The potential energy function for either one of the two atoms in a diatomic molecule is often approximated by U(x) = −a/x12 − b/x6 where x is the distance between the atoms.

a) At what distance of separation does the potential energy have a local minimum (not at x = [infinity])

Explanation:

If we want to find a local minimum, we can serach for the points where

U'(x) = 0

here we have that U(x) = -a/x^12 - b/x^6

then U'(x) = (-12)*-a/x^11 -(-6)*b/x^5

this must be zero.

12a/x^11 + 6b/x^5 = 0

12a/x^11 = 6b/x^5

12a = (6b/x^5)*x^11 = 6b*x^(11-5) = 6bx^6

x^6 = 12a/6b = 2a/b

This is the distance at wich the potential energy has a local minimum.

a charge 3q is at the origin, and a charge −2q is on the positive x axis at x=a. part a where on the x-axis would you place a third charge so it would experience no net electric force? enter an expression for the exact answer, which will involve a square-root sign. (do not enter an approximate decimal answer.)

seeing the graph would