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joshchab
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Hello, I'm attempting to calculate the magnetic field [itex]\mathbf{B}[/itex] outside of a coil with current density, [itex]\mathbf{J}[/itex], and whose volume is a section 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. Also, the origin of my coordinate frame is coincident with the spherical center of the coil - where the z and y axes cross in this diagram. ...Also, I've just realized that the diagram doesn't exactly show the volume produced by my integration bound, but that doesn't affect my question.)
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. Also, the origin of my coordinate frame is coincident with the spherical center of the coil - where the z and y axes cross in this diagram. ...Also, I've just realized that the diagram doesn't exactly show the volume produced by my integration bound, but that doesn't affect my question.)
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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