two models are shown in Fig. 6 [the plot was generated along the axial direction z = [−0.0015, 0.012] (m) with the radial distance r = 0.00825 (m) and azimuthal angle β = 4.957 (rad)]. Moreover, the results of 14 points with random axial coordinate z (mm) are extracted and listed in Table I. Figure 6 and Table I demonstrate that the results of the developed analytical expressions are in excellent agreement with those of the FEM (the maximum, average, and minimum errors between the two models are 0.51%, 0.14%, and 0.0057%, respectively, for the axial component; 0.17%, 0.05%, and 4.45 × 10^{−5}%, respectively, for the azimuthal component; and 0.99%, 0.26%, and 0.0027%, respectively, for the radial component).

Points (r = 0.00825 m, . | Axial component (gauss) . | Azimuthal component (gauss) . | Radial component (gauss) . | |||
---|---|---|---|---|---|---|

z m, β = 4.957 rad) . | Analytical model . | FEA . | Analytical model . | FEA . | Analytical model . | FEA . |

−0.8181 | −59.8687 | −59.8653 | −165.7066 | −165.7111 | 21.6651 | 21.5873 |

−0.2727 | −65.3803 | −65.4131 | −182.2708 | −182.2264 | 28.0656 | 28.0159 |

1.7727 | −85.8743 | −85.8113 | −257.5002 | −257.4356 | 63.5014 | 63.4997 |

9.8182 | 92.1019 | 92.5757 | −462.4400 | −462.4561 | 232.8043 | 232.7093 |

3.2727 | −96.3648 | −96.2581 | −324.2330 | −324.1736 | 102.8055 | 102.6393 |

4.0909 | −98.0242 | −98.0037 | −362.9352 | −363.2549 | 129.0379 | 128.7251 |

10.5 | 118.0807 | 117.7244 | −427.2725 | −426.9482 | 194.8541 | 194.5199 |

5.4545 | −89.7257 | −89.5719 | −425.9476 | −425.6513 | 178.4385 | 178.1625 |

6.1364 | −78.2050 | −78.2665 | −453.8830 | −454.6911 | 204.1356 | 203.8762 |

7.7727 | −22.2021 | −22.2167 | −496.6415 | −496.2163 | 255.2363 | 255.6377 |

2.5909 | −92.4938 | −92.4305 | −292.9522 | −292.9801 | 83.4821 | 83.2599 |

0.6818 | −75.2204 | −75.2368 | −214.8197 | −214.7843 | 42.1795 | 42.1099 |

10.091 | 104.2757 | 103.9000 | −449.4231 | −449.4233 | 218.8197 | 217.9128 |

−1.3636 | −54.5664 | −54.4806 | −150.5720 | −150.6286 | 16.3184 | 16.1575 |

Points (r = 0.00825 m, . | Axial component (gauss) . | Azimuthal component (gauss) . | Radial component (gauss) . | |||
---|---|---|---|---|---|---|

z m, β = 4.957 rad) . | Analytical model . | FEA . | Analytical model . | FEA . | Analytical model . | FEA . |

−0.8181 | −59.8687 | −59.8653 | −165.7066 | −165.7111 | 21.6651 | 21.5873 |

−0.2727 | −65.3803 | −65.4131 | −182.2708 | −182.2264 | 28.0656 | 28.0159 |

1.7727 | −85.8743 | −85.8113 | −257.5002 | −257.4356 | 63.5014 | 63.4997 |

9.8182 | 92.1019 | 92.5757 | −462.4400 | −462.4561 | 232.8043 | 232.7093 |

3.2727 | −96.3648 | −96.2581 | −324.2330 | −324.1736 | 102.8055 | 102.6393 |

4.0909 | −98.0242 | −98.0037 | −362.9352 | −363.2549 | 129.0379 | 128.7251 |

10.5 | 118.0807 | 117.7244 | −427.2725 | −426.9482 | 194.8541 | 194.5199 |

5.4545 | −89.7257 | −89.5719 | −425.9476 | −425.6513 | 178.4385 | 178.1625 |

6.1364 | −78.2050 | −78.2665 | −453.8830 | −454.6911 | 204.1356 | 203.8762 |

7.7727 | −22.2021 | −22.2167 | −496.6415 | −496.2163 | 255.2363 | 255.6377 |

2.5909 | −92.4938 | −92.4305 | −292.9522 | −292.9801 | 83.4821 | 83.2599 |

0.6818 | −75.2204 | −75.2368 | −214.8197 | −214.7843 | 42.1795 | 42.1099 |

10.091 | 104.2757 | 103.9000 | −449.4231 | −449.4233 | 218.8197 | 217.9128 |

−1.3636 | −54.5664 | −54.4806 | −150.5720 | −150.6286 | 16.3184 | 16.1575 |

### B. Validation of the analytical model to compute the magnetic field generated by a cone with arbitrarily uniform magnetization

As mentioned in Sec. II, based on the analytical model developed in this current study and that of the author’s previous work,^{15} the magnetic field generated by a cone with arbitrarily uniform magnetization can be computed analytically. To validate this, a model of a cone with the same geometrical parameters as those of the one implemented in Sec. III A was built in EMS integrated with INVENTOR. The magnetization of the cone at this time had a slope angle Ω = π/4, and its magnitude is J = $2$ T. Moreover, the analytical model, which can be represented as the algebraic summation of those developed in this study and those developed in Ref. 15 [Eqs. (12), (15), and (18)], was coded in MATLAB R2016b. The results of the comparison are plotted in Fig. 7 (the plot was generated along the axial direction z = [−0.0015, 0.012] m with the radial distance r = 0.008 25 m and azimuthal angle β = 4.957 rad). Furthermore, the results of the 14 random points that are used in Table I were extracted and are listed in Table II. From Fig. 7 and Table II, it is concluded that the results of the analytical model are in excellent agreement with those of the FEM (the maximum, average, and minimum errors between the two models are 0.62%, 0.24%, and 0.05%, respectively, for the axial component; 0.14%, 0.06%, and 0.008%, respectively, for the azimuthal component; and 0.69%, 0.27%, and 0.036%, respectively, for the radial component).

Points (r = 0.00825 m, . | Axial component (gauss) . | Azimuthal component (gauss) . | Radial component (gauss) . | |||
---|---|---|---|---|---|---|

z m, β = 4.957 rad) . | Analytical model . | FEM . | Analytical model . | FEM . | Analytical model . | FEM . |

−0.8181 | 21.6102 | 21.7200 | −165.7066 | −165.7938 | −225.2348 | −224.7978 |

−0.2727 | 6.7828 | 6.7561 | −182.2708 | −182.2500 | −241.5753 | −240.9652 |

1.7727 | −82.2723 | −81.7691 | −257.5002 | −257.4637 | −290.7283 | −290.3790 |

9.8182 | −391.1025 | −390.7076 | −462.4400 | −462.6739 | 611.9609 | 611.7414 |

3.2727 | −186.0311 | −185.5419 | −324.2330 | −324.0046 | −294.7771 | −294.1961 |

4.0909 | −255.9562 | −255.6254 | −362.9352 | −363.1533 | −275.4525 | −274.8153 |

10.5 | −244.9007 | −244.1744 | −427.2725 | −427.4012 | 681.2204 | 679.7142 |

5.4545 | −386.3896 | −385.4932 | −425.9476 | −425.3470 | −191.9615 | −191.4261 |

6.1364 | −452.0258 | −451.1190 | −453.8830 | −454.3709 | −118.8286 | −118.0120 |

7.7727 | −562.6418 | −562.2303 | −496.6415 | −496.1794 | 163.0489 | 163.9408 |

2.5909 | −134.7311 | −134.5163 | −292.9522 | −292.9299 | −298.0908 | −297.6009 |

0.68182 | −27.7009 | −27.7197 | −214.8204 | −214.9406 | −268.0691 | −267.5137 |

10.091 | −334.6861 | −332.9267 | −449.4231 | −449.8453 | 648.2057 | 645.7909 |

−1.3636 | 33.3573 | 33.3737 | −150.5720 | −150.7034 | −208.7058 | −208.2549 |

Points (r = 0.00825 m, . | Axial component (gauss) . | Azimuthal component (gauss) . | Radial component (gauss) . | |||
---|---|---|---|---|---|---|

z m, β = 4.957 rad) . | Analytical model . | FEM . | Analytical model . | FEM . | Analytical model . | FEM . |

−0.8181 | 21.6102 | 21.7200 | −165.7066 | −165.7938 | −225.2348 | −224.7978 |

−0.2727 | 6.7828 | 6.7561 | −182.2708 | −182.2500 | −241.5753 | −240.9652 |

1.7727 | −82.2723 | −81.7691 | −257.5002 | −257.4637 | −290.7283 | −290.3790 |

9.8182 | −391.1025 | −390.7076 | −462.4400 | −462.6739 | 611.9609 | 611.7414 |

3.2727 | −186.0311 | −185.5419 | −324.2330 | −324.0046 | −294.7771 | −294.1961 |

4.0909 | −255.9562 | −255.6254 | −362.9352 | −363.1533 | −275.4525 | −274.8153 |

10.5 | −244.9007 | −244.1744 | −427.2725 | −427.4012 | 681.2204 | 679.7142 |

5.4545 | −386.3896 | −385.4932 | −425.9476 | −425.3470 | −191.9615 | −191.4261 |

6.1364 | −452.0258 | −451.1190 | −453.8830 | −454.3709 | −118.8286 | −118.0120 |

7.7727 | −562.6418 | −562.2303 | −496.6415 | −496.1794 | 163.0489 | 163.9408 |

2.5909 | −134.7311 | −134.5163 | −292.9522 | −292.9299 | −298.0908 | −297.6009 |

0.68182 | −27.7009 | −27.7197 | −214.8204 | −214.9406 | −268.0691 | −267.5137 |

10.091 | −334.6861 | −332.9267 | −449.4231 | −449.8453 | 648.2057 | 645.7909 |

−1.3636 | 33.3573 | 33.3737 | −150.5720 | −150.7034 | −208.7058 | −208.2549 |

In addition, it is worth mentioning that simulated on a personal computer (Intel Core i7-6700 CPU, 3.40 GHz), using MATLAB R2016b, it took an average of less than 0.5 (ms) to execute each analytical expression of the magnetic field components at a single point (40 000 random points were implemented in this simulation) created by a cone with diametrical magnetization. Moreover, it took an average of less than 1 ms to execute each analytical expression of the magnetic field components generated by a cone with arbitrarily uniform magnetization. In contrast, the computation costs of simulation of the FEM on the same personal computer for both cases were 779 s.

## IV. MAGNETIC FIELD DISTRIBUTION ANALYSIS WITH DIFFERENT MAGNETIZATION DIRECTIONS

In this study, the magnetic field distribution, which is produced by an annular cone with arbitrarily uniform magnetization, is analyzed using the developed analytical expressions. The cone is assumed to be applied in magnetic resonance imaging for a human head.^{15} Figure 8 depicts both the schematic representation of this application and the geometrical parameters of the cone.

The cone has the height h (m); the inner vertex angle is 2φ_{2} (rad); the outer vertex angle is 2φ_{1} (rad); the inner and outer base radii are R_{i} (m) and R_{O} (m), respectively; the slope angle of its magnetization is Ω (rad) with the magnitude J (T); the human head has the average radius of R_{h} (m); the considered plane of the human head is the center plane Δ with a radius of R_{h} (m); this plane lies inside the cone, with a distance a (m) from the cone’s base surface. The values of these parameters are listed in Table III.

Parameters . | Values . |
---|---|

h (m) | 0.3 |

φ_{1} (rad) | 0.6174 |

φ_{2} (rad) | 0.4636 |

R_{i} (m) | 0.15 |

R_{O} (m) | 0.213 |

Ω (rad) | [0, π/6, π/3, π/2] |

J (T) | 1 |

R_{h} (m) | 0.094 |

a (m) | 0.02 |

Parameters . | Values . |
---|---|

h (m) | 0.3 |

φ_{1} (rad) | 0.6174 |

φ_{2} (rad) | 0.4636 |

R_{i} (m) | 0.15 |

R_{O} (m) | 0.213 |

Ω (rad) | [0, π/6, π/3, π/2] |

J (T) | 1 |

R_{h} (m) | 0.094 |

a (m) | 0.02 |

Due to the superposition principle, the magnetic field $H\u2192N(annular)$ created by the annular cone at any given point N in space can be computed using the following equation:

where $H\u2192N2\phi 1$ is the magnetic field created by a solid cone that has the same geometrical parameters as those of the outer cone (h, 2φ_{1}) and the same magnetization as that of the annular cone. Similarly, $H\u2192N2\phi 2$ is the magnetic field produced by a solid cone with the same geometrical parameters of the inner cone (h, 2φ_{2}) and the same magnetization as that of the annular cone.

The plots of the magnetic field distribution of the axial, azimuthal, and radial components with varying slope angle Ω are depicted in Figs. 9, 10, and 11, respectively. Figures 9(a), 10(a), and 11(a) show the three-dimensional (3D) plot of the magnetic field distribution of these components. On the other hand, Figs. 9(b), 10(b), and 11(b) illustrate the distribution along a two-dimensional (2D) plot along the radial distance r = [0, 0.094] m and the azimuthal angle β = π/6; Figs. 9(c), 10(c), and 11(c) depict the distribution along a two-dimensional plot along the azimuthal angle β = [0, 2π] rad and the radial distance r = 0.015 m.

For the axial component, Figs. 9(a) and 9(c) show that the distribution of the axial component is a cosine-like wave along the azimuthal angle β when the slope angle Ω is not equal to π/2. If Ω = π/2, the value of the magnetic flux density is constant along the azimuthal direction (Baxial = −500 G). Figure 9(b) demonstrates that the magnitude of the axial component decreases with the increase of the slope angle, and it is more uniformly distributed when Ω = π/2.^{15} For the azimuthal component, Figs. 10(a) and 10(c) show that the distribution of the azimuthal component is a sine-like wave along β when Ω is not equal to π/2, and the component is equal to zero when Ω = π/2.^{15} Figure 10(b) illustrates that this component tends to be more uniformly distributed along r when Ω increases. For the radial component, Figs. 11(a) and 11(c) show that the distribution of this component along β is a cosine-like wave and in the opposite direction from the axial component when Ω is not equal to π/2. When Ω = π/2, this component is a constant long β (Bradial = −75.29 G). Figure 11(b) demonstrates that along r, the distribution of the radial component becomes less uniformly distributed with the increase in Ω.

## V. CONCLUSION

In this research, fast-computed analytical expressions of the magnetic field generated by a cone with arbitrarily uniform magnetization are developed. First, the analytical model of a magnetic field generated by a cone with diametrical magnetization is developed based on the charge model, algebraic and geometrical analyses. Using this model, combining with the analytical model developed by the author in the previous publication for a cone with axial magnetization, the magnetic field created by a cone with arbitrarily uniform magnetization can be computed analytically due to the principle of superposition. The expressions were validated with a Finite Element Model (FEM). Moreover, as is demonstrated, the analytical model is much faster in terms of the computational cost than the FEM. Using the developed expressions, the magnetic field distribution of a cone with arbitrarily uniform magnetization, which is assumed to be applied in magnetic resonance imaging, is analyzed. As shown in this study, the distribution of the magnetic field produced by a cone with arbitrarily uniform magnetization is sinusoidal, except for the case where the slope angle is equal to π/2 (axial magnetization). Furthermore, with the changes in the slope angle Ω, the magnitude and uniformity of the magnetic field vary accordingly.

## ACKNOWLEDGMENTS

The author is grateful to the EMWORKS Company for providing the license for the EMS software to conduct the finite element analyses. The author would like to thank the School of Mechanical Engineering, University of Adelaide, Australia, for providing the equipment, facilities, and assistance for this research. The author acknowledges the financial support from Thuyloi University according to the Decision No. 989/QD-DHTL. The author is also grateful to Dr. Alison-Jane Hunter for her time helping to edit the English.

The data that support the findings of this study are available from the corresponding author upon reasonable request.