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Internal inductance: Difference between revisions
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On the other hand, the energy contained in the magnetic field produced by the loop is | On the other hand, the energy contained in the magnetic field produced by the loop is | ||
:<math>W = \int{\frac{B^2}{2\mu_0} d\vec r}</math> | :<math>W = \int{\frac{B^2}{2\mu_0} d\vec r}</math> | ||
It can be shown that<ref>P.M. Bellan, ''Fundamentals of Plasma Physics'', Cambridge University Press (2006) ISBN 0521821169</ref><ref>[[:Wikipedia:Inductance]]</ref> | It can be shown that<ref>P.M. Bellan, ''Fundamentals of Plasma Physics'', Cambridge University Press (2006) {{ISBN|0521821169}}</ref><ref>[[:Wikipedia:Inductance]]</ref> | ||
:<math>W = \frac12 L I^2</math> | :<math>W = \frac12 L I^2</math> | ||
== Internal inductance of a plasma == | == Internal inductance of a plasma == | ||
The ''internal'' inductance is defined as the part of the inductance obtained by integrating over the plasma volume ''P'' <ref name="Freidberg">J.P. Freidberg, ''Plasma physics and fusion energy'', Cambridge University Press (2007) ISBN 0521851076</ref>: | The ''internal'' inductance is defined as the part of the inductance obtained by integrating over the plasma volume ''P'' <ref name="Freidberg">J.P. Freidberg, ''Plasma physics and fusion energy'', Cambridge University Press (2007) {{ISBN|0521851076}}</ref>: | ||
:<math>\frac12 L_i I^2 = \int_P{\frac{B^2}{2\mu_0} d\vec r}</math> | :<math>\frac12 L_i I^2 = \int_P{\frac{B^2}{2\mu_0} d\vec r}</math> | ||
Its complement is the external inductance (''L = L<sub>i</sub> + L<sub>e</sub>''). | Its complement is the external inductance (''L = L<sub>i</sub> + L<sub>e</sub>''). | ||
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In a [[tokamak]], the field produced by the plasma current is the ''poloidal'' magnetic field ''B<sub>θ<sub>'', so only this field component enters the definition. | In a [[tokamak]], the field produced by the plasma current is the ''poloidal'' magnetic field ''B<sub>θ<sub>'', so only this field component enters the definition. | ||
In this context, it is common to use the ''normalized'' internal inductance<ref>K. Miyamoto, ''Plasma Physics and Controlled Nuclear Fusion'', Springer-Verlag (2005) ISBN 3540242171</ref> | In this context, it is common to use the ''normalized'' internal inductance<ref>K. Miyamoto, ''Plasma Physics and Controlled Nuclear Fusion'', Springer-Verlag (2005) {{ISBN|3540242171}}</ref> | ||
:<math>l_i = \frac{\left \langle B_\theta^2 \right \rangle_P}{B_\theta^2(a)} = \frac{2 \pi \int_0^a{B_\theta^2(\rho) \rho d\rho}}{\pi a^2 B_\theta^2(a)}</math> | :<math>l_i = \frac{\left \langle B_\theta^2 \right \rangle_P}{B_\theta^2(a)} = \frac{2 \pi \int_0^a{B_\theta^2(\rho) \rho d\rho}}{\pi a^2 B_\theta^2(a)}</math> | ||
(for circular cross section plasmas with [[Toroidal coordinates|minor radius]] ''a''), where angular brackets signify taking a mean value. | (for circular cross section plasmas with [[Toroidal coordinates|minor radius]] ''a''), where angular brackets signify taking a mean value. | ||