- #1

phos19

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- Homework Statement
- Let $$ \vec{B} =\dfrac{1}{4 \pi} \dfrac{-3}{r^4} ( 3\cos^2{\theta} - 1) \hat{u_r} + \dfrac{1}{4 \pi} \dfrac{1}{r^4} ( - 6 \cos{\theta} \sin{\theta} ) \hat{u_{\theta}} $$ (spherical unit vectors)

Find ##\vec{A}## such that ## \vec{B} = \nabla \times \vec{A}##

- Relevant Equations
- (The ##\vec{B}## is divergenceless !)

I've tried writing the curl A (in spherical coord.) and equating the components, but I end up with something that is beyond me:

\begin{equation}

{\displaystyle {\begin{aligned}{B_r = \dfrac{1}{4 \pi} \dfrac{-3}{r^4} ( 3\cos^2{\theta} - 1) =\frac {1}{r\sin \theta }}\left({\frac {\partial }{\partial \theta }}\left(A_{\varphi }\sin \theta \right)-{\frac {\partial A_{\theta }}{\partial \varphi }}\right)&\\B_{\theta}= \dfrac{1}{4 \pi} \dfrac{1}{r^4} ( - 6 \cos{\theta} \sin{\theta} ) ={}+{\frac {1}{r}}\left({\frac {1}{\sin \theta }}{\frac {\partial A_{r}}{\partial \varphi }}-{\frac {\partial }{\partial r}}\left(rA_{\varphi }\right)\right)&\\B_{\varphi}= 0={}+{\frac {1}{r}}\left({\frac {\partial }{\partial r}}\left(rA_{\theta }\right)-{\frac {\partial A_{r}}{\partial \theta }}\right)&\end{aligned}}}

\end{equation}

Is there a "trick" to solve this , or maybe some vector identity to simplify the problem ?

Any hints are greatly appreciated , thanks!

\begin{equation}

{\displaystyle {\begin{aligned}{B_r = \dfrac{1}{4 \pi} \dfrac{-3}{r^4} ( 3\cos^2{\theta} - 1) =\frac {1}{r\sin \theta }}\left({\frac {\partial }{\partial \theta }}\left(A_{\varphi }\sin \theta \right)-{\frac {\partial A_{\theta }}{\partial \varphi }}\right)&\\B_{\theta}= \dfrac{1}{4 \pi} \dfrac{1}{r^4} ( - 6 \cos{\theta} \sin{\theta} ) ={}+{\frac {1}{r}}\left({\frac {1}{\sin \theta }}{\frac {\partial A_{r}}{\partial \varphi }}-{\frac {\partial }{\partial r}}\left(rA_{\varphi }\right)\right)&\\B_{\varphi}= 0={}+{\frac {1}{r}}\left({\frac {\partial }{\partial r}}\left(rA_{\theta }\right)-{\frac {\partial A_{r}}{\partial \theta }}\right)&\end{aligned}}}

\end{equation}

Is there a "trick" to solve this , or maybe some vector identity to simplify the problem ?

Any hints are greatly appreciated , thanks!

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