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The method of images is used to study the charge distribution for cases where Coulomb’s law deviates from the inverse square law. This method shows that in these cases some of the charge goes to the surface, while the remainder charge distributed over the volume of the conductor. In accord with the experimental work, we show that the charge distribution will depend on the photon rest mass and is very sensitive to it; a very small value of the rest of mass of the photon will create deviation from Coulomb’s law.

One of the foundations of electrostatics is Coulomb’s law. Major electromagnetic laws are built upon this law. As a direct consequence of this law (or its equivalent, Gauss’s law), any excess charge placed on a conductor must lie entirely on its surface. According to Coulomb’s law, excess charges given to a conductor will move away from each other and distribute themselves about the conductor in such a manner as to reduce the total amount of repulsive forces within the conductor and that both the charge and the field inside the conductor will vanish [

Testing this law has been a subject for many experiments over the past two and a half centuries [^{−22} and detect a photon rest mass at the level of 9 × 10^{−50} grams [

In an interesting papers, Spencer [

In this paper we introduce the method of images to study the distribution of charges in cases where the potential is depending on the photon rest of mass. And give a theoretical extension work to the experimental results that detect a photon rest mass at the level of 9 × 10^{−50} grams and as a result a deviating from Coulomb’s Law. This paper is also important to understand physics of molecules and electron transport through a single molecule which offers a highly promising new technology for the production of electronic chip.

The reaction field of a point charge due to surrounding medium can be represented by the method of image charge. The method of images allows us to solve certain differential form of electric potential problem without specifically solving a differential equation of this problem.

The potential

If the sphere is grounded, then the potential at the surface of the sphere vanishes

where

More generally, the potential in the neighborhood of an uncharged grounded conducting sphere is given by Equation (4):

Let

Substitute Equation (5) in Equation (4) and then differentiate to get the actual induced charge density on the surface of the grounded uncharged conducting sphere:

Let

Then we get:

The total charge on the sphere may be found by integrating Equation (6) over all angles. The total surface induced charge is equal to the magnitude of the image charge for Coulomb potential. But in case of Yukawa potential the total surface induced charge is less than the value of the image charge. This result implies that small portion of the induced charge distributed itself inside the volume of the conducting sphere. The rest of the induced charge is distributed itself on the surface of the conducting sphere. Some values of the total induced surface charge on grounded conducting sphere are given in

Potential | Parameters | ||||
---|---|---|---|---|---|

^{ } | Total surface charge | Total volume charge | |||

Coulomb | 0 | 2.0 | 0.50000q | 0.5 | 0.0 |

Yukawa | 0.008 | 2.0 | 0.49697q | 0.4932 | 0.0037 |

Yukawa | 0.5 | 2.0 | 0.34592q | 0.2267 | 0.1192 |

Yukawa | 1.0 | 2.0 | 0.23931q | 0.1087 | 0.1306 |

We can generalize Equation (4) for an insulated conducting sphere. Consider insulated charged sphere with total charge

The surface charge density:

where

where

The charge density given by Equation (8) in units of

In accord with the experimental work we show that the charge distribution greatly depends on the photon rest

mass and is very sensitive to it; a very small value of the rest of mass will create deviation from Coulomb’s law. We have studied the distribution of charges on grounded spherical conductor and insulated charged spherical conductor by using the method of images. It is proven that using the image charge to study the distribution of charges on conductors is effective. Our results show that the charge distribution is depending greatly on