= Parallel current sheets with a resistive termination
{title2=$B=\mu_0I/w,\quad E=IR/d,\quad v_{\rm selected}=Rw/(\mu_0d)$}
Two broad perfect conductors of width $w$, separated by gap $d\ll w$, carry opposite steady <electric currents> $I$ and connect through a terminating <resistor> $R$. Away from edges, <Ampère's circuital law> and the <superposition principle> give gap <magnetic field> magnitude $\mu_0I/w$. Each conductor is an <equipotential>, and <Ohm's law> sets their <electric potential difference> to $IR$. Planar <symmetry> and charge-free <Gauss's law> then give uniform gap <electric field> magnitude $IR/d$. Injection parallel to the upper current makes the electric and magnetic <Lorentz forces> opposite. Their cancellation selects <speed> $Rw/(\mu_0d)$ independent of nonzero <electric charge> and of operating current; nonrelativistic <kinetic energy> is $m[Rw/(\mu_0d)]^2/2$.
Back to article page
