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Current sheet
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<h2 id="magnetic-field-of-an-infinite-current-sheet">Magnetic field of an infinite current sheet</h2> <p>An infinite current sheet can be modelled as an infinite number of parallel wires all carrying the same current. Assuming each wire carries current <i>I</i>, and there are <i>N</i> wires per unit length, the magnetic field can be derived using <a href="/facts/Amp%25C3%25A8re%2527s_law/B6WVCpc6">Ampère's law</a>: </p><p> ∮ R B ⋅ d l = μ 0 I enc {\displaystyle \oint _{R}\mathbf {B} \cdot \mathbf {dl} =\mu _{0}I_{\text{enc}}} ∮ R B cos ( θ ) d l = μ 0 I enc {\displaystyle \oint _{R}B\cos(\theta )\,dl=\mu _{0}I_{\text{enc}}} </p><p>R is a rectangular loop surrounding the current sheet, perpendicular to the plane and perpendicular to the wires. In the two sides perpendicular to the sheet, B ⋅ d s = 0 {\displaystyle \mathbf {B} \cdot d\mathbf {s} =0} since cos ( 90 ∘ ) = 0 {\displaystyle \cos(90^{\circ })=0} . In the other two sides, cos ( 0 ) = 1 {\displaystyle \cos(0)=1} , so if S is one parallel side of the rectangular loop of dimensions L × W, the integral simplifies to: 2 ∫ S B d s = μ 0 I enc {\displaystyle 2\int _{S}Bds=\mu _{0}I_{\text{enc}}} Since <i>B</i> is constant due to the chosen path, it can be pulled out of the integral: 2 B ∫ S d s = μ 0 I enc {\displaystyle 2B\int _{S}ds=\mu _{0}I_{\text{enc}}} The integral is evaluated: 2 B L = μ 0 I enc {\displaystyle 2BL=\mu _{0}I_{\text{enc}}} Solving for <i>B</i>, plugging in for <i>I</i>enc (total current enclosed in path <i>R</i>) as <i>I</i>×<i>N</i>×<i>L</i>, and simplifying: B = μ 0 I enc 2 L = μ 0 I N L 2 L = μ 0 I N 2 {\displaystyle {\begin{aligned}B&={\frac {\mu _{0}I_{\text{enc}}}{2L}}={\frac {\mu _{0}INL}{2L}}\\[1ex]&={\frac {\mu _{0}IN}{2}}\end{aligned}}} Notably, the magnetic field strength of an infinite current sheet does not depend on the distance from it. </p><p>The direction of B can be found via the <a href="/facts/Right-hand_rule/LJe0Qqh4">right-hand rule</a>. </p> <h2 id="harris-sheet">Harris sheet</h2> <p>A well-known one-dimensional current sheet equilibrium is the Harris sheet, which is a stationary solution to the Maxwell–Vlasov system.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> The magnetic field profile of a Harris sheet along y = 0 {\displaystyle y=0} is given by B ( y ) = B 0 tanh ( y δ ) x ^ , {\displaystyle \mathbf {B} (y)=B_{0}\tanh \left({\frac {y}{\delta }}\right)\mathbf {\hat {x}} ,} where B 0 {\displaystyle B_{0}} is the asymptotic magnetic field strength and δ {\displaystyle \delta } provides the thickness of the current sheet. The current density is given by J ( y ) = − B 0 μ 0 δ sech 2 ( y δ ) z ^ . {\displaystyle \mathbf {J} (y)=-{\frac {B_{0}}{\mu _{0}\delta }}\operatorname {sech} ^{2}\left({\frac {y}{\delta }}\right)\mathbf {\hat {z}} .} The plasma pressure is given by p ( y ) = B 0 2 2 μ 0 sech 2 ( y δ ) + p 0 , {\displaystyle p(y)={\frac {B_{0}^{2}}{2\mu _{0}}}\operatorname {sech} ^{2}\left({\frac {y}{\delta }}\right)+p_{0},} where p 0 {\displaystyle p_{0}} is the asymptotic pressure. </p> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Biskamp, Dieter (1997) Nonlinear Magnetohydrodynamics Cambridge University Press, Cambridge, England, page 130, ISBN 0-521-59918-0 <a href="https://books.google.com/books?id=OzFNhaVKA48C&pg=PA130" target="_blank">https://books.google.com/books?id=OzFNhaVKA48C&pg=PA130</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Biskamp, Dieter (May 1986) "Magnetic reconnection via current sheets" Physics of Fluids 29: pp. 1520-1531, doi:10.1063/1.865670 <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Low, B. C. and Wolfson, R. (1988) "Spontaneous formation of electric current sheets and the origin of solar flares" Astrophysical Journal 324(11): pp. 574-581 <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Hughes, W. J. (1990) "The Magnetopause, Magnetotail, and Magnetic Reconnection" (from the "Rubey Colloquium" held in March 1990 at U.C.L.A.) pp. 227-287 In Kivelson, Margaret Galland and Russell, Christopher T. (editors) (1995) Introduction to Space Physics Cambridge University Press, Cambridge, England, pages 250-251, ISBN 0-521-45104-3 <a href="https://books.google.com/books?id=V935mEEjoTIC&pg=PA250" target="_blank">https://books.google.com/books?id=V935mEEjoTIC&pg=PA250</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>