Static Electricity Legacy Problem #13 Guided Solution

Now here's a very difficult problem, one that's going to demand the use of the habits of an effective problem solver. One of the first things I'm going to do is really read this problem carefully and diagram the situation. There's a lot going on here, including three forces which are acting upon a soap bubble. So as I diagram the situation, I'm going to draw a picture of a cylindrical tube, a long tube, like a golf tube, a tube that holds golf clubs, and then I'm going to draw right underneath it an oblonged soap bubble. I'm going to write down everything and encourage you to do the same. There's going to be a lot to write down here. On this oblonged soap bubble, near the top of it, I'm going to draw a little positive sign to indicate the center of positive charge at the top of the soap bubble. The center of negative charge is near the lower end of that soap bubble. This separation or polarization of charge occurs because the presence of this charged tube induces electron movement within the soap bubble such that electrons move to get away from the tube. Now what I have are three forces acting upon this suspended soap bubble. One of the forces is the force of gravity. It's directed downwards, and I actually draw it on the diagram. It can be found by going mg, where g is 9.8 Newtons per kilogram. And the m, the mass of the soap bubble, is 22.8 milligrams, which is a nasty unit. I'm going to convert that to the standard unit of kilograms. It becomes 22.8 times 10 to the negative 6 kilograms. Now that's one of my forces of downforce of gravity on my soap bubble. There's another downforce of gravity, and it's the force of electrical repulsion between this negative tube and the center of negative charge. Opposites attract and likes repel, and so the negative tube repels the negative center of charge on the soap bubble. I can write an expression for this force, but I can't actually calculate it because there's one thing I don't know, and that's the amount of charge on the soap bubble. Now this expression would be written as k times the q of the tube. I just write it qt, times the q of the negative charge. I just write that q subscripted n, divided by d squared, where this d squared is 16.9 centimeters. Again, a nasty unit. I need to convert that to meters, 0.169 meters, and that gets squared. Now for the q of the t, the q of the tube, it's negative 45.6 nanocoulombs. Again, a nasty unit. I'm going to skip the negative. That's not important. I'm just going to write 45.6 times 10 to the negative 9th coulombs. Now there's one final force, and that's an upforce. I draw it on my diagram, and I label it the force of attraction, the attraction between the negative charge on the tube and the positive center of charge on the soap bubble. Its expression would be written again as k times qt, q of the tube, multiplied by qp, q of the positive charge, divided by the distance squared of separation, where the d here is 0.114 meters, and that gets squared. Now I don't do any substituting quite yet. I'm going to do some algebra on this equation, or on these equations. What I'm going to do is I'm going to say since the soap bubble is suspended motionless, all the forces must balance, and so of the three forces, the one up must balance the one down. I'm going to write an expression that goes something like this. You probably want to write it down if you're having troubles following this. It goes the force up, k, qt times qp, divided by distance squared, where this distance is the 0.114 meters, and it gets squared. That whole expression is equal to mg, we've already talked about that, plus k times q of tube, times q of the negative charge, divided by the distance squared, where that second distance is equal to the 0.169 meters, and that gets squared. Now my goal in this problem is to solve for the q of the positive charge and the q of the negative charge, and since these q's are the result of movements of electrons from one side to the other side, the q of the positive charge is equal to the q of the negative charge, and I'm just simply going to call it q. And I'm going to try to get the two q terms by themselves on the same side of the equation. So what I would do is I would subtract the repulsive force from the right side, since it shows up as a minus term on the left side. So now I have kq2 times q positive over this 0.114 squared, minus k times qt times q, divided by the 0.169 squared is equal to mg. I'm going to factor out a q, that's the q of the positive charge and the q of the negative charge, and then I'm going to evaluate what's left on the left side of the equation. So what I'm evaluating is kq2 divided by the 0.114 squared, minus k times q2 divided by the 0.169 squared. That evaluates to 17,190.5807. That's a lot of digits. I'm giving you more than you need. And so q times that number is equal to mg. If I divide both sides of the equation by 17,190.5807, I'm able to get the value of q, and it comes out to be 1.2998 times 10 to the negative 8th Coulombs. I would use three significant digits in this answer, so it becomes 1.30 times 10 to the negative 8th Coulombs, or converted to nanocoulombs, 13.0 nanocoulombs.