PHYSICS 272 Electric & Magnetic Interactions Lecture 10 Electric Potentials [EMI 17.5-17.8]

For a conductor in equilibrium: The potential difference is zero between any two locations inside.

Potential Difference in Metal In static equilibrium What is E inside metal? E = 0 In static equilibrium the electric field is zero at all locations along any path through a metal. i f What is the potential difference (Vf – Vi)? The potential difference is zero between any two locations inside the metal, and the potential at any location must be the same as the potential at any other location. Is V zero everywhere inside a metal? No! But it is constant 3

Example: metal inserted into capacitor In static equilibrium Insert a 1 mm thick metal slab into the center of the capacitor. d =3 mm +Q1 -Q1 1 mm Metal slab polarizes and has charges +Q2 and -Q2 on its surfaces. What are the charges Q1 and Q2? Q2=Q1 E inside metal is zero  2000V/m Now we have 2 capacitors instead of one E inside metal slab is zero! V inside metal slab is zero! V = 4 V There is no “conservation of potential”! Charges +Q2 and –Q2 4

Clicker Question 1 300 V/m 0 V/m 300 V/m A B 0.02m 0.03m 0.04m What is VB-VA? 270 V -270 V -18 V 6 V -6 V 5

Clicker Question 1 300 V/m 300 V/m 0 V/m A B 0.02m 0.03m 0.04m D What is VB-VA? 270 V -270 V -18 V 6 V -6 V D 6

Clicker Question 2 A 7

Clicker Question 2 A 8

= Two adjacent regions with different fields From A to C, ∆V1 = -E1x(xC - xA) From C to B, ∆V2 = -E2x(xB - xC) From A to B, ∆V = ∆V1 + ∆V2 C In general, =

Example: x Q > 0, ∆V < 0 If xB → , VB → 0

Recall Physics I: The potential energy differences depend only on the initial and final states of a system and are independent of path.

Example: Different Paths near Point Charge 1. Along straight radial path: rf ri +q For final r greater than initial r, the change in potential is less than zero as expected since the path direction is the same as that of the electric field. Likewise, if we go from a larger r to a small r, the potential increases. 13

Example: Different Paths near Point Charge 2. Special case iA: AB: BC: + Cf: 14

Superposition Principle for Potential Cut the charge distribution into pieces Calculate the contribution to the potential due to each piece Use superposition to get the total potential Qi We typically pick reference point of potential at infinity to be 0: Potential due to point charge Q (at origin) at any given point ( ) in space:

Potential of a Uniformly Charged Ring Q Method 1: Divide into point charges and add up contributions due to each charge Superposition Principle for Electric Potential 16

Potential of a Uniformly Charged Ring Q Method 2: Integrate electric field along a path Note that we integrate from an initial z=infinity to a final z so that V represents the energy per unit charge required to move a point charge in from infinity to z. 17

Potential of a Uniformly Charged Ring Q What is V for z>>R ? Is it unexpected? The same as for a point charge! 18

Potential Inside a Uniformly Charged Hollow Sphere =0 Outside (r>R), looks like point charge 19

Potential Difference in an Insulator 1 2 3 4 5 Electric field in capacitor filled with insulator: Enet=Eplates+Edipoles Eplates=const (in capacitor) Edipoles=f(x,y,z) Edipoles,A Edipoles,B A B Travel from B to A: Edipoles is sometimes parallel to dl, and sometimes antiparallel to dl Situation in insulator is more complex than in metals. Polarized molecules contribute to net electric field A … inside ”capacitor” B … between two “capacitors” 20

Potential Difference in an Insulator Instead of traveling through inside – travel outside from B to A: Edipoles, average A B Effect of dielectric is to reduce the potential difference. 21

Dielectric Constant Electric field in capacitor filled with insulator: Enet=Eplates-Edipoles K – dielectric constant Alpha is small. 22

Dielectric Constant Inside an insulator: Dielectric constant for various insulators: vacuum 1 (by definition) air 1.0006 typical plastic 5 NaCl 6.1 water 80 strontium titanate 310 23

Potential Difference in Partially Filled Capacitor K -Q +Q Talk if time permits – skip with no consequences s A B x 24

Electric Potential Energy of Two Particles Fint q2 r12 q1 The potential energy of a pair of particles is defined as: 25