Hello, I'm attempting to calculate the magnetic field [itex]\mathbf{B}[/itex] of a coil with current density, [itex]\mathbf{J}[/itex], whose volume is a portion of a sphere. Here is a diagram of the cross-section of the coil:
(The wire ridges are aesthetic and aren't considered in the calculations. I'm assuming a solid volume.)
I'm trying to apply the Biot-Savart law:
[tex]\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi}\int_{r_a}^{r_b}\int_{\theta_a}^{\theta_b}\int_{-\pi}^\pi\ \frac{\mathbf{J} \times (\mathbf{r}-\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|^3} \, r'^2\,\sin \theta \,d\phi\,d\theta\,dr'[/tex]
As can be seen from this triple integral, I'm using spherical components to simplify things.
Because of the spherical nature of this problem, I figure it's best to use a spherical coordinate frame to describe [itex]\mathbf{r}'[/itex] and [itex]\mathbf{J}[/itex]. However, I confuse myself when trying to do this. Am I correct in thinking:
[tex]\mathbf{r}' = \begin{pmatrix} r' \\ 0 \\ 0 \end{pmatrix} \quad \mathrm{and} \quad \mathbf{r} = \begin{pmatrix} r \\ 0 \\ 0 \end{pmatrix}[/tex]?
Additionally, I want my current vector to be tangential to the axis of symmetry of the coil and perpendicular to the cross-sectional plane in the diagram and am unsure if this would be a correct description:
[tex]\mathbf{J} = J\begin{pmatrix} 1 \\ \pi/2 \\ \phi \end{pmatrix}[/tex]
Also for reference, I'm confident that these vector quantities are as follows in the Cartesian frame:
[tex]\mathbf{r}' = \begin{pmatrix} r \sin\theta\cos\phi \\ r \sin\theta\sin\phi \\ r\cos\theta \end{pmatrix} [/tex]
[tex]\mathbf{J} = J \begin{pmatrix} -\sin\phi \\ \cos \phi \\ 0 \end{pmatrix} [/tex]
Just to recap, can someone please advise me about my vectors and their description in a spherical frame? Also, do you think a spherical frame is advantageous? Thanks!
(The wire ridges are aesthetic and aren't considered in the calculations. I'm assuming a solid volume.)
I'm trying to apply the Biot-Savart law:
[tex]\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi}\int_{r_a}^{r_b}\int_{\theta_a}^{\theta_b}\int_{-\pi}^\pi\ \frac{\mathbf{J} \times (\mathbf{r}-\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|^3} \, r'^2\,\sin \theta \,d\phi\,d\theta\,dr'[/tex]
As can be seen from this triple integral, I'm using spherical components to simplify things.
Because of the spherical nature of this problem, I figure it's best to use a spherical coordinate frame to describe [itex]\mathbf{r}'[/itex] and [itex]\mathbf{J}[/itex]. However, I confuse myself when trying to do this. Am I correct in thinking:
[tex]\mathbf{r}' = \begin{pmatrix} r' \\ 0 \\ 0 \end{pmatrix} \quad \mathrm{and} \quad \mathbf{r} = \begin{pmatrix} r \\ 0 \\ 0 \end{pmatrix}[/tex]?
Additionally, I want my current vector to be tangential to the axis of symmetry of the coil and perpendicular to the cross-sectional plane in the diagram and am unsure if this would be a correct description:
[tex]\mathbf{J} = J\begin{pmatrix} 1 \\ \pi/2 \\ \phi \end{pmatrix}[/tex]
Also for reference, I'm confident that these vector quantities are as follows in the Cartesian frame:
[tex]\mathbf{r}' = \begin{pmatrix} r \sin\theta\cos\phi \\ r \sin\theta\sin\phi \\ r\cos\theta \end{pmatrix} [/tex]
[tex]\mathbf{J} = J \begin{pmatrix} -\sin\phi \\ \cos \phi \\ 0 \end{pmatrix} [/tex]
Just to recap, can someone please advise me about my vectors and their description in a spherical frame? Also, do you think a spherical frame is advantageous? Thanks!
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