Static Electricity Legacy Problem #19 Guided Solution

One thing which gives students fits when they're doing electric field problems is that there's two different equations for calculating electric field and for manipulating electric field quantities. One of the equations expresses the electric field in terms of the variables which are used to test its strength, and that's the equation E equal F per Q. And when we speak of testing the strength of the electric field surrounding a source charge, what we do is we place another object with a charge on it and we measure the force upon that charged object. We simply take the ratio of the force per quantity of charge on the so-called test charge used to test the strength of the electric field. I usually represent that charge, quantity of charge, by little q, and the equation is E equal F per little q, the force per charge on the test charge. The other equation is E equal K times big Q divided by D squared, and this is the equation that expresses electric field strength in terms of the variables which affect the amount of strength of the electric field. And those two variables are the quantity of charge on the source charge and the distance you are from the source charge. So here we have a small balloon which serves as a test charge used to test the strength of the electric field, and we have another object which is the source of charge and its charge, big Q value, is 1.28 microcoulombs or 1.28 times 10 to the negative 6 coulombs. What we wish to know is what is the electric field at a distance of 41 centimeters or 0.41 meters from that source charge. So here we have to use the equation E equal K times big Q, 1.28 times 10 to the negative 6 coulombs, divided by D squared. Now using the Coulomb's Law constant of K value of 8.99 times 10 to the 9th, that's in units of newtons per centimeter squared per coulomb squared, we can solve for the electric field intensity at this location, 41 centimeters from the source charge. I get 6.8454 times 10 to the 4th newtons per coulomb, and not all those digits are significant. So I round to three significant digits, 6.8454 times 10 to the 4th newtons per coulomb. Now in part B of the problem, what I wish to do is calculate the electric force on the test charge at that location. So now I need to use the other equation, the E equal F per little q, where little q here is 4.36 times 10 to the negative 9th coulombs. My E value is simply what I calculated from part A, that's the E value at this location. And so now I can rearrange my equation to say F equal E times q, I can take my E from part A and my little q value of 4.36 times 10 to the negative 9th coulombs, and I should be able to solve for the F, the force upon this balloon.