Theory

1. Coulomb's law

In order to obtain both the magnitude and direction of the force on a charge, q1 at position r1, experiencing a field due to the presence of another charge, q2 at position r2, in free space, the Coulomb's law is stated as below.



where r is the separation of the two charges. This is simply the scalar definition of Coulomb's law with the direction given by the unit vector , parallel with the line from charge q2 to charge q1.

If both charges have the same sign (like charges) then the product q1q2 is positive and the direction of the force on q1 is given by ; the charges repel each other. If the charges have opposite signs then the product q1q2 is negative and the direction of the force on q1 is given by ; the charges attract each other.



The proportionality constant 1/4πε₀, called the Coulomb constant (k) (sometimes called the Coulomb force constant), is related to defined properties of space.
k = 1/4πε₀
ε₀=8.85x10^-12 C²/N-m² ( ε₀ is the "permittivity of free space")
hence k=9x10⁹


If both charges have the same sign (like charges) then the product q₁q₂ is positive and the direction of the force on q₁ is shown in above figure and the charges repel each other. If the charges have opposite signs then the product q₁q₂ is negative and the direction of the force on q₁ is shown in above figure and the charges attract each other.
force on q₂ due to q₁ is F₂₁ = -F₁₂ and it is consistent with Newtons 3^rd law.

In this experiment, initially F ~ 1/R² will be verified using graph :

On keeping the value of charges fixed, if value of R ,i.e. distance between 2 point charges is changed we get a graph which shows F ~ 1/R²

Hence it can be easily seen that F ~ 1/R²
After verifying F ~ 1/R², now F ~ q₁q₂ will be verified:

On keeping R, i.e. distance between 2 point charges constant, now if values of q₁ or q₂ or both are varied it can be seen from the graph that change in value of F will be according to the change in product of charges.

It can be seen that F ~ q₁q₂
Now concept of Principle of Superposition will be discussed

2. Principle of Superposition

Coulomb's law can be applied to any pair of point charges. When more than 2 charges are present, the net force on any one of the charge is the vector sum of forces exerted on it by other charges. For ex:
If 3 charges are present the net force experienced by q₃ due to q₂ and q₁ will be
F₃=F₁₃ + F₂₃

. For a system of N charges the net force by the j ^th particle would be


where F_ij denotes the force between particle i and j.

Example 1) Three charges are arranged as shown in the figure. Find force on the charge q₃assuming that q₁ = 6.0x10⁶ ,q₂=-q1, q₃= q₁/2 , and r=2.0x10^-2 m.


F₃ = F₁₃ + F₂₃


a_r13 = cosθ a_x + sinθ a_y,
a_r13 = (a_x +a_y )/2^1/2, 
a_r23=a_x

Hence F₃=3.0N

φ=tan^-1(F_3,y/F_3,x) =151.3⁰.