What is Field energy, what does it mean?

[tsuwal]

I am a bit confused the concept of energy density. I was taught that, in vacum:

$\frac{1}{2}\int_{V}^{ } \rho P dV = \frac{1}{2}\varepsilon _0 \int_{V}^{ } E^{2} dV$

where P is the eletric potencial.

This is true because:

$\rho P=\bigtriangledown \cdot (PE)+E^{2}$

where the first term can be ignored because if we integrate in the whole space and use the divergence theorem we find that this term goes to zero.

This expression only makes sense if we integrate in the whole space right? (because we ignored a term in the dedution on this basis). Similarly, can we do the same to gravitic fields? This is:

$\frac{1}{2}\int_{V}^{ } \rho P dV = \frac{1}{2}\varepsilon _g \int_{V}^{ } F_g^{2} dV$

where $\rho$ is the mass density $\epsilon_g=\frac{1}{4\pi G}$ is the "gravitic permitivity" and $F_g$ is gravitacional force?
If this is true, then if we consider a sphere with a mass $m$ and compute its energy by this formula should we get $mc^{2}$? (I tried it but didn't get this)

 

[Greg Bernhardt]

Field energy is the energy that is stored in the electromagnetic field of the earth. It is a part of the earth’s total energy. The earth’s total energy is the sum of the field energy and the kinetic energy of the earth’s matter. The earth’s matter moves in circular orbits about the sun.

The distance of the Earth from the sun is determined by the force of gravity between them. This force is proportional to the product of their masses and inversely proportional to the square of the distance. In the nineteenth century, it was discovered that the energy of moving bodies increased with velocity. In addition, it was discovered that kinetic energy is equal to one-half the mass of the body times the square of the velocity.

The equation that expresses the relationship between the kinetic energy and velocity for a body is: Kinetic energy = ½ mv2 To find the kinetic energy of the earth, we must begin with the mass of the earth. The earth’s mass is 6.0 x 10^24 kilograms.