Electric Potential IIElectric Potential II

Physics 2102

Jonathan Dowling

Physics 2102 Physics 2102 Lecture: 09 MON 02 FEBLecture: 09 MON 02 FEB

Ch24.6-10

PHYS2102 FIRST MIDTERM EXAM!

6–7PM THU 05 FEB 2009

Dowling’s Sec. 2 in Lockett Hall, Room 6

YOU MUST BRING YOUR STUDENT ID!

The exam will cover chapters 21 through 24, as covered in homework sets 1, 2, and 3. The formula sheet for the exam can be found here:http://www.phys.lsu.edu/classes/spring2009/phys2102/formulasheet.pdf

THERE WILL BE A REVIEW SESSION 6–7PM WED 04 FEB 2009 in Williams 103

Conservative Forces, Work, and Potential Energy

€

W = F r( )∫ dr Work Done (W) is Integral of Force (F)

€

U = −WPotential Energy (U) is Negative of Work Done

€

F r( ) = −dU

dr

Hence Force is Negative Derivative of Potential Energy

Coulomb’s Law for Point Charge

P1P2

q1q2

P1P2

q2

€

F12 =kq1q2

r2

€

F12 = −dU12

dr

Force [N]= Newton

€

U12 =kq1q2

r

€

U12 = − F12dr∫

Potential Energy [J]=Joule

ElectricPotential [J/C]=[V]=Volt

€

V12 =kq2

r

€

V12 = − E12dr∫

€

vE 12 =

kq2

r2

ElectricField [N/C]=[V/m]

€

E12 = −dV12

dr

Electric Potential of a Electric Potential of a Point ChargePoint Charge

2

f P

i

RR

V E ds E ds

kQ kQ kQdr

r r R

∞

∞∞

=− ⋅ =− =

=− =+ =+

∫ ∫

∫

r r

Note: if Q were anegative charge,V would be negative

Electric Potential of Many Point Electric Potential of Many Point ChargesCharges

• Electric potential is a SCALAR not a vector.

• Just calculate the potential due to each individual point charge, and add together! (Make sure you get the SIGNS correct!)

q1

q5

q4q3

q2

∑=i i

i

r

qkV

r1

r2

r3r4

r5

P

Electric Potential and Electric Potential and Electric Potential Electric Potential

EnergyEnergyappU W q VΔ = = ΔappU W q VΔ = = Δ

+Q –QaWhat is the potential energy of a dipole?

+Q

• First: Bring charge +Q: no work involved, no potential energy.

• The charge +Q has created an electric potential everywhere, V(r) = kQ/r

–Qa

• Second: The work needed to bring the charge –Q to a

distance a from the charge +Q is Wapp = U = (-Q)V = (–Q)

(+kQ/a) = -kQ2/a• The dipole has a negative potential energy equal to -

kQ2/a: we had to do negative work to build the dipole

(electric field did positive work).

Positive Positive WorkWork

Negative Negative WorkWork

+Q +Qa

+Q –Qa

Potential Energy of A System of Potential Energy of A System of ChargesCharges

• 4 point charges (each +Q and equal mass) are connected by strings, forming a square of side L

• If all four strings suddenly snap, what is the kinetic energy of each charge when they are very far apart?

• Use conservation of energy:– Final kinetic energy of all four charges = initial potential energy stored = energy required to assemble the system of charges

+Q +Q

+Q +Q

Do this from scratch!

Potential Energy of A Potential Energy of A System of Charges: SolutionSystem of Charges: Solution

• No energy needed to bring in first charge: U1=0

2

2 1

kQU QV

L= =

• Energy needed to bring in 2nd charge:

2 2

3 1 2( )2

kQ kQU QV Q V V

L L= = + = +

• Energy needed to bring in 3rd charge =

• Energy needed to bring in 4th charge =

+Q +Q

+Q +QTotal potential energy is sum of all the individual terms shown on left hand side =

2 2

4 1 2 3

2( )

2

kQ kQU QV Q V V V

L L= = + + = + €

kQ2

L4 + 2( )

So, final kinetic energy of each charge =

€

kQ2

4L4 + 2( )

L

€

2L

ExampleExample

Positive and negative charges of equal

magnitude Q are held in a circle of

radius R.

1. What is the electric potential at the

center of each circle?

• VA =

• VB =

• VC =

2. Draw an arrow representing the

approximate direction of the electric

field at the center of each circle.

3. Which system has the highest electric

potential energy?

–Q

+Q

A

C

B€

k +3Q − 2Q( ) /r = +kQ /r

∑=i i

i

r

qkV

€

k +2Q − 4Q( ) /r = −2kQ /r

k +2Q−2Q( ) / r =0

UB =Q•Vb has largest un-canceled charge

Electric Potential of a Dipole Electric Potential of a Dipole (on axis)(on axis)

( ) ( )2 2

Q QV k k

a ar r

= −− +

What is V at a point at an axial distance r away from the midpoint of a dipole (on side of positive charge)?

a

r–Q +Q( ) ( )

2 2

( )( )2 2

a ar r

kQa a

r r

⎛ ⎞+ − −⎜ ⎟= ⎜ ⎟

⎜ ⎟− +⎝ ⎠

)4

(42

20

ar

Qa

−=

πε2

04 r

p

πε=

Far away, when r >> a:

V

p

Electric Potential on Electric Potential on Perpendicular Bisector of Perpendicular Bisector of

DipoleDipole

You bring a charge of Qo = –3C from infinity to a point P on the perpendicular bisector of a dipole as shown. Is the work that you do:

a) Positive?b) Negative?c) Zero?

a

-Q +Q

–3C

P

U = QoV = Qo(–Q/d+Q/d) = 0 

d

Summary:Summary:• Electric potential: work needed to bring +1C from infinity; units [V] = Volt

• Electric potential uniquely defined for every point in space -- independent of path!

• Electric potential is a scalar — add contributions from individual point charges

• We calculated the electric potential produced by a single charge: V = kq/r, and by continuous charge distributions :

V = ∫ kdq/r• Electric potential energy: work used to build the system, charge by charge. Use W = qV for each charge.