[Ishwar]

Hi, I'm kinda new here, ok i have this problem frm a book :

Consider the hydrogen atom to be a proton embedded in a cavity of radius (Bohr radius) whose charge is neutralized by the addition of an electron to the cavity in a vacuum, in a vacuum, infinitly slowly. Estimate the average total energy of an electron in its ground state as the work done in the above neutralization process.
Also, if the average kinetic energy is half of the magnitude of the average potential energy, find the average potential energy.


Ok, someone gave me the answer like this:
but I don't think it is right, because if you look at (ii) down below, shoudn't it be : KZe² / a₀ = mv² instead of minus KZe² / a₀ = mv²


Potential energy = - KZe² / a₀²

(a₀ = bohr's radius)

Kinetic energy = ½ mv²

Total energy = -Ke²/a₀² + ½ mv² ………………. (i)


(electrostatic force = centripetal force)

- KZe² / a₀² = mv² / a₀

- KZe² / a₀ = mv² ……………. (ii)


By substituting (ii) into (i)
E = - KZe² / a₀ + ½(- Ke² / a₀ ) = -3Kez² / 2a₀

[Yggdrasil]

Remember that the force (F) is related to potential (V) by the equation:

F = -dV/dr

So if you define your potential to be negative, then your force is positive.  In other words:

Potential energy: V = -KZe²/a₀
Force: F = KZe²/a₀²

These should give you the correct numbers.

[edit:  since this is the high-school forum, I guess I shouldn't assume you know calculus.  If the first equation doesn't make sense, it's because it involves calculus which you may or may not have taken yet]

[Ishwar]

Ok, thanks for that Yggdrasil,

but please tell me how negative potential energy (- KZe² / a₀) equals positive centripetal force  (mv²) , as suggested in (ii)

[Yggdrasil]

In (ii) you are setting the Coulombic force (not potential) equal to the centripetal force.  Therefore:

KZe²/a₀² = mv²/a₀

which gives:

KZe²/a₀ = mv²

[Ishwar]

In (ii) you are setting the Coulombic force (not potential) equal to the centripetal force.  Therefore:

KZe2/a02 = mv2/a0

which gives:

KZe2/a0 = mv2

So the answer is wrong ?

Shudn't it be :

(electrostatic force = centripetal force)

KZe² / a₀² = mv² / a₀

KZe² / a₀ = mv² ……………. (ii)


By substituting (ii) into (i)
E = - KZe2 / a₀ + ½(Ke² / a₀ ) = -Kez² / 2a₀

[Ishwar]

Thanks for all your help, Yggdrasil, the problem is resolved now,

i had a misconception, and that confused me,

anyways Thanks