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Defintion, Formula and Vector Form of Coulomb’s Law
We know that two like charges repel each other and unlike charges attract each other. The force between two charges is given by Coulomb’s law.
This law is valid
- only for point charges, i.e., when the sizes of the charged bodies are negligible in comparison to their distance of separation,
- only for distances greater than 10-15m (nuclear distance) and
- for charges at rest with respect to the observer.
Coulomb’s law: The force of attraction or repulsion between two point charges at rest is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
The force acts along the line joining the two charges. It depends on the nature of the intervening medium.
Suppose two point charges q1 and q2 are at rest at a distance r from each other [Fig.]. If F is magnitude of the electrostatic force acting between them, then according to Coulomb’s law,
F ∝ q1q2, when r is constant
and F ∝ \(\frac{1}{r^2}\), when q1 and q2 are constants
∴ F ∝ \(\frac{q_1 q_2}{r^2}\), when q1, q2 and r are variables
or, F = k’\(\frac{q_1 q_2}{r^2}\) …….. (1)
where k’ is a constant of proportionality. Its value depends on the nature of the intervening medium and also on the units in which the quantities F, q1, q2 and r are measured. Sometimes the constant k’ is referred to as electrostatic constant or Coulomb constant.
i) In CGS system,
F = \(\frac{q_1 q_2}{r^2}\) (for vacuum or air) …… (2)
ii) In SI, For vacuum or air, k’
k’ = \(\frac{1}{4 \pi \epsilon_0}\)
where ε0 is the permittivity of vacuum or air or permit-tivity of free space.
Hence, F = \(\frac{1}{4 \pi \epsilon_0} \cdot \frac{q_1 q_2}{r^2}\) (for vacuum of air) ……… (3)
i.e., ε0 = \(\frac{1}{4 \pi} \cdot \frac{q_1 q_2}{F r^2}\) ….. (4)
In this system charge is expressed in coulomb (C), force in newton (N) and distance in metre (m).
In this case ε0 = 8.854 × 10-12 C2 . N-1 . m-2
∴ \(\frac{1}{4 \pi \epsilon_0}\) = 9 × 109 N . m2 . C-2
Hence, from equation (3) F = 9 × 109 \(\frac{q_1 q_2}{r^2}\) newton ……… (5)
For any other intervening medium,
F = \(\frac{1}{4 \pi \epsilon} \cdot \frac{q_1 q_2}{r^2}\) …… (6)
where ε is the permittivity of the medium.
Again, ε = Kε0, where K is the relative permittivity or dielectric constant of the medium.
In CGS system, ε0 = 1, so, ε = K , i.e., permittivity of a medium is numerically equal to its dielectric constant. But permittivity has the unit statC2 ᐧ dyn-1 ᐧ cm-2. For example, dielectric con-stant of mica is 5.4 and permittivity of mica is 5.4 stateC2 ᐧ dyn-1 ᐧ cm-2.
But in SI, the value of ε0 is 8.854 × 10-12C2 ᐧ N-1 ᐧ m-2 and hence the magnitudes of the dielectric constant and the permit-tivity of a medium are not the same.
So, in SI, permittivity of mica,
ε = kε0 = 5.4 × 8.854 × 10-12C2 ᐧ NT-1 ᐧ m-2
= 4.78 × 10-11 C2 ᐧ N-1 ᐧ m-2
Note that, permittivity of any medium is greater than that of vacuum. So the force of attraction or repulsion between two charges in any medium is less than that in vacuum. Only dry air has experimentally been observed to have almost the same permittivity (ε0) as that of vacuum.
Thus, the general formula of Coulomb’s law for any medium is given by
F = \(\frac{1}{4 \pi \kappa \epsilon_0} \cdot \frac{q_1 q_2}{r^2}\) …… (7)
From equation (4) we come to know the dimension of ε0,
It may be noted that in static electricity Coulomb’s law is an analogue of Newton’s law of gravitation. Of course in case of gravitation, this force is always attractive in nature and the proportionality constant (G) is universal.
Vector form of Coulomb’s law: Suppose, \(\vec{r}_1\) and \(\vec{r}_2\) are the position vectors of charges q1 and q2, respectively [Fig.].
So the force acting on q2 due to q1,
……… (8)
where \(\vec{r}_{21}\) is the position vector of q2 with respect to q1, i.e., \(\vec{r}_{21}\) = \(\vec{r}_2\) – \(\vec{r}_1\) = \(r_{21} \hat{r}_{21}\)
Similarly, the force acting on q1 due to q2,
Each of the equations (8) and (9) represent the vector form of Coulomb’s law.
As \(\vec{r}_{12}\) = –\(\vec{r}_{21}\), and \(\hat{r}_{12}\) = –\(\hat{r}_{21}\). we have \(\vec{F}_{12}\) = –\(\vec{F}_{21}\).
Comparison between electrostatic force and grav-itational force: Both electrostatic and gravitational forces act between two charged bodies. Similarities and dissimilarities between these two forces are given below.
Similarities:
- Both the forces are inversely proportional to the square of the distance between the two bodies.
- Both the forces are conservative, i.e., work done by these forces are independent of path.
- Both the forces act in vacuum as well.
- Both the forces are central force where the force is expressed as \(\vec{F}\) = -F(r)\(\hat{r}\).
- Both the forces are mutually interactive, i.e., \(\vec{F}_{21}\) = –\(\vec{F}_{12}\).
- Both the forces are spherically symmetric [function of r only].
Dissimilarities:
Gravitational Force | Electrostatic Force |
1. This force is very weak. | 1. This force is remarkably stronger, relative strength being of the order of 1038 |
2. This force is always attractive in nature. | 2. According to the nature of the charges, this force is either attractive or repulsive. |
3. This force does not depend on the nature of the intervening medium. | 3. This force depends on the nature of the intervening medium. |