TITLE: Why doesn't electric charge immediately leak off charged objects?
QUESTION [2 upvotes]: I will focus my question with a particular example: a metal sphere, surrounded by vacuum, is given a negative charge. I know that when this charge is great enough, electrons will be emitted from the sphere, but why is the threshold for this so high? As I understand it, the reason an electron stays on the negative sphere despite the electric repulsion is because of the metal's work function. But the work function of metals are typically ~4 eV. Wouldn't this suggest that -4 Volts would be the threshold for electron emission from the sphere in a vacuum? (Or a voltage even closer to zero, because of the thermal distribution of electron energies in the metal.) This seems way too small and I would think the threshold would concern a minimum field strength rather than minimum voltage.

REPLY [1 votes]: As an electron makes its way outside a metal surface, there is immediately
an image charge (positive virtual charge) on the other side of that surface
that attracts it back.   The 'work function' is the kinetic energy that the
free electron needs if it is to escape to a large distance, and has
to include both surface-binding-energy and the long-range attractive force
that a conductor exerts on a nearby charge.
The sphere, if it is very large, might have a local field completely dominated
by the (attractive) image charge, and not by the net (repulsive) charge on the sphere.  Net charge is one sphere radius away, and image charge is two
surface-to-electron steps away.


TITLE: Busy beaver candidate elimination: Minimum space requirements
QUESTION [4 upvotes]: I'm currently enrolled in a course that introduces Turing machines. As I wanted to play around a bit, I wrote a little TM engine and had it search for busy beavers (it successfully found the 4-state 2-symbol BB listed in Wikipedia).
To more quickly eliminate bad candidates, I'm searching for conditions that imply indefinite runtime. I found a couple of filters already, but the number of indefinite runs is still rather high (indefinite, as in, either seemingly running forever or sometimes checked manually).
In this questions in particular I would like to inquire what space requirements a BB must have after a certain number of shifts. If it accessed less of the tape than this number, it can be dismissed directly. For this purpose, consider the tape length to extend exactly as far as the head went at most in either direction.
As simple upper bound I guess something like $N_{shifts} \leq {N_{TapeLength}}^{|Q|}$ could work, though I am not sure this is even correct. Also, it should certainly be possible to improve this boundary considerably.
Edit: I currently employ the following eliminations:
Static

Fixed first/last transition (A0-B1R,xx-H1R)
State enumeration ordering
All states in transitive closure of A
H in transitive closure of all states
Trap states
Equivalent states
Inability to change the tape

Dynamic

At either end of the tape, current state's transitive closure (using only 0-transitions) always directs further away

REPLY [4 votes]: There are many methods for detecting that specific TMs will run infinitely. As Marzio mentioned, Heiner Marzen's page and papers provide almost all of the currently used methods.
The method you describe is a great simple requirement. Specifically, if we know that the TM has only moved around on a small tape of size $N_{tape}$, then the exact configuration at every step so far can be described by (1) the symbols at each of these $N_{tape}$ cells, (2) the position of the reading head and (3) the TM state. Thus there are $|Q| \cdot N_{tape} \cdot |S|^{N_{tape}}$ possible configurations (Where $|Q|$ is the number of states and $|S|$ is the number of symbols). So, if the TM has taken more than that many steps of computation, you know that it must have been in one configuration at least twice and thus that it will infinitely repeat.
However, I don't think this method works very often in practice because it is more common for TMs to run off infinitely in one direction than to stay on a small section of tape.