Physics Friday 40

What is the distribution of induced charge on a conducting sphere of radius R placed in a uniform electric field E₀?

Monday Math 39

Suppose we have a group of n people (n≥2). The name of each person is written on a separate slip of paper. These n slips are put in a box, and each person then draws a slip from the box. What, then, is the probability that nobody draws their own name? What happens to this probability when n becomes large?

Physics Friday 39

Let us consider two point charges of charge -q and q (q>0) separated by a distance d. Let d be the vector from the negative to the positive charge. Choosing the point halfway between the charges as the origin of our coordinate system, then the electric potential is given by:

and the electric field is
.

First, let us examine the field far from the charges (r≫d). To simplify things, let θ denote the angle between d and r. Then the law of cosines tells us that
and
. Thus, for r≫d, we see

and

So we have
.
Using , we see that for distant points, the field is approximated by
.
Defining , and using the unit vector in the direction of r , we have
, or, in terms of θ,
, so we see the magnitude of the distant field goes as the inverse cube of distance. The vector p is the dipole moment, and if one lets d go to zero while holding p constant, the field approaches the one above at all points, and one has a point dipole.

Now, let us instead consider the potential and field near the point between the charges (r≪d). In this case, , and

Thus

and so
, where is the unit vector in the direction of d. Note that this approximation is a constant vector. Thus, if we have while holding constant, our electric field appoaches a uniform electric field.

Happy Constitution Day

The United States Constitution was adopted by the Constitutional Convention in Philadelphia 221 years ago today. [My thanks go to Eugene Volokh for the reminder].

Schneier on the Boston Case

Bruce Schneier discusses full disclosure and the recent Boston court decision concerning the security researchers who found a vulnerability in the Boston subway’s fare-card system.

Olypics per Capita

Dr. John Wilkins analyzes countries’ Olympic medal counts in proportion to their respective population sizes.

Enzyte

As I noted previously, Berkeley Premium Neutraceuticals founder Steve Warshak was convicted earlier this year on 93 counts of fraud, conspiracy, and money laundering. Well, he’s now been sentenced to 25 years in jail.
[Via Dispatches fron the Culture Wars]

Catch Up

I’ve been rather remiss in keeping this blog updated lately. Hopefully I’ll be able to take a good chunck out of the backlog today.

Anticommons

James Surowiecki discusses the Tragedy of the Anticommons in The New Yorker: “The Permission Problem.”

A Question…

Why do they have suicide watches for death row inmates?