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: \oint_{R} \mathbf{B}\cdot\mathbf{dl} = \mu_0 I_\text{enc} \oint_{R} B \cos(\theta) \, dl = \mu_0 I_\text{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, \mathbf{B} \cdot d\mathbf{s} = 0 since \cos (90^\circ) = 0. In the other two sides, \cos (0) = 1, so if S is one parallel side of the rectangular loop of dimensions L × W, the integral simplifies to: 2\int_{S} B ds = \mu_0 I_\text{enc} Since B is constant due to the chosen path, it can be pulled out of the integral: 2B \int_{S} ds = \mu_0 I_\text{enc} The integral is evaluated: 2BL = \mu_0 I_\text{enc} Solving for B, plugging in for Ienc (total current enclosed in path R) as K×N×L, and simplifying: \begin{align} B &= \frac{\mu_0 I_\text{enc}}{2L} = \frac{\mu_0 K N L}{2L} \\[1ex] &= \frac{\mu_0 K N}{2} \end{align} 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. == Harris sheet ==

A well-known one-dimensional current sheet equilibrium is the Harris sheet, which is a stationary solution to the Maxwell–Vlasov system. The magnetic field profile of a Harris sheet along y = 0 is given by \mathbf{B}(y) = B_0 \tanh\left(\frac{y}{\delta}\right)\mathbf{\hat{x}}, where B_0 is the asymptotic magnetic field strength and \delta provides the thickness of the current sheet. The current density is given by \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) = \frac{B_0^2}{2\mu_0} \operatorname{sech}^2\left(\frac{y}{\delta}\right) + p_0, where p_0 is the asymptotic pressure. == References ==