8.2.P14

To try to get some understanding of how a water molecule looks to an ion electrically, let’s map out a piece of the electric potential produced by the charges in water. We’ll use a toy (easy to calculate model) of water: Two positive charges of +e (hydrogen) and a negative charge of -2e (oxygen) with the bonds at a right angle as shown in the figure. Take the separation of each + from the – charges as the bond length d. (Just to get a sense of scale, real water molecules have a value of d ~ 0.1 nm. We won't use this in this problem. Instead, we will construct dimensionless forms of the potential to study.)   A. Calculate the electric potential, V(A), that a test charge would feel if placed at the point A. Express your answer as a number times the combination of symbols, kce/d.

B. Calculate the electric potential, V(B), that a test charge would feel if placed at the point B. Express your answer as a number times the combination of symbols, k_ce/d.

C. Write an equation to express the electric potential, V(x) that a test charge would feel if placed at a position x on the positive x-axis. By dimensional analysis, we know that the final equation must have the form 

V(x) = (k_Ce/d) f(X)

where X is the dimensionless function of the dimensionless variable, X = x/d. Write your result in this form by using legitimate algebraic manipulations to identify the function f(X).

D. Use a spreadsheet or graphing calculator to make a plot of the function f(X) and include this plot in your assignment. Identify qualitative features of your function (Does it go to 0 anywhere? Does it go to infinity anywhere? Does it have a peak anywhere?) and explain whether these are reasonable given the physical situation.

E. If a positive ion approached the molecule along the x axis from the right, would it be attracted or repelled from the molecule? Is this consistent with the information in your graph? Explain.

Joe Redish 2/20/20