** When a test charge q _{0} is placed
in an electric field E, the electric force acting on the test charge is q_{0}E.**

For an infinitesimal displacement ds the work done by the electric field on the charge is Fds = q_{0}Eds. As this work is done by the field, the potential energy of the field-charge system is reduced in a quantity dU = - q_{0}Eds.

For a charge displacement between points A and B, the change in potential energy of the system, U_{B} - U_{A} is

**The integration is performed along the path
described by q _{0}. This integral is called line integral.**

**The potential energy per unit charge U/q _{0}
does not depend on the value of q_{0}, and has a unique value in each
point in an electric field. U/q_{0} receives the name electric potential,
or simply potential, and is denoted by V, V = U/q_{0}. **

**The
potential difference V _{B} - V_{A} between any two points
A and B in an electric field is defined as the change in potential energy
divided by the test charge q_{0}:**

**The SI unit for the electric potential
(and for the potential difference) is joule/coulomb, a unit called a volt,
abbreviated V,
1 V = 1 J/C.**

**Electric Potential and Potential Energy
Due to Point Charges.**

**We know the electric field E due to a point
charge q is E = k _{e}q/r^{2}. Replacing in equation A), changing
ds by dr, we obtain**

Defining V

V = k

The electric potential energy associated to a pair of point charges separated by a distance r

U = Vq

**Electric Potential Energy Example.**

**Two non conductor spheres with 0.4 cm and
0.6 cm radius, 0.2 kg and 0.8 kg mass and -3 micro Coulombs and 4 micro Coulombs
charge, respectively, are released from rest when their centers are separated
0.9 m. Calculate each sphere speed when they collide. The charge is uniformly
distributed on each sphere.**

**The electric potential energy for point
charges is
U = k**

**Exercises, Potential and Electric Potential Energy**

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