The force of attraction/repulsion between two point charges is given by Coulomb's Law:

If the charges are of like sign, then there well be a repulsive force between the two. Alternatively, if the net force is positive, it is repulsive; if it is negative, it is attractive.

Therefore, the force of repulsion between the two charges is:

The force of attraction/repulsion between two point charges is given by Coulomb's Law:

If the charges are of like sign, then there well be a repulsive force between the two.

Work is given as the dot product of force and distance. However, in this case, force is also dependent on distance.

The amount of work required to move a charge an incremental distance, , is given as:

The negative sign in this case is to account for repulsion.

The total work to change distances between charges can then be found by taking the integral with respect to distance:

Since  are constants, they can be factored out of the integral:

Coulomb's law in vector notation is given as:

, where  is Coulomb's constant,  and  are the two charges,  is the distance between the charges squared, and  is the unit vector going from one charge to another. 

To write this in vector notation, we have to know the unit vector going from the negative to the positive charge, since we're trying to determine the force on the positive charge. Since they are both sitting on the -axis, with the negative charge to the left of the positive, the unit vector will be going in the direction of positive :

We know that  

We know that , , and . Putting this together:

We can rewrite this as:

To determine this, we have to solve for  in Coulomb's law and then determine its constants. 

Recall that the magnitude of the electrostatic force between point charges is given as:

,  is the force given in ,  and  are the charges given in  and  is distance given in 

Solving for ,

Writing out the terms on the left in their units:

Therefore,  is given in 

Coulomb's law shows that the force between two charged particles is inversely proportional to the square of the distance between the particles.

If the distance between the charges is reduced by , that means the  is squared in the denominator and the  will flip up to the top to give  time the original force. More explicitly, if we plug in the given information the initial force will be:

Use Coulomb's law:

, where  is Coulomb's constant,  are charges of the two points and  is the distance between the charges. 

The work done by an electric field on a charged particle is:

Where  is the angle between the electric field vector and the displacement vector. Plug in the given values.

This answer makes sense since an electric field does work on a positive charge when going in the direction of the field lines.

The electrostatic force between two charges is:

The force that the 3rd charge feels from the  charge is:

The force that the 3rd charge feels from the  charge is:

We can also rewrite the distance between the charges as  and . In this explanation, we will use  as the distance between the  and the third unknown charge making the distance between the  and the 3rd charge . Finally, we solve for a net force of 0 giving us this relationship between the two opposing forces.

Solve for .

Charge is fundamentally conserved. In order to give the glass rod a positive charge, the silk cloth removes electrons from the glass rod. Every electron that leaves the glass rod leaves behind the proton (positively charged particle) that used to balance the negative charge of the electron (negatively charged particle). Every electron that leaves the glass rod goes onto the silk cloth, giving it the same amount of charge as the glass rod, but with a negative sign (since electrons are negatively charged). Only a tiny fraction of the total electrons in the glass rod are removed, even with vigorous charging.

Remember that the formula for the electric potential is given by:

Where  is Coulomb's constant,  is the source charge, and  is the distance. This formula indicates that electric potential is unaffected by changing the test charge.