Physics 202, Lecture 4

Today’s Topics

!! More on Gauss’s Law

!! Electric Potential

!! Electric Potential Energy and Electric Potential

!! Electric Potential and Electric Field

Gauss’s Law: Review

!E =!Eid!A"" =

qencl

#0

Use it to obtain E field for highly

symmetric charge distributions.

Method: evaluate flux over carefully chosen “Gaussian surface”

spherical cylindrical planar

(point charge, uniform

sphere, spherical shell,…) (infinite uniform line of

charge or cylinder…) (infinite uniform sheet

of charge,…)

Example: Thin Spherical Shell

Find E field inside/outside a uniformly charged thin

spherical shell, surface charge density !, total charge q

Gaussian Surface

for E(outside)

Gaussian Surface

for E(inside)

Er<a = 0

Er>a =

q

4!"0r2

qencl = 0qencl = q Trickier examples

(holes, conductors,…)

Gauss’s Law: Examples

Planar symmetry. Example: infinite uniform sheet of charge.

x

y

!E indep of x,y, in z direction

z

Gaussian surface: pillbox, area of faces=A

!Eid!A = 2EA =

qencl

!0

"" =#A

!0

!E =

!

2"0

z!

Examples: multiple charged sheets, infinite slab

Electric Potential Energy and Electric Potential

Kinetic Energy (K). Potential Energy U: for conservative forces (can be defined since work done by F is path-independent)

If only conservative forces present in system, conservation of mechanical energy: constant

Conservative forces:

"! Springs: elastic potential energy

"! Gravity: gravitational potential energy

"! Electrostatic: electric potential energy (analogy with gravity)

Nonconservative forces

"! Friction, viscous damping (terminal velocity)

Review: Conservation of Energy (particle)

K =1

2mv

2

U(x, y, z)

K +U =

U = kspringx22

Electric Potential Energy Given two positive charges q and q0: initially very far apart,

choose

To push particles together requires work (they want to repel).

Final potential energy will increase!

If q fixed, what is work needed to move q0 a distance r from q?

Note: if q negative, final potential energy negative

Particles will move to minimize their final potential energy!

q q0 U

i= 0

q q0

q q0 !!

!

Ur "U# = "! F d! l

#

r

$ = " kq0q

% r 2

d % r =#

r

$ kq0q

r

!

"U =Uf #Ui = #"W

Electric Potential Energy

Electric potential energy between two point charges:

"! U is a scalar quantity, can be + or -

"! convenient choice: U=0 at r= "

"! SI unit: Joule (J)

Electric potential energy for system of multiple charges:

sum over pairs:

r

q0

q

U(r) =kq

0q

r

U(r) =kqiqj

rijj

!i< j

!

This is the work required to assemble charges.

Electric Potential

Charge q0 is subject to Coulomb force in electric field E.

Work done by electric force:

independent of q0

Electric Potential Difference:

Often called potential V, but meaningful only as potential difference

Customary to choose reference point V=0 at r = !

(OK for localized charge distribution)

Units: Volts (1 V = 1 J/C)

!

"U = #"W = #! F d! l = #q

0i

f

$! E d! l

i

f

$

!

"V #"U

q0

= $! E d! l

i

f

%

Electric Potential and Point Charges

For point charge q shown below, what is VB-VA ?

A

B

rA

rB

q

VB !VA = ! E(r)dr = !A

B

" kqdr

r2

A

B

" = kq1

rB!1

rA

#

$%&

'(

Many point charges: superposition

V (r) =kq

r

V (r) = kqi

rii

!

Potential of point charge:

independent of path b/w A and B!

Equipotential Lines

Equipotentials:lines of constant potential

Electric Potential: Continuous Distributions

Two methods for calculating V:

1. Brute force integration (next lecture)

2. Obtain from Gauss’s law and definition of V:

V (r) = !!E(r)id

!l

ref

r

"

dV = kdq

r,V = dV!

Obtaining the Electric Field From

the Electric Potential

#! Three ways to calculate the electric field

"! Coulomb’s Law

"! Gauss’s Law

"! Derive from electric potential !

#! Formalism

Ex= !

"V

"x , E

y= !

"V

"y ,E

z= !

"V

"z or

!E = !#V

!

"V = #! E

A

B

$ d! l

!

dV = "! E d! l = "Exdx " Eydy " Ezdz

A Picture to Remember

$! Field lines always point towards lower electric potential

$! In an electric field:

"! positive charges are always subject to a force in the

direction of field lines, towards lower V

"! negative charge is always subject a force in the

opposite direction of field lines, towards higher V

E

higher V lower V

+q

-q