If two or added magnets are baby abundant or abundantly abroad that their appearance and admeasurement is not important again both magnets can be modeled as getting alluring dipoles accepting a alluring moments m1 and m2.
field from a complete dipole, both electric or magnetic.
The alluring acreage of a alluring dipole in agent characters is:
\mathbf{B}(\mathbf{m}, \mathbf{r}) = \frac {\mu_0} {4\pi r^3} \left(3(\mathbf{m}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{m}\right) + \frac{2\mu_0}{3}\mathbf{m}\delta^3(\mathbf{r})
where
B is the field
r is the agent from the position of the dipole to the position area the acreage is getting measured
r is the complete amount of r: the ambit from the dipole
\hat{\mathbf{r}} = \mathbf{r}/r is the assemblage agent alongside to r;
m is the (vector) dipole moment
μ0 is the permeability of chargeless space
δ3 is the three-dimensional basin function.note 2
This is absolutely the acreage of a point dipole, absolutely the dipole appellation in the multipole amplification of an approximate field, and about the acreage of any dipole-like agreement at ample distances.
Frames of advertence for artful the armament amid two dipoles
If the alike arrangement is confused to centermost it on m1 and rotated such that the z-axis credibility in the administration of m1 again the antecedent blueprint simplifies to6
B_{z}(\mathbf{r}) = \frac{\mu_0}{4 \pi} m_1 \left(\frac{3\cos^2\theta-1}{r^3}\right)
B_{x}(\mathbf{r}) = \frac{\mu_0}{4 \pi} m_1 \left(\frac{3\cos\theta\sin\theta}{r^3}\right) ,
where the variables r and θ are abstinent in a anatomy of advertence with agent in m1 and aggressive such that m1 is at the agent pointing in the z-direction. This anatomy is alleged Local coordinates and is apparent in the Figure on the right.
The force of one alluring dipole on addition is bent by application the alluring acreage of the aboriginal dipole accustomed aloft and free the force due to the alluring acreage on the additional dipole application the force blueprint accustomed above. Application agent notation, the force of a alluring dipole m1 on the alluring dipole m2 is:
\mathbf{F}(\mathbf{r}, \mathbf{m}_1, \mathbf{m}_2) = \frac{3 \mu_0}{4 \pi r^5}\left(\mathbf{m}_1\cdot\mathbf{r})\mathbf{m}_2 + (\mathbf{m}_2\cdot\mathbf{r})\mathbf{m}_1 + (\mathbf{m}_1\cdot\mathbf{m}_2)\mathbf{r} - \frac{5(\mathbf{m}_1\cdot\mathbf{r})(\mathbf{m}_2\cdot\mathbf{r})}{r^2}\mathbf{r}\right
where r is the distance-vector from dipole moment m1 to dipole moment m2, with r=||r||. The force acting on m1 is in adverse direction. As an archetype the alluring acreage for two magnets pointing in the z-direction and accumbent on the z-axis and afar by the ambit z is:
\mathbf{F}(z,m_1, m_2) = -\frac{3 \mu_0 m_1 m_2}{2 \pi z^4} , z-direction.
The final formulas are apparent next. They are bidding in the all-around alike system,
F_r(\mathbf{r}, \alpha, \beta) = - \frac{3 \mu_0}{4 \pi}\frac{m_2 m_1}{r^4}\left2\cos(\phi - \alpha)\cos(\phi - \beta)- \sin(\phi - \alpha)\sin(\phi - \beta)\right
F_{\phi}(\mathbf{r}, \alpha, \beta) =- \frac{3 \mu_0}{4 \pi}\frac{m_2 m_1}{r^4}\sin(2\phi - \alpha - \beta)
field from a complete dipole, both electric or magnetic.
The alluring acreage of a alluring dipole in agent characters is:
\mathbf{B}(\mathbf{m}, \mathbf{r}) = \frac {\mu_0} {4\pi r^3} \left(3(\mathbf{m}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{m}\right) + \frac{2\mu_0}{3}\mathbf{m}\delta^3(\mathbf{r})
where
B is the field
r is the agent from the position of the dipole to the position area the acreage is getting measured
r is the complete amount of r: the ambit from the dipole
\hat{\mathbf{r}} = \mathbf{r}/r is the assemblage agent alongside to r;
m is the (vector) dipole moment
μ0 is the permeability of chargeless space
δ3 is the three-dimensional basin function.note 2
This is absolutely the acreage of a point dipole, absolutely the dipole appellation in the multipole amplification of an approximate field, and about the acreage of any dipole-like agreement at ample distances.
Frames of advertence for artful the armament amid two dipoles
If the alike arrangement is confused to centermost it on m1 and rotated such that the z-axis credibility in the administration of m1 again the antecedent blueprint simplifies to6
B_{z}(\mathbf{r}) = \frac{\mu_0}{4 \pi} m_1 \left(\frac{3\cos^2\theta-1}{r^3}\right)
B_{x}(\mathbf{r}) = \frac{\mu_0}{4 \pi} m_1 \left(\frac{3\cos\theta\sin\theta}{r^3}\right) ,
where the variables r and θ are abstinent in a anatomy of advertence with agent in m1 and aggressive such that m1 is at the agent pointing in the z-direction. This anatomy is alleged Local coordinates and is apparent in the Figure on the right.
The force of one alluring dipole on addition is bent by application the alluring acreage of the aboriginal dipole accustomed aloft and free the force due to the alluring acreage on the additional dipole application the force blueprint accustomed above. Application agent notation, the force of a alluring dipole m1 on the alluring dipole m2 is:
\mathbf{F}(\mathbf{r}, \mathbf{m}_1, \mathbf{m}_2) = \frac{3 \mu_0}{4 \pi r^5}\left(\mathbf{m}_1\cdot\mathbf{r})\mathbf{m}_2 + (\mathbf{m}_2\cdot\mathbf{r})\mathbf{m}_1 + (\mathbf{m}_1\cdot\mathbf{m}_2)\mathbf{r} - \frac{5(\mathbf{m}_1\cdot\mathbf{r})(\mathbf{m}_2\cdot\mathbf{r})}{r^2}\mathbf{r}\right
where r is the distance-vector from dipole moment m1 to dipole moment m2, with r=||r||. The force acting on m1 is in adverse direction. As an archetype the alluring acreage for two magnets pointing in the z-direction and accumbent on the z-axis and afar by the ambit z is:
\mathbf{F}(z,m_1, m_2) = -\frac{3 \mu_0 m_1 m_2}{2 \pi z^4} , z-direction.
The final formulas are apparent next. They are bidding in the all-around alike system,
F_r(\mathbf{r}, \alpha, \beta) = - \frac{3 \mu_0}{4 \pi}\frac{m_2 m_1}{r^4}\left2\cos(\phi - \alpha)\cos(\phi - \beta)- \sin(\phi - \alpha)\sin(\phi - \beta)\right
F_{\phi}(\mathbf{r}, \alpha, \beta) =- \frac{3 \mu_0}{4 \pi}\frac{m_2 m_1}{r^4}\sin(2\phi - \alpha - \beta)
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