Static Electricity Legacy Problem #32 Guided Solution

This is a very difficult problem and one in which you're going to have to employ the habits of an effective problem solver, the most important habit of which is to do a good deal of strategizing and thinking before you ever pick up your calculator. In fact, as I solve this problem, I won't pick up my calculator until the very end when I get ready to calculate my answer. Most of what I do will either be thinking or algebra. Here what we have are two charges which lie along an axis and lie 50 centimeters apart. Charge Q1 is charged positively and charge Q2 is charged negatively. And what we wish to do is determine the location along the axis where the net electric field is zero. Now each of these individual objects create their own electric field. The direction of that electric field is created by the object is the direction that the object would push a positive test charge. So with charge Q1 being negative, it creates an electric field that is directed towards itself. And with Q2 being positive, it creates an electric field with the direction being away from itself. So if you were to consider a location in between Q1 and Q2 in which the net electric field was zero, it would be impossible to have such a position because in order to have the net electric field be zero, the E1 has got to be equal in magnitude and opposite in direction to E2. And if you consider a location in between Q1 and Q2, the direction of both electric fields would be towards the left and they could never cancel each other. So we have to consider a location either to the left of Q1 or to the right of Q2. So which general location would it be? So if I begin to think about this, the electric field due to Q1 is going to be by nature greater just based on the fact that the charge of Q1 is greater than the charge of Q2. So for Q2 to make up for what it loses in terms of charge, we would have to consider a location that is closer to Q2. It has to make up for its lack of charge by being closer in distance. So the location where the electric field is zero is going to be somewhere to the right of charge Q2. So I'm going to put a little dot along the axis, a little bit to the right, some distance. The right doesn't matter what distance, I don't know that distance. But I'm just going to put a dot on that axis to the right of Q2. And then I'm going to call the distance from Q2 to that point, to that dot, x. And the distance from Q1 to that dot would be 50 centimeters plus x. In such a situation, x is in units of centimeters. Then I'm going to make the algebraic statement that E1 is equal to E2. And I'm going to expand the statement by saying K times Q1 divided by distance 1 squared is equal to K times Q2 divided by distance 2 squared. Now the distance 1 is simply 50 plus x and the distance 2 is simply x. So I'm going to rewrite my equation as K times Q1 over 50 plus x, the quantity squared, is equal to K times Q2 over the quantity x squared. Now in this equation you have K on both sides of the equation, so you can divide through by K and cancel your K. Leaving you with Q1 divided by 50 plus x, the quantity squared, is equal to Q2 divided by x squared. What I wish to do is solve for x and ultimately find the location along the axes where this condition of E1 equal the opposite of E2 is true. So to solve for x, one of the next things I'm going to do is I'm going to take the square root of both sides of the equation. Doing so eliminates the possibility of me having to use a quadratic in the solution of my problem. So this leaves me square root of Q1 divided by just 50 plus x now is equal to the square root of Q2 divided by x. Now if I cross multiply here I'm going to end up with square root of Q1 times x is equal to the square root of Q2 times 50 plus x. Then I can group my two x terms by themselves which would leave me with square root of Q1 times x minus square root of Q2 times x is equal to the square root of Q2 times 50. My next algebraic step is to factor out an x from the left side of the equation which would leave me with square root of Q1 minus square root of Q2. That quantity multiplied by x is equal to the square root of Q2 times 50. Now if I divide through by the quantity square root of Q1 minus square root of Q2 I'll end up with x is equal to the square root of Q2 times 50 divided by the quantity square root of Q1 minus square root of Q2. Now finally I get to use my calculator and I take my values of Q1 and Q2, 38 and 18, and I substitute them into that equation ignoring the negative sign on the Q2 value because that has nothing to do with quantity only with time. Now solving for x I end up getting the value 110.38 centimeters and as we've defined x that's the distance from Q2 out to this location where the net electric field is zero. So the answer to the question what's the x coordinate at which E is equal to zero is going to be the 50 centimeters plus the 110.38 centimeters which comes out to be approximately 160 centimeters.