Current sheet

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, and there are N wires per unit length, the magnetic field can be derived using Ampère's law:

${\displaystyle \oint _{R}\mathbf {B} \cdot \mathbf {dl} =\mu _{0}I_{enc}}$

${\displaystyle \oint _{R}Bdl\cos {\theta }=\mu _{0}I_{enc}}$

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, ${\displaystyle \mathbf {B} \cdot \mathbf {ds} =0}$ since ${\displaystyle \cos(90)=0}$. In the other two sides, ${\displaystyle \cos(0)=1}$, so if S is one parallel side of the rectangular loop of dimensions L x W, the integral simplifies to:

${\displaystyle 2\int _{S}Bds=\mu _{0}I_{enc}}$

Since B is constant due to the chosen path, it can be pulled out of the integral:

${\displaystyle 2B\int _{S}ds=\mu _{0}I_{enc}}$

The integral is evaluated:

${\displaystyle 2BL=\mu _{0}I_{enc}}$

Solving for B, plugging in for I_enc (total current enclosed in path R) as I*N*L, and simplifying:

${\displaystyle B={\frac {\mu _{0}I_{enc}}{2L}}}$

${\displaystyle B={\frac {\mu _{0}INL}{2L}}}$

${\displaystyle B={\frac {\mu _{0}IN}{2}}}$

Notably, the magnetic field strength of an infinite current sheet does not depend on the distance from it.

The direction of B can be found via the right-hand rule.

A well-known one-dimensional current sheet equilibrium is Harris current sheet, which is a stationary solution to the Maxwell-Vlasov system.[6] The magnetic field profile is given by

${\displaystyle \mathbf {B} =B_{0}\tanh(x/L)\mathbf {e} _{z}}$

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