How much work would it take to push two protons very slowly from a separation of 2.00×10−10 m (a typical atomic distance) to 3.00×10−15 m (a typical nuclear distance)?

We can visualize the problem in another way, which is equivalent but easier to solve: let's imagine we hold one proton in the same place, and we move the other proton from a distance of 2.00×10−10 m to a distance of 3.00×10−15 m from the first proton. How much work is done?

The work done is equal to the electric potential energy gained by the proton:

where is the charge of the proton and is the potential difference between the final position and the initial position of the proton. To calculate this , we must calculate the electric potential generated by the proton at rest at the two points, using the formula:

where is the Coulomb constant and Q is the proton charge. Substituting the initial and final distance of the second proton, we find

Therefore, the work done is

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