Coulomb’s Law

The Coulomb law or Coulomb law is the basis of electrostatics. It describes the force between two point charges or electrical charges distributed in a spherical symmetry. The magnitude of this force is proportional to the product of the two charge quantities and inversely proportional to the square of the distance between the ball centres. Depending on the sign of the charges, the force acts as an attraction or repulsion force in the direction of the connecting line of the centres. So in the gravitational case it behaves quite correspondingly like the force between two point masses according to the law of gravity.

For more than two charges, the individual force vectors are added according to the superposition principle.

Coulomb’s law is the basis of influenza.

Coulomb force

Coulomb’s law was discovered by Charles Augustin de Coulomb around 1785 and confirmed in extensive experiments. Within the international system of units, in scalar form and in vacuum, the force is therefore

   ball symmetrically distributed charge quantities

          Distance between the centres of the load quantities

     electrical field constant

vector shape

The general vectorial notation of discrete charges in any matter ()  is provided by the Coulomb force field, to which a test load  is subjected in the field of a second charge , as follows:

(i. e.: Power on the sample load , caused by the charge ; and: the vector from the point of charge to the point of the test load )

Here are the following  the unit vector obtained from (along the connecting line of both load centres) in direction  ; and  the position vectors of the two cargo centres. As you can see, charges of the same name, i. e. of the same sign, must repel each other according to the definition above, because the force  in such a case, the same orientation as  while charges with an unequal sign (unequal charges) attract each other, because the force  then (analogous to Newtonian gravity law) the opposite orientation of  possesses.

If the coordinate origin is moved to the position of the load  the equation above is simplified:

Next is then

the vector of the field strength of the field strength of the central charge  generated electric field at the place , i. e. at a distance  from the source. If the central charge generating the field is used  by a cloud of charges distributed in space with the charge distribution  replaces the formula given above for the Coulomb force on the test load  the integral.

The coulomb’s law in the form given at the beginning is included in this formula as a special case for a point-shaped charge distribution. Conversely, this more general form can also be derived from Coulomb’s law by means of the principle of superposition.

The term appearing in the above equations

is also known as the Coulomb constant.

In vacuum ()  applies:

where c is the speed of light.

In Gaussian units and in CGS units, the coulomb’s law is used to define the electrical charge. The electrical base unit of the unit systems SI, CGS-ESU and CGS-EMU differs in principle only by the specification of the following units: SI, CGS-ESU and CGS-EMU 

  • In the CGS-ESU there is . Therefore, the Coulomb constant in this unit system has the value .
  • Im CGS-EMU ist . Therefore, the Coulomb constant in this unit system has the value .

coulomb potential

As long as there is no temporal change in the magnetic field, the electric field is vortex-free and the energy difference during the transfer of a charge from A to B is therefore in this case independent of the actual distance travelled. Correspondingly, the electric field and the electrical force can also be described by a potential.

In the case of the simple Coulomb force, the Coulomb potential is obtained, which can be described for a single point charge Q as follows:

The integration constant C typically becomes zero, so that the potential disappears in the infinite. The potential difference between two points is the voltage drop U between these two points. The Coulomb potential applies only to stationary charges. For moving point charges, on the other hand, where magnetic fields also come into play, the Coulomb potential becomes a Liénard-Wiechert potential.

The potential electrical energy  ist ebenfalls ein Potential, nun bezüglich der elektrischen Kraft:

Here, too, it is customary to select the boundary condition in such a way that the potential energy becomes infinite zero,   is zero here too.

Coulomb force in a medium

Coulomb’s law can easily be extended to the case of charges in homogeneous, isotropic, linear media. The material surrounding the charges must have these properties in good approximation:

  • It is electrically neutral.
  • It fills the space between and around charges homogeneously.
  • The polarizability of the medium is direction-independent.
  • The polarization is proportional to the electric field generated by the charges.

In particular, homogeneity demands that the atomic character of matter is negligible compared to the distance between charges.

For such media, coulomb’s law is written in the same form as in a vacuum, with the only difference being that  by   is replaced:

The relative permittivity  is a material constant for isotropic media which takes into account the polarizability of the medium. It can be obtained both by measurements and from theoretical considerations.

Conversely, the following applies in vacuum .

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