Theory of magnetic insulation

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Abstract
An investigation is made of the behavior of a high voltage diode for the situation where the cathode‐anode gap is initially filled with a transverse magnetic field. An exact relativistic self‐consistent equilibrium is found for the electron sheath which is expected to form under the condition where the applied magnetic field is sufficiently strong to prevent electrons from flowing between the electrodes. The condition on the magnetic field for “insulation” is found to be (eB y 0 d/mc 2 ) 2 > 2|eV 0 /mc 2 | + (eV 0 /mc 2 ) 2 , where B y 0 is the applied magnetic field, V 0 is the voltage across the diode, d is the cathode‐anode separation in the x direction, and — e and m are the electron charge and rest mass, respectively.