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Unit of Charge : Defintion, Meaning and Relation
CGS unit: Coulomb’s law, in CGS system is expressed as, F = \(\frac{q_1 q_2}{r^2}\). If we put F = 1 dyne, r = 1 cm, and q1 = q2 = q, we get from equation (2),
q2 = 1 or, q = ±1
Definition: If an electric repulsive force of 1 dyne arises between two similar point charges kept 1 cm apart in air, then each point charge is called a unit charge. This unit is known as 1 esu or 1 statcoulomb (statC).
SI: Coulomb’s law in SI for vacuum or air is expressed as
F = \(\frac{1}{4 \pi \epsilon_0} \cdot \frac{q_1 q_2}{r^2}\)
Now, putting F = 9 × 109 N, r = 1 m,
q1 = q2 = q and \(\frac{1}{4 \pi \epsilon_0}\) = 9 × 109 N ᐧ m2 ᐧ C-2 we get,
9 × 109 = 9 × 109 × \(\frac{q \times q}{(1)^2}\)
or, q2 = 1 or, q = ±1
Definition: If an electric repulsive force of 9 × 109 N arises between two similar point charges kept 1 m apart in vacuum or air, the amount of each charge is 1 C.
Relation between the units of charge :
1 C = 3 × 109 state (precisely, 2.99792458 × 109 esu)
Electromagnetic unit or emu is another unit of charge, defined in the discussions of electromagnetism. Its relations with the other units are,
1 emu of charge = 10 C = 3 × 1010 statC
In atomic physics, the charge of an electron or a proton is taken as the unit of measurment of charge. This is called electronic charge unit (e).
1 electronic unit of charge is equal to the charge of an electron or a proton.
1e = 4.8 × 10-10 statC = 1.6 × 10-19 C
±e is the least amount of charge that can exist in nature. No charge exists which is not an integral multiple of ±e.
Force between Multiple Charges: Superposition Principle
The principle of superposition: The total force acting on a charge due to a number of interacting charges around it is the vector sum of the individual forces acting on that charge due to each of the other charges. Individual forces are calculated from Coulomb’s law. The force with which two charges interact does not change by the presence of other charges.
Coulomb’s law gives the electrostatic force between two point charges. In this case it is assumed that other charges are not present near the two charges. If there are a number of interacting charges, the force on a particular charge may be obtained by the principle of superposition.
Suppose a system contains n number of point charges q1, q2, q3, ………, qn placed in vacuum or in air [Fig.]. So by the principle of superposition the total force acting on q1 exerted by all the other charges is given by,
\(\vec{F}_1\) = \(\vec{F}_{21}\) + \(\vec{F}_{31}\) + \(\vec{F}_{41}\) + ….. + \(\vec{F}_{n1}\)
Here, \(\vec{F}_{21}\) = force acting on q1 due to q2
\(\vec{F}_{31}\) = force acting on q1 due to q3
…………………………………………….
\(\vec{F}_{n 1}\) = force acting on q1 due to qn
According to Coulomb’s law,
\(\vec{F}_{21}\) = \(\frac{1}{4 \pi \epsilon_0} \frac{q_1 q_2}{r_{12}^2} \hat{r}_{12}\) ;
\(\hat{r}_{12}\) is the position vector of charge q1 with respect to q2.
Similarly,
Similarly, the total force acting on q2 exerted by all the other charges is given by,
In general, if \(\vec{F}_i\) be the total force on the i th charge of the con-figuration, then,
Here, \(\hat{r}_{j i}\) is the position vector of the charge qi with respect to qj.
In CGS system the above equation is n