Jan 15 2011

## Permittivity of Free Space

It is interesting, to say the least, the way some of the constants of physics mix and match.  There are lots of examples of this in text books and on web sites, including this one, so there is no point in reiterating any now which do not relate specifically to this entry.

Some constants, nevertheless, turn out not to be constants, including the permittivity of free space, termed ε0.  At the surface of the earth, and at any point not far enough away to discern a difference, we have ε0 = 8.85 x 10-12 C2/(N-m2) when measured in a vacuum.  One of the ways in which the permittivity of free space relates to other physical entities is in the makeup of the Coulomb constant, k = 1/(4πε0), which is then also not really a constant in all locations, the difference relating to the local density of gravitons.

The speed of light in a vacuum can be written as: c = sqrt(k/km) = 1/sqrt(ε0 µ0) = 2.998 x 108 m/s, which is a constant throughout the universe.  One may be tempted to say then that µ0 is an inverse function to ε0 when evaluated at a given point, except that this is so unlikely with present understanding so as to be unimaginable.  Along with the constancy of the speed of light in a vacuum in any reference frame, the fact that it equals 1/sqrt(ε0 µ0) in our locale can remain somewhat of a mystery.

Jan 12 2011

## Gradient of a Scalar Potential

In Thomas’ Calculus we find the definition: “If F is a vector field defined on D and F = del f for some scalar function f on D, then f is called a potential function for F.” (, pg 921)

Kaplan states that the “gravitational field is the gradient of the scalar f = kMm/r” (, Prob. 4, pg 180), which for General Relativity is indeed the case.  With gamma ray energy exchange however, the scalar is f = k1M/r, in form much like the scalar potential k2q/r with electrodynamics.  This tells us that both gravitational and Coulomb forces are of independent action, and the corresponding fields are gradients of scalar potentials, which can be considered as more evidence for unification.

The vector fields we are looking at have units of N/kg for gravity, and of course E = N/C for an electric field.  A gravitational field calculation is most applicable when the object in the field is small compared to, or far away from, the source of the field, such as Jupiter compared to the sun.

Aside from the King James Version of the Bible, Wilfred Kaplan’s Advanced Calculus is still my favorite book.  By the way, we used Thomas’ Calculus book at the University of Wisconsin in 1976 and 1977.

 Thomas, George B., as revised by Weir, Maurice D. and Hass, Joel, Thomas’ Calculus, Twelfth Edition, Addison-Wesley of Pearson Education, Inc., 2010, 2005, 2001

 Kaplan, Wilfred, Advanced Calculus, Fifth Edition, Addison-Wesley of Pearson Education, Inc., 2003

Jan 05 2011