Tensors for Physics

8.3 Volume Integrals, Gauss 139

Therelation(8.77) is referred to as the differential form of theGauss law. The integral

of (8.77) over a volumeVand use of the Gauss theorem, with d^2 sμ=nμd^2 s, yields:

∮

∂V

Dμd^2 sμ=

∫

V

ρd^3 r=QV. (8.78)

This is the Gauss law of electrodynamics. It means that the flux of theD-field through

the closed surface∂Vis equal to the charge contained within the volumeV.The

Coulomb law for the force between two charges, located in vacuum, follows from

the Gauss law (8.78). This is seen as follows.

In general, one hasD=ε 0 E+P, whereε 0 is the dielectric permeability of the

vacuum. Its numerical value depends on the choice of the basic physical units for

length, time, mass and charge. In the system of physical units originally introduced

by Gauss, where no independent basic unit for the charge occurs,ε 0 is equal to 1.

The vector fieldPis the electric polarization. In vacuum,P=0 applies. Thus in

vacuum, the electric fieldEis related to the charge density via

∮

∂V

Eμd^2 sμ=

1

ε 0

∫

V

ρd^3 r=

1

ε 0

QV. (8.79)

Now letρbe a charge density with spherical symmetry, centered aroundr=0.
Then the electric field is parallel (or anti-parallel) torμ, thus it can be written as
Eμ=Êrμ.
Now the volume integration is performed over a sphere with radiusr, then one has
̂rμd^2 sμ=d^2 s, andEis constant on the surface of the sphere. The surface integral
of (8.79) yieldsEtimes the surface 4πr^2 of the sphere. Assuming that the charge
density is completely contained within this sphere and denoting the total charge by
Q, one obtains 4πr^2 E=ε^10 Q, and

Eμ=

Q

4 πε 0

1

r^2

̂rμ=

Q

4 πε 0

1

r^3

rμ. (8.80)

This is the electric fieldEμ(r)at the positionr, located in vacuum, caused by the

chargeQatr=0. A “test” chargeq, placed atr, experiences the forceF=qE(r).

Thus the force between these charges is theCoulomb force

F=

qQ

4 πε 0

1

r^3

r. (8.81)

The strength of the Coulomb force decreases with increasing distancerbetween the

charges like 1/r^2 , just like the gravitational force. Gravitation is always attractive.

The Coulomb force is repulsive or attractive, depending on whether the charges have

equal or opposite sign.