Today’s agenda: Electric potential of a charge distribution.

1. Today’s agenda: Electric potential of a charge distribution. You must be able to calculate the electric potential for a charge distribution. Equipotentials. You must be able to sketch and interpret equipotential plots. Potential gradient. You must be able to calculate the electric field if you are given the electric potential. Potentials and fields near conductors. You must be able to use what you have learned about electric fields, Gauss’ law, and electric potential to understand and apply several useful facts about conductors in electrostatic equilibrium.

2. Example 1: potential and electric field between two parallel conducting plates. Assume V0<V1 (so I have a direction to draw the electric field). Also assume the plates are large compared to their separation, so the electric field is constant and perpendicular to the plates. Also, let the plates be separated by a distance d. E V0 V1 d

3. y x E z dl V0 V1 d |V|=Ed The famous “Mr. Ed equation!*” I’ll discuss in lecture why the absolute value signs are needed. *2004, Prof. R. E. Olson.

4. E Important note: the derivation of d E d did not require rectangular plates, or any plates at all. It works as long as E is uniform and parallel or antiparallel to d. If is uniform but not parallel or antiparallel to , then use. E d

5. Example 2:A rod* of length L located along the x-axis has a total charge Q uniformly distributed along the rod. Find the electric potential at a point P along the y-axis a distance d from the origin. (*“rod” means “really thin”) y * P r dq=dx d Q dq x dx x =Q/L L *What are we assuming when we use this equation?

6. y A good set of math tables will have the integral: P r d Q dq x dx x L note: ln(a) – ln(b) = ln(a/b) Include the sign of Q to get the correct sign for V. What is the direction of V?

7. Example 3:Find the electric potential due to a uniformly charged ring of radius R and total charge Q at a point P on the axis of the ring. dq Every dq of charge on the ring is the same distance from the point P. r R P x x Q

8. Could you use this expression for V to calculate E? Would you get the same result as I got in Lecture 3? dq r R P x x Q You must also derive an equation for the potential at the center of a ring if you need it for homework! In the next slide I will show you how easy the derivation is. Homework hint: you must derive this equation in tomorrow’s homework! Include the sign of Q to get the correct sign for V.

9. Example 4:Find the electric potential at the center of a uniformly charged ring of radius R and total charge Q. dq R Every dq of charge on the ring is the same distance from the point P. P Q

10. Example 4: A disc of radius R has a uniform charge per unit area  and total charge Q. Calculate V at a point P along the central axis of the disc at a distance x from its center. dq Start with r′ P The disc is made of concentric rings. The area of a ring at a radius r′ is 2r′dr′, and the charge dq on each ring is (2r′dr′), where =Q/R2. x x R Q Each ring is a distance from point P.

11. dq r′ P x x R Q This is the (infinitesimal) potential for an (infinitesimal) ring of radius r′. On the next slide, just for kicks I’ll replace k by 1/40.

12. dq r′ P x x R Q

13. Could you use this expression for V to calculate E? Would you get the same result as I got in Lecture 3? dq r′ P x x R Q