66. Units of electric charge

The selection of the unit of electric charge, as well as the units of other physical variables, can be made at will. The point here is in expediency of this or that choice.

It is practically impossible to create a macroscopic standard of charge similar to the standard of length - meter, because of the inevitable leakage of charge. Naturally, it would be possible to take the charge of the electron as a unit of charge (it is done now in atomic physics). But at the time of Coulomb nothing was known about the existence of the electron in nature. Moreover, the charge of the electron is too small and therefore it is not always convenient to use it as a unit of charge.

It is possible to set a unit of charge using the law of Coulomb. This unit will be a derivative (not a main unit) and no reference is needed for it. Let's put the coefficient \(k\) in the formula \((8-1)\) \( F ~= ~k\frac{q_1 \,q_2}{r^2} \) equal to one. Then the unit of charge \((q)\) should be taken as a point charge, which acts in a vacuum on the same charge, located at a distance of \(1 \,cm\) with a force of \(1 \,din ~(10^{-5}N)\). After all at one unit distance, with one unit force and \(q_1 = q_2\) charges according to \((8-1)\) will be equal to one unit if \(k = 1\). The law of the Coulomb at this choice of unit of charge will be written as follows

\( F ~= ~\frac{q_1 \,q_2}{r^2} \)

The unit of charge installed in this way is called the absolute electrostatic unit of charge or, for short, the unit of charge of the CGSE. The letters "CGSE" indicate that the CGS system (centimetre-gram-second) is used as the basis, while the letter "E" reminds us that the unit of charge is set by the basic electrostatic law. The absolute value of the electron charge

In the international system of units (SI), the unit of charge is also derived. However, it is set in a different way. In the SI system along with meter, second and kilogram another main unit is introduced - the unit of electric current force - ampere. The standard value of ampere is set in the way described in the first book on physics. The unit of charge, the coulomb, is set using a unit of electric current. One coulomb \((q)\) - is a charge that passes for \(1 ~second\) through the cross-section of the conductor, which is a direct current of \(1 ~ampere\). The coefficient \(k\) in the law of Coulomb \((8-1)\) in this system of units is not only not equal to a one unit, but is called \(k_e\) Coulomb constant. Write it down as it is accepted

\( k_e ~= ~\frac{1}{4 \pi \,\varepsilon{_0}} \)

Magnitude \(\varepsilon{_0}\) is called the electric constant. Multiplier \(4\pi\) is introduced to simplify basic equations for an electromagnetic field.

Using the notation \(k_e \) let's write down the Coulomb law in SI units

\( F ~= ~\frac{1}{4 \pi \,\varepsilon{_0}} \cdot \frac{q_1 \,q_2}{r^2} \) (8-2)

In this formula the force is expressed in newtons, the distance in meters, and the charge in coulomb. The quantity of \(\varepsilon{_0}\) can be determined experimentally. It is enough to measure the charges, distance and force in the SI units. Then

\( \varepsilon{_0} ~= ~\frac{q_1 \,q_2}{4 \pi \,F \,r^2} \)

The resulting quantity \(\varepsilon{_0}\) equals

\( \varepsilon{_0} ~= ~\frac{1}{4 \pi \,\cdot 9 \,\cdot 10^9} ~\scriptsize{SI units}\)

Using the Coulomb law, we can find that the electric constant \(\varepsilon{_0}\) is measured in \(a^2 \cdot sek^4/kg \cdot m^3\).

Knowing the value of \(\varepsilon{_0}\), you can find how many units of charge CGSE are contained in one coulomb. Two charges by one coulomb interact in a vacuum with force

\( F ~= ~\frac{1}{4 \pi \,\varepsilon{_0}} \cdot \frac{1 \,q \,\cdot \,1 \,q}{1 \,m \,\cdot \,1 \,m } ~= ~9 \cdot 10^9 ~N\) (8-3)

On the other hand, these same charges interact with a force (calculated in the CGSE system) equal to

\( F ~= ~\frac{q^2}{10^4} ~\scriptsize{din} ~= ~q^2 \cdot 10^9 ~N\) (8-4)

since \(1 ~din ~= ~10^{-5} ~N\).

By equating the right parts \((8-3)\) and \((8-4)\), we will find

\(1 ~q ~= ~3 \cdot 10^9 ~\scriptsize units ~of ~charge ~CGSE\)

The charge of \(1 ~q\) is very high. Two such charges at a distance of \(1 km\) would be pushed away from each other with a force slightly less than one ton. Therefore to create at a small body (the size about meters) charge in one coulomb is impossible. Repulsing from each other, the charged particles would not be able to hold on the body. There are no other forces in nature that would be able to compensate for the coulomb repulsion under these conditions. But to set in motion in a conductor, which is generally neutral, the quantity of electricity equal to one coulomb is not difficult. After all, through a normal light bulb of \(100 ~{Watt}\) at \(127 ~{Volt}\) goes current, a little less than one ampere.