self-force on time independent current distribution

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A current distribution $\mathbf{J}(\mathbf{r})\,$ in a magnetic field $\mathbf{B}\,$ experiences a force

$\mathbf{F} = \int\!d^3r\, \mathbf{J}\times\mathbf{B}\,$ .

Supposing that there are no external sources for $\mathbf{B}\,$ , so that it is entirely due to $\mathbf{J}\,$ , then the Biot-Savart law gives

$\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int\!d^3r' \mathbf{J}(\mathbf{r}') \times \frac{ (\mathbf{r}-\mathbf{r}') }{|\mathbf{r}-\mathbf{r}'|^3 }\,$ .

Thus

$\mathbf{F}\,$	$= \int\!d^3r\, \mathbf{J}\times\mathbf{B}\,$ ,
	$= \frac{\mu_0}{4\pi} \int\!d^3r d^3r'\, \mathbf{J}(\mathbf{r}) \times \left( \mathbf{J}(\mathbf{r}') \times \frac{ (\mathbf{r}-\mathbf{r}') }{\|\mathbf{r}-\mathbf{r}'\|^3 } \right)\,$ ,
	$= \frac{\mu_0}{4\pi} \int\!d^3r d^3r'\, \left[ \mathbf{J}(\mathbf{r}') \left( \mathbf{J}(\mathbf{r}) \cdot \frac{ (\mathbf{r}-\mathbf{r}') }{\|\mathbf{r}-\mathbf{r}'\|^3 } \right) - \frac{ (\mathbf{r}-\mathbf{r}') }{\|\mathbf{r}-\mathbf{r}'\|^3 } \left( \mathbf{J}(\mathbf{r}) \cdot \mathbf{J}(\mathbf{r}') \right) \right] \,$ , (BAC-CAB identity)
	$= \frac{\mu_0}{4\pi} \int\!d^3r d^3r'\, \mathbf{J}(\mathbf{r}') \left( \mathbf{J}(\mathbf{r}) \cdot \frac{ (\mathbf{r}-\mathbf{r}') }{\|\mathbf{r}-\mathbf{r}'\|^3 } \right) \,$ , (as the integrand is antisymmetric in $\mathbf{r}\,$ and $\mathbf{r}'\,$ )
	$= -\frac{\mu_0}{8\pi} \int\!d^3r d^3r'\, \mathbf{J}(\mathbf{r}') \left( \mathbf{J}(\mathbf{r}) \cdot \boldsymbol{\nabla}_{r} \frac{ 1 }{\|\mathbf{r}-\mathbf{r}'\|^2 } \right) \,$ ,
	$= -\frac{\mu_0}{8\pi} \int\!d^3r'\, \mathbf{J}(\mathbf{r}') \int\!d^3r\, \mathbf{J}(\mathbf{r}) \cdot \boldsymbol{\nabla}_{r} \frac{ 1 }{\|\mathbf{r}-\mathbf{r}'\|^2 } \,$ ,
	$= \frac{\mu_0}{8\pi} \int\!d^3r'\, \mathbf{J}(\mathbf{r}') \int\!d^3r\, \boldsymbol{\nabla}_r \left(\frac{ \mathbf{J}(\mathbf{r}) }{\|\mathbf{r}-\mathbf{r}'\|^2 } \right)\,$ , (since $\boldsymbol{\nabla}\cdot\mathbf{J} = 0\,$ by the continuity equation)
	$= \frac{\mu_0}{8\pi} \int\!d^3r'\, \mathbf{J}(\mathbf{r}') \int\!dS\, \hat\mathbf{n} \cdot \left(\frac{ \mathbf{J}(\mathbf{r}) }{\|\mathbf{r}-\mathbf{r}'\|^2 } \right)\,$ , (by the Divergence theorem)
	$= 0\,$ ,

since, by assumption, the region of integration encloses all currents. Thus a static current distribution exerts no net force on itself.

Retrieved from "http://www.physics.thetangentbundle.net/wiki/Classical_electrodynamics/self-force_on_time_independent_current_distribution"

$\mathbf{F}\,$	$= \int\!d^3r\, \mathbf{J}\times\mathbf{B}\,$ ,
	$= \frac{\mu_0}{4\pi} \int\!d^3r d^3r'\, \mathbf{J}(\mathbf{r}) \times \left( \mathbf{J}(\mathbf{r}') \times \frac{ (\mathbf{r}-\mathbf{r}') }{\|\mathbf{r}-\mathbf{r}'\|^3 } \right)\,$ ,
	$= \frac{\mu_0}{4\pi} \int\!d^3r d^3r'\, \left[ \mathbf{J}(\mathbf{r}') \left( \mathbf{J}(\mathbf{r}) \cdot \frac{ (\mathbf{r}-\mathbf{r}') }{\|\mathbf{r}-\mathbf{r}'\|^3 } \right) - \frac{ (\mathbf{r}-\mathbf{r}') }{\|\mathbf{r}-\mathbf{r}'\|^3 } \left( \mathbf{J}(\mathbf{r}) \cdot \mathbf{J}(\mathbf{r}') \right) \right] \,$ , (BAC-CAB identity)
	$= \frac{\mu_0}{4\pi} \int\!d^3r d^3r'\, \mathbf{J}(\mathbf{r}') \left( \mathbf{J}(\mathbf{r}) \cdot \frac{ (\mathbf{r}-\mathbf{r}') }{\|\mathbf{r}-\mathbf{r}'\|^3 } \right) \,$ , (as the integrand is antisymmetric in $\mathbf{r}\,$ and $\mathbf{r}'\,$ )
	$= -\frac{\mu_0}{8\pi} \int\!d^3r d^3r'\, \mathbf{J}(\mathbf{r}') \left( \mathbf{J}(\mathbf{r}) \cdot \boldsymbol{\nabla}_{r} \frac{ 1 }{\|\mathbf{r}-\mathbf{r}'\|^2 } \right) \,$ ,
	$= -\frac{\mu_0}{8\pi} \int\!d^3r'\, \mathbf{J}(\mathbf{r}') \int\!d^3r\, \mathbf{J}(\mathbf{r}) \cdot \boldsymbol{\nabla}_{r} \frac{ 1 }{\|\mathbf{r}-\mathbf{r}'\|^2 } \,$ ,
	$= \frac{\mu_0}{8\pi} \int\!d^3r'\, \mathbf{J}(\mathbf{r}') \int\!d^3r\, \boldsymbol{\nabla}_r \left(\frac{ \mathbf{J}(\mathbf{r}) }{\|\mathbf{r}-\mathbf{r}'\|^2 } \right)\,$ , (since $\boldsymbol{\nabla}\cdot\mathbf{J} = 0\,$ by the continuity equation)
	$= \frac{\mu_0}{8\pi} \int\!d^3r'\, \mathbf{J}(\mathbf{r}') \int\!dS\, \hat\mathbf{n} \cdot \left(\frac{ \mathbf{J}(\mathbf{r}) }{\|\mathbf{r}-\mathbf{r}'\|^2 } \right)\,$ , (by the Divergence theorem)
	$= 0\,$ ,

self-force on time independent current distribution

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