4.2.4.P21

One of the most important electrical structures in chemistry and cellular biology is the dipole. The simplest example is a bound pair of equal and opposite charges separated by a small distance as shown in the figure at the right.  (Though this is the simplest example, many molecules have non-symmetric charge distributions and behave electrically like a dipole when you're not too close to it. The water molecule is one important example.)

Since Coulomb's law only holds between point charges, when a charge such as an ion approaches a dipole, the electric force it feels doesn't satisfy a simple Coulomb's law, but rather the result of the sum of two Coulomb's law forces. This has some interesting and important effects. In order to see how a dipole works electrically, in this problem you'll calculate the magnitude of the force an ion feels from a dipole using a spreadsheet and create a graph of the dipole-ion force as a function of distance.

To do this, we will simplify the calculation by suppressing units in order to get a sense of how the electric force from a dipole behaves as a function of distance. We will take Coulomb's law to have the form F = qQ/r² and we will treat q, Q, and r for the purpose of the spreadsheet as pure numbers.

A.1 Use a spreadsheet to create a graph of the force that a point charge Q = 5 creates on a point charge q = 1 at a distance ranging from r = 1 to 100. Make a graph of F vs r and a log-log plot of F vs r and include these graphs in your solution.

A.2 Your log-log plot should look like a straight line with a slope of -2. Does it? Explain why it looks like this. (Use math!)

B.1 Now use your spreadsheet to create a graph of the force that a dipole consisting of two point charges produce on a charge. Put a charge Q1 = -5 at a position -½, and a charge Q2 = +5 at a position +½. Write an equation for the force, F, between the dipole and a charge q = 1 if it is a distance r away. (Your equation should look like F = function of r with some numbers.)

B.1 Now use your spreadsheet to create a graph of the force that a dipole consisting of two point charges produce on a charge. Put a charge Q₁ = -5 at a position -½, and a charge Q₂ = +5 at a position +½. Write an equation for the force, F, between the dipole and a charge q = 1 if it is a distance r away. (Your equation should look like F = function of r with some numbers.)

B.2 For the force between q and the dipole, use your spreadsheet to create a graph of F vs r and a log-log plot of F vs r and include these graphs in your solution.

B.3 Does your log-log plot look like a straight line? Does it tend become a straight line for large values of r? If so, what slope does it have? Can you explain why it looks like that and not like what we found in A?

Joe Redish 10/3/19