1. AP Physics ST Electric Potential and Potential Energy Due to a Point Charge, “E from V” www.ccp1.ac.uk

2. Electric Potential Due to a Point Charge Recall that every location on an equipotential surface has the same electric potential. – Meaning NO work is needed to move along that specific surface. www.pstcc.edu

3. Electric Potential Due to a Point Charge We’ve defined the electric potential due to a uniform electric field as… But what is creating this field?... (a charge) So now we endeavor to define the electric potential (and thus energy) associated with the charge itself.

4. Electric Potential Due to a Point Charge Consider a point charge with equipotential surfaces identified… – Electric field lines would necessarily be perpendicular to these surfaces at all points. Consider the movement from point A to point B along any path. +q A B rArA rBrB r ds

5. Electric Potential Due to a Point Charge Because you are traveling between two different equipotential surfaces the electric potential changes. To determine what this change depends on focus on a differential element (ds) of the motion along the path between points A and B highlighted in orange. +q A B rArA rBrB r ds

6. Electric Potential Due to a Point Charge Note that theta is the angle between vectors r hat and ds. What then is the electric potential V a distance r from the source charge? ds r dr θ

7. Electric Potential Due to a Point Charge Begin with the general electric potential expression… – Recall the negative sign means that B is at a lower potential than A. Recall from Gauss’s Law:

8. Electric Potential Due to a Point Charge Recall that is a unit vector…magnitude of 1 thus showing direction only. represents the angle between and ds. represents the component of r hat in the direction of ds projection of ds onto r (both vectors)

9. Electric Potential Due to a Point Charge We only care about the change in radial displacement because we already know that movement along the equipotential surface does not require work … meaning no change in energy… meaning no change in electric potential. radial displacement

10. Electric Potential Due to a Point Charge The guts of the integral thus simplifies to … Further simplification…

11. Electric Potential Due to a Point Charge Following integration… – Notice the negative sign cancels out resulting from integration. Potential difference between 2 pts is independent of the path taken! – This is a proof that the electric field is a conservative field!

12. Electric Potential Due to a Point Charge Potential difference depends ONLY on the radial components of A and B. It is quite typical for point A to be located at infinity thus defining the zero level.

13. Electric Potential Due to a Point Charge The Electric Potential due to a Point Charge is therefore… The awesome thing is that electric potential is a scalar quantity! – Speaks only to the magnitude of energy per charge at any location (not direction).

14. Electric Potential Due to a Point Charge Sooooo… if multiple charges exist you can just add ‘em!! (superposition principle)

15. Electric Potential ENERGY Due to a Point Charge Consider the potential energy involved (work) to bring q 2 from infinity to location r 12 through electric potential V 1 To bring q 2 to this location WITHOUT accelerating requires work by an external force. q1q1 q2q2 r 12 from ∞ to this location P V1V1

16. Electric Potential ENERGY Due to a Point Charge The magnitude of work done (thus energy transferred) is equal to the magnitude of charge q 2 moved to the location where the electric potential is V 1 Thus the energy at point r 12 resulting from two point charges is described by….

17. Electric Potential ENERGY Due to a Point Charge If the charges are the same: – +U  external force must do positive work to bring two charges together (without accelerating) If charges are opposite: – -U  external force must do negative work to bring two charges together (without accelerating).

18. Electric Potential ENERGY Due to a Point Charge This is SOOOOO cool… – It says that when brought from infinity THIS is the magnitude of work that was done! – This expression acknowledges the fact that the electric potential varies with distance but depends ONLY on the beginning and ending radial location points. – Further simplified by setting the zero level at infinity.

19. Electric Potential ENERGY Due to a Point Charge Multiple charges present???... Total energy of the system of charges… q1q1 q3q3 q2q2 r 12 r 13 r 23

20. Obtaining E from V To max out the level of awesimity (yes, it is a word and yes this really is that cool!) we can even determine the magnitude of the electric field if we know JUST the potential… … I knooowwww, too cool! Seeeeee… there’s the electric field!

21. Obtaining E from V The crux is too know that V is a scalar quantity and E is a vector quantity so one must note that there is a directional component for the E. To isolate E begin by differentiating both sides…

22. Obtaining E from V Recognize that E has an x-, y- and z-component, thus allow the “ds” to reflect the appropriate direction by changing variables. – Dot-product simplifies because E is specifically along the identified axis

23. Obtaining E from V x-component y-component: z-component:

24. Obtaining E from V “Full circle”… – Recall that the electric potential due to a point charge… – To determine the E… – This should be a familiar result for the electric field of a point charge!

25. Lesson Summary (THINK! What are the BIG, MAIN, GLOBAL lesson ideas?)

27. Example #1: 87-1cd, (a,b optional)

28. Example #2: 25-37 Serway 5 th ed Over a certain region of space, the electric potential is V = 5x – 3x 2 y + 2yz 2. a.Find the expressions for the x, y, and z components of the electric field over this region. b.What is the magnitude of the field at the point P, which has coordinates (1,0,-2)m?

29. Example #3 Two point charges, q 1 = +5µC and q 2 = -2µC, are separated by 50 cm. Where along a line through both charges, is V = 0?