Static Electricity Legacy Problem #10 Guided Solution

Many students of physics figure that physics problems are difficult because the mathematics, meaning the algebra, is difficult. Yet, in many instances, students get stuck before they even get a chance to do the mathematical or algebraic manipulations. They get stuck very early in the problem, well before the math takes place. Their difficulty is more in the area of the conceptual nature of physics and not the mathematical manipulation of physics formulas. That could be the case in this problem if you're having difficulty with it. In situations like this, it's very important that a student of physics resort to the habits of an effective problem solver. That they begin to read the problem very carefully and develop a mental picture of what's going on. They begin to write down what's known and begin to plot out a strategy before they ever begin to use their calculator. Applied to this problem, we have a problem about a pair of electrons. We're told that they begin to repel because of their light charge. But because they have mass, they begin to attract one another. What we're asked is what mass must a pair of electrons have in order for this usually greater force of repulsion to be balanced out by an equal size gravitational force of attraction. So what we have to do is we have to remember some information from a previous unit on gravitation. We have to remember that the value of the force of gravitation can be calculated as big G, the gravitational constant, multiplied by the mass of one object and the mass of the other object, divided by the distance squared. And then we need to recognize that what we have is a problem in which we're being asked, what is the m-value, m, the mass of the electron, for which this f-grav expression is equal to the f-electrical expression. So I'm going to write down to start this problem out that f-grav equals f-electrical. Now we're talking about the gravitational attraction between two electrons of unknown mass. And we're talking about the electrical repulsion of two electrons of known charge. Now I'm going to write big G times m times m over d-squared for the force of gravitational attraction. I'm going to write k times q1 times q2 over d-squared for the force of electrical attraction. Now I've written this as an equation. I'm actually using paper to do this. And I notice that in my equation I have a d-squared in the denominator on the left side of the equal sign and a d-squared in the denominator on the right side of the equal sign. If I multiply through by d-squared, these two quantities, these two values, d-squared, d-squared, will cancel out. And I'm left with big G times m times m equal k times q times q. Now my mass of the electron, the first electron, is equal to the mass of the second electron. So I'm going to say m1 times m2 is equal to m-squared, such that the left side becomes big G times m-squared. And the right side would then become k times q1 times q2. But since the q of electron 1 is equal to q of electron 2, I can say q1 times q2 is equal to q-squared. And so the right side becomes k times q-squared. Now if I want to get my mass of the electron by itself, I would divide through each side of the equation by big G, such that my equation becomes m-squared equal k times q-squared over big G. Now on the right side of the equation, I know all the quantities. k, Coulomb's Law constant, 8.99 times 10 to the 9th. Big G, given in this problem, is 6.67 times 10 to the negative 11th. And q, the charge of an electron, I'll have to make sure I square it, but the value of q is 1.60 times 10 to the negative 19th. Substituting into this equation, I can solve for m-squared, and it would come out to be 3.45 times 10 to the negative 18th kilograms. If I take the square root of that, I'll have the value for the mass of an electron, and it is 1.8575 times 10 to the negative 9th kilograms. I can round that to three significant digits. Now before completing this audio file, think about it a bit. The mass of an electron is actually on the order of 10 to the negative 31st kilograms, meaning that in order to get the gravitational force to equal the electrical force, we would have to really supersize the mass of this electron. It's for that reason that we often conclude that in general, electrical forces are dominant forces compared to gravitational forces.