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We have calculated (probably measured) the speed, temperature and voltage of lightning so does lightning have a thickness?

Note: When a lightning occur it heats up the air rapidly causing the air to become plasma, I'm not referring to the thickness of ionized air.

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The primary "bolt" of lightning carrying the most current is only a few centimeters across. However, the entire "tube" that is carrying a significant current can be upwards of 10 meters across. If the lightning hits the ground, the current spreads out across the surface in all directions (not necessarily uniformly) increasing the radial extent of danger. This is one of the many reasons why you need not be hit by the main bolt to still be struck by lightning.

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