A doubly ionized carbon atom (with charge 2e) is located at the origin of the x

Question

A doubly ionized carbon atom (with charge 2e) is located at the origin of the x axis, and an electron (with charge -e) is placed at x = 5.76 cm. There is one location along the x axis at which the electric field is zero. Give the x coordinate of this point in cm.

Assume that the potential is defined to be zero infinitely far away from the particles. Unlike the electric field, the potential will be zero at multiple points near the particles. Find the two points along the x axis at which the potential is zero, and express their locations along the x axis in cm, starting with the point which is farther away from the origin.

I want an exaplination with the answer

Explanation / Answer

Let's define the positive 2e charge as charge 1 and the electron (negative e) as charge 2. Since we know we need them to be equal and opposite in order to get a zero electric field, we set up our equation like this:

k(q1)/((r1)^2) + k(q2)/((r2)^2) = 0; or
k(q1)/((r1)^2) = - k(q2)/((r2)^2)

We can cancel the k's and substitute in the actual charges to get:

(2e)/((r1)^2) = e/((r2)^2); or
2/((r1)^2) = 1/((r2)^2); or
2 = (r1^2)/(r2^2)

What else do we know? Well, we know where the two point charges are on the x axis. This tells us that whatever r2 is, r1 must be 5.76 greater, because charge 1 is 5.76cm further to the left:

r2 = r1 - 5.76

We use this to make a substitution for r2:

2 = (r1^2)/((r1-5.76)^2)

This simplifies to:

2(r1^2 - 10.02*r1 + 5.76^2) = r1^2; or
r1^2 - 20.04*r1 + 66.3552 = 0

This equation has two roots (possible values for r1).

r1 = 15.855 , 4.185 cm

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