1. Electric Potential
AP Physics C

2. Conservative Forces and Energy
Remember from our lecture on work and energy that we defined a category of forces called conservative forces.
If a force is conservative, we can make a few claims:
There is a potential energy associated with the force.
The work done by the force is path independent.
We can express the work done by the conservative force as the negative change in potential energy:
𝑊 𝑐𝑜𝑛𝑠 =−∆𝑈
In addition, the total mechanical energy is conserved in the absence of non-conservative forces.

3. Gravitational Potential Energy
Remember back to gravitational potential energy. As an object falls to the Earth, gravity does positive work on the object.
As the object falls, the object gains kinetic energy and looses potential energy, meaning that our change in potential energy is negative.
𝑊 𝑔𝑟𝑎𝑣 =−∆ 𝑈 𝑔𝑟𝑎𝑣
We will make this analogous to a charged particle moving through an electric field.

4. Electric Potential Energy – Uniform Electric Field
Inside a parallel plate capacitor (two oppositely charged plates) there is an electric field present. If a positive charge is placed between the plates, there is a force exerted on the charge by the field. That force then does work on the charge.
𝐹=𝑞𝐸 𝑊=𝐹𝑑𝑐𝑜𝑠 𝜃 𝑊 𝑒𝑙𝑒𝑐 =𝑞𝐸𝑑
Because electric forces are conservative, we can determine the formula for electric potential energy.
𝑊=−∆𝑈 𝑞𝐸𝑑=−∆𝑈→𝑈= 𝑈 𝑜 −𝑞𝐸𝑑

5. Electric Potential Energy – Uniform Electric Field
As charges approach the plate with opposite charge, they loose potential energy and gain kinetic.
This is because the particles are being accelerated!

6. Electric Potential Energy – Point Charges
Consider two positive charges. If we wish to move the charges closer to each other, we must do work on the charges.
Just as we did with gravity, we need to choose a reference point, and we will again choose infinity (the potential at that point will be zero).
Because electric forces are conservative, we can integrate coulombs law to find the potential between these two charges.
𝑊=− 𝐹𝑑𝑟 =− ∞ 𝑟 1 4𝜋 𝜀 𝑜 𝑞 1 𝑞 2 𝑟 2 𝑑𝑟 =− 𝑞 1 𝑞 2 4𝜋 𝜀 𝑜 ∗ −1 𝑟 + 1 ∞ 𝑈 𝑒𝑙𝑒𝑐 = 1 4𝜋 𝜀 𝑜 𝑞 1 𝑞 2 𝑟

7. Electric Potential
Often we are not concerned so much with the electric potential energy of a system, but rather the electric potential. I know, it sounds like the same thing, but there is a slight difference.
Electric potential (often called potential difference or voltage) is the amount of electric potential energy per unit charge.
The SI unit for potential is the volt.

8. Electric Potential
A test charge q is used as a probe to determine the electric potential, but the value of V is independent of q.
The electric potential, like the electric field, is a property of the source charges, not the test charge.
𝑉 𝑝𝑜𝑖𝑛𝑡 𝑐ℎ𝑎𝑟𝑔𝑒 = 𝑈 𝑒𝑙𝑒𝑐 𝑞 𝑡𝑒𝑠𝑡 = 1 4𝜋 𝜀 𝑜 𝑄 𝑟
𝑈 𝑒𝑙𝑒𝑐 = 𝑞 𝑡𝑒𝑠𝑡 𝑉

9. Potential and Superposition
If you have multiple point charges, you can find the electric potential at different points in space by calculating the potential from each charge.
This means that electric potential obeys the principle of superposition!
Electric potential is a scalar value as well, so calculations are drastically easier than electric fields!
𝑉 𝑡𝑜𝑡𝑎𝑙 = 1 4𝜋 𝜀 𝑜 𝑄 𝑟

10. Example
Calculate the electric potential at the point indicated.

11. Electric Potential
Positive charges tend to move towards areas of lower potential, which is analogous to a ball falling to the earth.
Negative charges are the opposite, they tend to move towards higher potentials.
In both cases, the charge moves towards the area with an opposite voltage, converting potential energy into kinetic.

12. Electric Potential in a Parallel Plate Capacitor
We said that the electric potential energy of a point charge in a parallel plate capacitor is:
𝑈=𝑞𝐸𝑑
Using our idea of voltage, we can conclude that the potential difference between the plates in the capacitor is:
𝑉= 𝑈 𝑞 =𝐸𝑑
Again, notice that the potential difference is independent of the test charge.

13. Equipotential Lines and Electric Field Lines
Any charged object emits an electric field, and there is a potential difference associated with it.
We can see in these two pictures that there are lines along which the electric potential does not change. These are called equipotential lines.
You can move a charge along these lines and no work is done, because the potential energy does not change.
The electric field lines are always perpendicular to these equipotential lines.

14. Choice of Zero Potential
Because only differences in potential are relevant, the choice of zero potential is completely arbitrary.
The three scenarios below represent the same situation, even though their zeros are at different points.

15. Making Comparisons
We are able to relate force and potential energy (when the force is conservative), and we can do the same thing with electric fields and electric potential.
𝑊=− 𝐹𝑑𝑟 →V=− 𝐸𝑑𝑟
To find the potential difference between charged objects, we again choose a reference at infinity and integrate our electric field function.
𝑉=− ∞ 𝑟 𝐸𝑑𝑟

16. Gauss’s Law and Potential
Often we will have problems where you will need to find the electric field and electric potential as functions of position for some charged object.
Gauss’s law will allow us to find the field in all regions of space, and that field function can be integrated to find the potentials at different points in space.
We will often graph these fields and potentials as functions of r.