Two point charges q_{1} = q_{2} = 10^{6} C are located respectively at coordinates (1, 0) and (1, 0) (coordinates expressed in meters). Calculate:

 The electric potential due to the charges at both point A of coordinates (0,1) and B (0,1).
 The work done by the electric force to move the electric charge q_{0} = – 2 10^{9} C from point A to point B.
 If the charge q_{0} were initially at rest at point A, would it get to point B?
 What would be the work done by the electric field to bring the charge q_{0} from infinity to point A? If q_{0} has a mass m = 10^{10} kg and it is initially at rest, determine the speed it will reach at point A.
Givens: k = 9 10^{9} N m^{2}/C^{2}
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Solution:
We are going to see step by step how to calculate the electric potential at any point P in space due to multiple charges. We will also see how to relate this potential with the electrostatic potential energy of a point charge.
You can see how to calculate step by step the electric field due to the charges q_{1} and q_{2} here.
First, we will represent the charges and points A and B in a Cartesian coordinate system.
The electrostatic potential due to multiple charges at any point is the sum of the individual electrostatic potentials due to each charge at this point.
We will now calculate the distances r (they are all the same in this problem) between the charges and the points A and B:
After substituting the value of the charges, of k and the distances in the potential expressions we obtain:
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The work done by the electrostatic force to move a point charge from a point A to a point B is given by:
And as in this case the electrostatic potential has the same value at point A and point B, this work is zero.
The work done by the electric force is equal to the change in the kinetic energy of q_{0}:
The first member of the previous equation is zero, so the speed of the charge cannot change between points A and B. Therefore, the charge q_{0} cannot reach point B and will stay at rest at point A.
The work done by the electric force to bring the charge q_{0} from infinity to the point A is:
The electrostatic potential at infinity is set to be zero:
Therefore the work is:
And after relating it with the change of the kinetic energy of q_{0}, we get:
Check the units of measurement page to know more about the prefixes used in Physics to express the multiples and submultiples of the SI units.
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