1. Prof. David R. Jackson ECE Dept. Spring 2016 Notes 14 ECE 3318 Applied Electricity and Magnetism 1

2. Potential Calculation from Field 2 In this set of notes we explore calculating the potential function  ( x,y,z ) (assuming that we already know the electric field). Note: The electric field is found first, either from Coulomb’s law (superposition) or from Gauss’s law.

3. Choose: Potential Formula 3 or Then we have:

4. Potential Formula (cont.) C r R 4 1) Pick a reference point (if it is not already given). 2) Integrate in a coordinate system of your choice (remember that any path can be chosen in statics). Recipe:

5. Path Independence of Voltage Drop Consider first a point charge: By superposition, the potential from any set of charges is path independent in statics. The integral is path independent (the answer only depends on the endpoints). 5 y z q Point charge x A C B

6. Example Given: R = ,  ( R ) = 0 Choose a radial path for convenience. Find the potential function y z q r R C Point charge x 6

7. Example (cont.) 7 Hence, we have

8. Summary for a Point Charge 8 y z q r x r These results for a point charge form the building blocks for all other charge distributions (using superposition). This is done in Notes 15.

9. Example Given: R is at  = b,  ( R ) = 0 Find the potential function Choose a radial path Infinite line charge 9 x y z  l0 [C/m] r R (  = b ) C  b

10. Example (cont.) Note: b cannot be chosen as  (there is an infinite voltage drop between  =  and  =  ). 10 In 2D problems (infinite in z ) the reference point cannot be put at infinity.

11. Example x y z a  l0 [C/m] r ( 0,0,z ) Given:  (  ) = 0 Note: We only know E on the z axis (from a previous example), so we MUST choose a path on the z axis. Find the potential function on the z axis Circular ring of line charge 11

12. Example (cont.)  (  ) = 0 C x y z r R (  ) Choice of path: 12

13. Example (cont.) From Coulomb’s law, we know that on the z axis the field is 13 so

14. Example (cont.) Hence, The result is: 14

15. Adding a Constant to  Proof: C1C1 R1R1 r C2C2 R2R2 15 Assume two solutions: Two different solutions for the potential function can only differ by a constant.

16. Adding a Constant to  (cont.) Conclusion: Valid potential functions can only differ by the addition of a constant. Adding a constant to a valid potential function gives a another valid potential function (this does not change the electric field). 16

17. Example Given :  (z = 1 [m]) = 10 [V] Find the potential function on the z axis Start with our previous solution, which has zero volts at infinity, and add a constant to it: Circular ring of line charge x y z a  l0 [C/m] r ( 0, 0, z ) R ( 0, 0, 1 ) 10 [ V ] 17

18. Example (cont.) Set: x y z a  l0 [C/m] r ( 0, 0, z ) R ( 0,0,1 ) 10 [ V ] 18 Hence we have: The solution is thus: