Discrete vs Continuous

So far as we have considered coulomb's law, we assumed charged particles, discrete entity with some amount of charge. Now we are going to consider more realistic case, continuous distribution of charges.

Distribution of charges means, some amount of charge spread over a region of space. If the region of space is

• 0 dimensional, then it is a point charge (charged particle).
• 1 dimensional, then it is linear distribution. 
• 2 dimensional, surface distribution. 
• 3 dimensional, volume distribution. 

1 dimensional case: This is the case of linear distribution of charge. If the charge is spread along a line with line charge density, $$\lambda (s)$$

the form of field equation would be, 

$$E = \dfrac{1}{4\pi{\epsilon_0}} \int_p \dfrac{\lambda (s)}{{r_{st}}^2} dl$$

2 dimensional case: This is the case of surface distribution of charge. If  the charge is distributed over a surface with surface charge density, $$\sigma (s)$$

the form of field equation would be, 

$$E = \dfrac{1}{4\pi{\epsilon_0}} \int_a \dfrac{\sigma (s)}{{r_{st}}^2} da$$

3 dimensional case: This is the case of volume distribution of charge. If  the charge is distributed over a volume with surface charge density, $$\rho (s)$$

the form of field equation would be, 

$$E = \dfrac{1}{4\pi{\epsilon_0}} \int_v \dfrac{\rho (s)}{{r_{st}}^2} dv$$

Field Concept #1

Coulomb's law says:

$$F = \dfrac{1}{4\pi{\epsilon_0}}. \dfrac{Q_t . Q_s}{{r_{st}}^2}$$

Just consider the test charge as unity. We are left with,

$$F = \dfrac{1}{4\pi{\epsilon_0}}. \dfrac{ Q_s}{{r_{st}}^2}$$

Making a slight change in notation ( replacing F with E ) we get, 

$$E = \dfrac{1}{4\pi{\epsilon_0}}. \dfrac{ Q_s}{{r_{st}}^2}$$

Conclusion: Mathematically speaking, this is what we call definition of  "Electric field".



  

Coulomb's law #1

Consider two charge particles. One is test particle another one is source particle.

Test particle: $$Q_t$$
Source particle: $$Q_s$$

Now assuming both of the charges are static, we can write according to coulomb's law:

$$F = \dfrac{k . Q_t . Q_s}{{r_{st}}^2}$$

Where ' k ' is the proportionality constant. In SI units value of  ' k ' is

$$8.9875517873681764 * 10^{-9} N . M^2 . C^{-2}$$

It is represented in SI units as

$$k = \dfrac{1}{4\pi{\epsilon_0}}$$

Where,

$${\epsilon_0}$$

is the permittivity of free space.