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Advice I was given when growing up: in a thunderstorm, stay away from tall, pointy things.

It's pretty obvious why lightning would prefer to strike tall objects, but pointy ones?

Is there any truth to this, or is it simply that tall objects tend to be pointy? I imagine a 100ft copper spike would be less receptive to strikes than a 100ft copper sphere.

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Yes they are. The reason is that for an electrically charged object the electric field at the surface is inversely proportional to the curvature radius. When an object is very sharp, the electric field is hight and can induce a lightning channel by attracting ar repelling electrons in the air.

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The short answer is yes. I saw this demonstrated at a local Science Center using a Van der Graff to generate charge. A wand, connected to the Van der Graff was used along with various shaped grounded objects. The high voltage electrical discharge obviously 'preferred' the sharp, pointed object, even though other shapes were closer to the wand, the electrical discharge would travel further to the pointed object. This was both an interesting and dramatic demonstration.

This is why lightning rods are pointed. Lightning rods have a sharp, pointed tip, since electric lines of force are more highly concentrated around pointed objects. If lightning strikes you want it to strike the lightning rod and not the structure the rod is there to protect.

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Whether the pointedness of the object really makes any difference I don't know, but there is an argument to suggest that it might.

The negative charge in the cloud is going to induce a positive charge on the ground and anything sticking up from it like your copper rod. If you take any (conducting) object and charge it then the field gradient will be higher than the average at any pointy areas on the object. This is responsible for the corona discharge that you see when a pointed object is highly charged. This happens because the field gradient at the point is high enough to ionise air molecules.

So the argument is that the pointed object will have a higher field gradient near it and will therefore attract the lightening bolt. You certainly see this effect in the lab with a Van de Graaff generator. My only reservation is that I suspect the path of a lightening bolt is controlled by many factors and the pointedness of the lightening conductor will be only one of the factors.

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