Week 3: Potential Energy and Potential 121
depending on thedetailsof how the charge we assemble is distributed. For a conducting sphere
(where all the charge resides on the outside) or spherical shell ofcharge, for example, it will be 1/2.
See if you can show this.
As a final note of interest, observe how the potential energy of the ball of charge scales with its
radius! As any fixed amount of charge is compressed into smaller andsmaller balls so thatR→0,
we see thatU(R→0)→ ∞! If we forget the factor of 3/5, or 1/2 (which depends on thedetails
of the charge distribution) and focus on the rest, we can computea couple of extremely interesting
quantities that give us insight into nuclear physics and certain properties of electrons.
To compute the first, assume thatQ= +eandR= 10−^15 meters (one fermi) – a model for
theprotonas a ball of charge. If one computeskee/Rfor this in SI units (Volts) and multiplies
by the remaining +eto getkee^2 /Rin eV, one gets +1.44 MeV – theorder of magnitudeof the
energy bound up in the electrostatic field of the charge of a proton. Nuclear forces that glue all of
this charge together (withgluons) must be much stronger than electrostatic forces to make thetotal
energy negative or a proton would not be a stable bound state, andthey are. Electronic energy
levels in atoms are scale eV, nuclear energy levels are scale MeV (and higher) which explains why
stars burn slowly and release far, far more energy than can be explained by “atomic” electronic
bonding (conventional burning). Nuclear fusion releases order often million times as much energy
per interaction than does e.g. burning one carbon atom into carbondioxide.
The second requires a “true fact” (that is, fortunately, fairly common knowledge): Mass and
energy are interchangeable, and the “rest mass” of an object corresponds to a “rest energy” ofmc^2
wherec= 3× 108 meters/second is the speed of light. Now we suppose that an electron’s rest mass is
all due to its electrostatic energy of confinement, the energy tiedup in the chargeeconfined tosome
radius, and we seek that radius, which we will call “the classical radius of the electron”^42. This is
the same computation as above, only backwards – we know the energy already, we knowkeand the
charge−e, we solve forre. If you do this, usingU= 0.5 MeV for an electron, one gets 2. 8 × 10 −^15
meters. Note well that this is somewhatlargerthan the size of a proton (as the electron has less
energy). The classical radius of the electron turns out to be an important quantity in determining
the properties of electromagnetic radiation from point charges.
(^42) Wikipedia: http://www.wikipedia.org/wiki/Classical Electron Radius.