1. Electric Potential Energy of the System of Charges
Potential energy of a test charge q0
in the presence of other charges
Potential energy of the system of charges
(energy required to assembly them together)
Potential energy difference can be equivalently described as a work done by external force required to move charges into the certain geometry (closer or farther apart).
External force now is opposite to
the electrostatic force

2. Electric Potential Energy of System
The potential energy of a system of two point charges
If more than two charges are present, sum the energies of every pair of two charges that are present to get the total potential energy

3. Electric potential is electric potential energy per unit charge
Finding potential (a scalar) is often much easier than the field
(which is a vector). Afterwards, we can find field from a potential
Units of potential are Volts [V]
1 Volt=1Joule/Coulomb
If an electric charge is moved by the electric
field, the work done by the field
Potential difference if often called voltage

4. Two equivalent interpretations of voltage:
Vab is the potential of a with respect to b, equals the work done
by the electric force when a UNIT charge moves from a to b.
2. Vab is the potential of a with respect to b, equals the work that must
be done to move a UNIT charge slowly from b to a against the
electric force.
Potential due to the point charges
Potential due to a continuous
distribution of charge
Finding Electric Potential through Electric Field

5. Some Useful Electric Potentials
For a uniform electric field
For a point charge
For a series of point charges

6. Potential of a point charge
Moving along the E-field lines means moving in the direction of decreasing V.
As a charge is moved by the field, it loses it potential energy, whereas if the charge is moved by the external forces against the E-field, it acquires potential energy

7. Negative charges are a potential minimum
Positive charges are a potential maximum

8. Positive Electric Charge Facts
For a positive source charge
Electric field points away from a positive source charge
Electric potential is a maximum
A positive object charge gains potential energy as it moves toward the source
A negative object charge loses potential energy as it moves toward the source

9. Negative Electric Charge Facts
For a negative source charge
Electric field points toward a negative source charge
Electric potential is a minimum
A positive object charge loses potential energy as it moves toward the source
A negative object charge gains potential energy as it moves toward the source

10. Unit: 1 Volt= 1 Joule/Coulomb (V=J/C)
Electron Volts
Electron volts – units of energy
1 eV – energy a positron (charge +e) receives when it goes
through the potential difference Vab =1 V
Unit: 1 Volt= 1 Joule/Coulomb (V=J/C)
Field: N/C=V/m
1 eV= 1.6 x J
Just as the electric field is the electric force per unit charge, the electrostatic potential is the potential energy per unit charge.

11. Examples
A small particle has a charge -5.0 mC and mass 2*10-4 kg. It moves from point A, where the electric potential is fa =200 V and its speed is V0=5 m/s, to point B, where electric potential is fb =800 V. What is the speed at point B? Is it moving faster or slower at B than at A? E A B F
In Bohr’s model of a hydrogen atom, an electron is considered moving around a stationary proton in a circle of radius r. Find electron’s speed; obtain expression for electron’s energy; find total energy.

12. Calculating Potential from E field
To calculate potential function from E field

13. When calculating potential due to charge distribution, we calculate potential explicitly if the exact distribution is known.
If we know the electric field as a function of position, we integrate the field.
Generally, in electrostatics it is easier to calculate a potential (scalar) and then find electric field (vector). In certain situation, Gauss’s law and symmetry consideration allow for direct field calculations.
Moreover, if applicable, use energy approach
rather than calculating forces directly
(dynamic approach)
Example: Solid conducting sphere
Outside: Potential of the point charge
Inside: E=0, V=const

14. Potential of Charged Isolated Conductor
The excess charge on an isolated conductor will distribute itself so all points of the conductor are the same potential (inside and surface).
The surface charge density (and E) is high where the radius of curvature is small and the surface is convex
At sharp points or edges  (and thus external E) may reach high values.
The potential in a cavity in a conductor is the same as the potential throughout the conductor and its surface

15. At the sharp tip (r tends to zero), large electric field is present even for small charges.
Corona – glow of air due to gas discharge near the sharp tip. Voltage breakdown of the air
Lightning rod – has blunt end to allow larger charge built-up – higher probability of a lightning strike

16. Example: Potential between oppositely charged parallel plates
From our previous examples
Easy way to calculate surface charge density
Remember! Zero potential doesn’t mean the conducting object has no charge! We can assign zero potential to any place, only difference in potential makes physical sense

17. Calculating E field from Potential
Remembering E is perpendicular to equipotential surfaces

18. Example: Charged wire We already know E-field around the wire
only has a radial component
Vb = 0 – not a good choice as it follows
Why so?
We would want to set Vb = 0 at
some distance r0 from the wire
r - some distance from the wire

19. Example: Sphere, uniformly charged inside through volume
Q - total charge
- volume density of charge
This is given that at infinity j =0