Static Electricity Legacy Problem #20 Guided Solution

You've likely seen a Van de Graaff generator at a museum or perhaps even in your own physics classroom. What it is is a big, metal, spherical ball that accumulates a great deal of charge. That charge generally remains right there upon that static ball. Unless the insulating ability of the air somehow breaks down, and it would break down if the quantity of charge on the generator became so big that the electric field in the space surrounding it was 3.0 times 10 to the 6th newtons per coulomb. Now in a situation such as this, it will discharge and you'll see a lightning bolt, so to speak, that protrudes out from the big static ball towards an object such as somebody's hand. Now what we wish to do is we wish to determine what quantity of charge would be on the Van de Graaff sphere in order to break down the insulating ability of air up to a distance of 50 centimeters away. So what I need to do is use the equation E equals K times big Q divided by D squared where the D squared here, or at least the D value, is 50 centimeters, but I'm going to substitute it in in units of meters such as 0.50 meters. My E value is 3.0 times 10 to the 6th newtons per coulomb, and of course K is what it always is, the constant value of 8.99 times 10 to the 9th newtons times meters squared per coulomb squared. Now what I wish to do is solve for Q, the quantity of charge, on the Van de Graaff generator. So I need to substitute in my numbers and do rearranging and solve for Q, and it's pretty straightforward algebra, and it ends up being a value of 8.342 times 10 to the negative 5th coulombs, and that's the value for the quantity of charge on the Van de Graaff generator that would create that electric field 50 centimeters or 0.50 meters away.