Example: Coulomb's law

Understanding the situation

The electric attractions and repulsions of the ions on the surface of complex molecules play a critical role in many biochemical reactions. An interesting component of some of these processes is that chemical reactions or electron transfers result in a change of the shape of a molecule, exchanging chemical energy for electrical energy, and resulting in a different surface configuration and therefore different interactions. Let's consider the force on one configuration of a complex molecule in order to see how to calculate forces using Coulomb's law.

Carefully drawn vector diagrams will help us get to an answer.

Presenting a sample problem

Consider the electric force between four ions on the surface of two large molecules as shown in the figure at the right. The heavy black lines running out of the frame indicate bonds to other atoms in the molecule that we will not consider here. The purple ion in the molecule on the right has a charge +e, the red ion in the molecule on the left has a charge -2e, and the two green molecules in the molecule on the left each have charges +e. A schematic diagram of these ions is shown on the left below indicating relative distances.

A. Qualitatively (without doing a calculation), would you expect the force of the ions (ABC) on ion D to be attractive or repulsive? Explain your reasoning.

B. Now do the calculation to find the net force between (ABC) and D. Express your result as a multiple of the combination $F_0 = k_Ce^2/d^2$.

Solving this problem

A. I would expect if to be attractive. The three forces that the charges in the molecule are exerted on D is shown in the figure at the right. The attractive force from A is twice as big as each of the forces from B and C because the charge is bigger. But the forces from B and C are also smaller than the force from A since they are farther away. Even more, the forces from B and C are not in the same direction, so they should partially cancel each other. As a result, A should definitely produce a stronger (attractive) force than both B and C produce (repulsive) taken together.

This part of the problem illustrates how you can use Coulomb's law qualitatively to analyze forces by understanding how it scales by the charge and by distance. (See Reading the content in Coulomb's law.)

In the next part, we see how to calculate the force quantitatively.

B. The force between any two charges is given by Coulomb’s law,

$$\overrightarrow{F}^E_{Q\rightarrow q} = \frac{k_CqQ}{r^2_{qQ}}  \hat{r}_{Q \rightarrow q}$$

with the direction along (or opposite to) the line between the two charges. Since A has charge -2e and D has charge e, the force between A and D has the magnitude

$$F^E_{A\rightarrow D} = \frac{k_C(2e)(e)}{d^2}  = 2\bigg(\frac{k_Ce^2}{d^2}\bigg) = 2F_0$$

The magnitudes of the forces from B and C on D, shown in blue on the right, only have factors of 1e in the charge (both have magnitude e), but their distance is larger. By the Pythagorean theorem they are a distance of square root of 2 further away than A. Since Coulomb's law tells us that the magnitude of the force falls as the square of the distance, we get 

$$F^E_{B\rightarrow D} = F^E_{C\rightarrow D} = \frac{k_C(e)(e)}{(\sqrt{2}d)^2}  = \bigg(\frac{k_Ce^2}{2d^2}\bigg) = ½ F_0$$

But the forces from B and C on D are vectors and the don't point in the same direction. So we can't just add them. We have to do the adding vectors thing. One way is to complete the parallelogram as shown in the figure at the right. Since the angles are all right angles and the sides are the same, this is a square. The length of the diagonal is square root of 2 times as big as the sides, so the sum of the two vectors points horizontally to the right and has a magnitude of √2 F₀/2. The result is therefore the difference — an attraction pointing left with magnitude

$$F^E_{net} = 2F_0 - \sqrt{2}F_0/2. = (2- \sqrt{2}/2) F_0 = 1.29 F_0.$$

(You can also break the two vectors up into vertical and horizontal components. The vertical ones cancel, the horizontal ones add.)

In general, we use a combination of quantitative and qualitative reasoning to decide what to do with the various forces.

 The most common error in finding electric forces is to forget that they are vectors and just add the magnitudes! (Perhaps because the Coulomb force formula looks messy and students focus on getting the value of the magnitude right.)This only gives the correct answer if all the charges on along a single line.

Joe Redish 4/30/19