# Work Done and Potential Energy in Electrostatics

This is a question from class 12 Physics NCERT Part 1:

I am having so much trouble with this, I can't figure out where I'm going wrong. I solved part (a) here:

(a) U= -27.2 eV

(b) Here, the kinetic energy is given to be half of the potential energy, and kinetic energy is always positive, therefore, K= +13.6 eV Total energy= -27.2 +13.6 = -13.6 eV

Now this is where I'm going wrong. To free an electron, the final potential energy must be zero and also work done is change in potential energy. From that, W= 0- (-27.2) = +27.2 eV. But the answer is +13.6 eV according to my book. What am I doing wrong?

(c) In this part,

U= (U at 0.53 Å) – (U at 1.06 Å)

U= -27.2 eV -(-13.6eV)

U= -13.6 eV

therefore, K= +6.8 eV

and, Total energy= -13.6 +6.8 = -6.8 eV

But my book says that the total energy here is zero. How? Sorry for this long question, but I've been trying to solve this for so long. Any help is appreciated. Thank you so much.

• "To free an electron, the final potential energy must be zero" - ??? – Alfred Centauri Aug 16 at 11:40
• We were taught that freeing an electron implies separating it from the proton by infinite distance. If we are considering the potential energy at infinite separation to be zero, I assumed that the final potential energy is zero. Can you please explain how this is wrong? – laksheya Aug 16 at 12:11
• "We were taught that freeing an electron implies separating it from the proton by infinite distance" - I hope that's just a misunderstanding. In these types of problems, a better terminology is bound and unbound. Assuming the zero of potential is 'at infinity' (so that the potential energy is always negative), the electron is bound if the total energy (potential + kinetic) is negative and unbound (free) otherwise. – Alfred Centauri Aug 16 at 13:04
• Oh, this really clears things up. Thank you so much. – laksheya Aug 16 at 13:08

1. To free an electron (or any particle), the requirement is that the particle should be able to run off to indefinitely large distances. This means that the electron should have sufficient kinetic energy so that it can fight against the potential and reach the spatial infinity. Thus, the electron should at least have as much kinetic energy as the difference between the potential at its initial position and the potential at the spatial infinity. If you take the reference point (datum) for your potential at spatial infinity then this means that the total energy of the particle should be greater than or equal to zero. So, since the total energy of the electron in your case is $$-13.6$$ $$\text{eV}$$, you have to at least provide it $$13.6$$ $$\text{eV}$$ energy so as to make its total energy non-negative.