# 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 '19 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 '19 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 '19 at 13:04
• Oh, this really clears things up. Thank you so much. – laksheya Aug 16 '19 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.

2. Why would change in the reference point for your potential energy change the kinetic energy of the electron? The kinetic energy is a physical quantity, the reference point for the potential is of no physical significance in itself. You can set it anywhere you want--the kinetic energy would remain the same. The reference point of the potential energy is simply a figment of the imagination of the physicist, a very useful one, but purely non-physical. Following the same logic, changing your reference point shouldn't change anything about how much energy is required to set the electron free.

• Thank you so much! I didn't really grasp that concept about kinetic energy. Can you help me understand why the formula (work done= change in potential energy) is not applicable here? – laksheya Aug 16 '19 at 12:36
• @laksheya Work done by the conservative force is indeed equal to the change in potential energy. The point is that the change in potential energy between a pair of points would remain the same no matter where you decide to put the reference point. You would subtract the reference point value from the value of the potential at both points--keeping the difference unchanged. Does this address what you are confused about? Let me know if it is something else. – Dvij Mankad Aug 16 '19 at 12:39
• I understood that the change in potential energy is the same between two points irrespective of the reference, and therefore the work done would be the same in both cases. But you arrived at the answer +13.6 eV from total energy of the electron. This is what is confusing me. Can you explain what I'm missing? Thanks! – laksheya Aug 16 '19 at 13:06
• In the convention where we set the reference point of potential energy to be zero at infinity, the requirement that the electron should have sufficient kinetic energy so as to reach the spatial infinity translates into the requirement that the total energy of the electron should be non-negative. This is a rather general result. A system is understood to be bound if its total energy is negative and it is understood to be free if its total energy is non-negative. So, since the total energy of the electron is -13.6, we need to add exactly +13.6 into it to make the total energy non-negative. – Dvij Mankad Aug 16 '19 at 13:13
• Ahhh, I get it now. You're honestly a lifesaver, thank you so much! – laksheya Aug 16 '19 at 13:18