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I want to know how a material's resistivity changes after the dielectric breakdown voltage is applied and crossed. Let's say for a circuit after air becomes a conductor for E=30kv/mm what would be the current flowing in that circuit and how to find it?

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I want to know how a material's resistivity changes after the dielectric breakdown voltage is applied and crossed.

Dielectric breakdown can permanently degrade the resistivity of solid electrical insulation, the degree to which depends on the material. Self heating due to the high localized breakdown current can cause permanent thermal damage which reduces resistivity. This is particularly the case for polymeric (plastic) insulating materials.

Let's say for a circuit after air becomes a conductor for E=30kv/mm what would be the current flowing in that circuit and how to find it?

The breakdown current would depend on the ampacity of the high voltage source, as the air essentially becomes a near short circuit when breakdown occurs.

Hope this helps.

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