Chapter 29 (Electric Potential)

• apply concept of energy to electric phenomena

• use conservation of energy and electric potential energy to analyze motion of charged particles

• calculate potential of charge distributions

• chapter 30: relate electric field to electric potential

• chapter 31, 32: currents; circuits (practical applications of electric field and electric potential)

Lecture 21

Outline for today

• electric potential energy: uniform field; point charges; dipole

• electric potential

Mechanical Energy• analogy between gravitational and electric forces: inverse square

law; uniform field (near earth and in capacitor)

• For conservative forces (work done independent of path e.g. gravitational and electric):

Constant force,linear displacement:W = F̄ .!r̄ = F!r cos !In general:W =

!j (Fs)j !sj

!" sf

siFsds =

" sf

siF̄ .ds̄

!Emech = !K + !U = 0K = "Ki, Ki = 1

2miv2i

Potential energy U is interaction energy of system:!U = Uf ! Ui = !Winteraction forces

Electric Potential Energy in uniform field

• Analogy with gravitational field:

• Work done on charge by E

Wgrav = w!r cos 0! = mg (yi ! yf )!Ugrav = !Wgrav(i" f) # Ugrav = U0 + mgy

Welec = F!rcos0! = qE (si ! sf )..."

Potential energy of Point Charges I

• Analogy with mass-spring system:

• Work done by charge 1 on charge 2

Wsp =! xf

xiFdx = !k

! xf

xixdx... " Usp = 1

2kx2

Welec =! xf

xiF1 on 2dx = Kq1q2

! xf

xi

1x2 = Kq1q2

"! 1

xf+ 1

xi

#..."

(positive for like charges...)

Potential energy of point charges II

• charges released from rest will move in direction of decreasing potential energy (like charges move apart...)

• Two like charges shot towards each other reach minimum separation

• Two opposite charges shot apart...slow down...reach maximum...

Potential energy of point charges III

• Electric force is conservative: work done independent of path(potential energy can be defined): divide path into...

• : think of zero as “no interaction”

• negative energies: less energy that for infinitely far apart, at rest

• (bound system); (can escape, barely for E = 0)E1 < 0 E2 > 0

Uelec ! 0 as r !"

Potential Energy of Multiple Charges/Dipole• each pair counted once:

• work done as dipole rotates:

• energy diagram of dipole

dW+ = F̄+.ds̄ = F+ds cos !with ds = rd" =

!12d

"d" (pivot at center of dipole)...

Welec = !pE# !f

!isin"d" = pE (cos "f ! cos "i)..."

Uelec =!

i<jKqiqj

rij

Example• Two 1.0 g beads, each charged to +5.0 nC, are 2.0 cm

apart. A 2.0 g bead charged to -1.0 nC is exactly halfway between them. The beads are released from rest. What is the speed of the positive beads, when they are very far apart?

Electric Potential

• Analogy with E

• q does “separate out”:

• Units: 1 volt (V) = 1 J/C

• V depends only on sources: ability to have interaction if charge q “shows up”

• source charges exert influence via V: “ignore” sources once V is known

force on q = [charge q] ! [alteration of space by source]potential energy of q + sources =[charge q] ! [potential for interaction of sources]

V ! Uq+sources

q

Using Electric Potential

• If V is changing, particle slows down...

• potential difference (voltage): !V = Vf ! Vi

A proton with speed 2!10!5 m/s passes thru’ a region with potential di!erenceof 100 V. What is its final speed?Kf "Ki + q"V = 0 gives 1

2mv2f = 1

2mv2i " q"V #

vf =!

v2i "

2qm "V = 1.44! 10!5 m/s