what you already kno · 2007-09-12 · what you already know Îcoulomb’s law Îelectric fields...

PHY2049: Chapter 24 1

What You Already KnowCoulomb’s law

Electric fields

Gauss’ law

Electric fields for several configurationsPointLinePlane (nonconducting)Sheet (conducting)Ring (along axis)Disk (along axis)SphereCylinderDipole (along || and ⊥ axes)


Chapter 24: Electric Potential

Electric Potential Energy

Electric Potential

Equipotential Surfaces

Potential of Point Charge

Potential of Charge Distribution

Calculating the Field from the Potential

Potential Energy from a System of Charges

Potential of Isolated Charged Conductors


Reading Quiz: Chapter 24An equipotential surface is:

a) a surface where the electric field is constant b) always parallel to the electric fieldc) a surface where the potential is zerod) always perpendicular to the electric fielde) a surface where the electric field is zero


Reading Quiz: Chapter 24The volt is a unit of:

a) potential energyb) electric fieldc) potentiald) force


Reading Quiz: Chapter 24Electric potential is:

a) a scalar quantityb) a vector quantityc) can be either scalar or vector


Electric Work and Potential EnergyPoint charges q1, q2: Work moving charge q2 from A → B

Path independence: “conservative force”Define potential energy of two point charge

( )1 22

1 2 1 2 1 22

ˆ ˆB B

AB A A

BAB A A B

kq qW F ds r ds r ds drr

kq q kq q kq qW drr rr

= ⋅ = ⋅ ⋅ =

= = −

∫ ∫

∫• Depends only on endpoints• Path independent• Like gravitation

1 2B A AB

kq qU U W Ur

− ≡ − ⇒ =


Electric Force is ConservativeHolds in all electrostatic situations (not just point charge)

Proof: integrate over any charge distribution

Work done by electric field moving charge q from i to fCalculate from difference of potential energies

Chargesi f

elec fi i fW U U U= −Δ = −Work:

q


Problem: Electric Potential Energy Two identical +12 mC point charges are initially spaced 5 cm from each other. If they are released at the same instant from rest, how fast will they be moving when they are very far from each other? Assume m1 = m2 = 1.0 g.

i i f fK U K U+ = +

( )2 2

2120 2 0f f

i i

kq kqmv vd md

+ = + ⇒ =

( )( )

( )( )

295

3

9 10 0.0121.6 10 m/s

10 0.05fv

−

×= = ×


Gravitational & Electric Potential Energy Gravity

G A BhW mg U U= = − E A BdW qE U U= = −

Electric

d

+++++++

-------

A

B A B

Point B at lower potential energy than point A (q>0)

h


Electric Potential

Potential = PE per unit charge

Potential difference: general E field

Potential difference: constant E

Potential higher at + charges and “falls” to lower value at − charges

+q: Moves from higher to lower V−q: Moves from lower to higher V

/V U q=

a bV V Ed− =

bb a a

V V E ds− = − ⋅∫

b

a +++++++++++

-------------------

+QEd


Units for V and EUnits of potential: “volt”

V = U/q ⎯→ Volt = Joule / Coulomb

Units of electric fieldF = Eq ⎯→ E = F/q → Newton / CoulombV = Ed ⎯→ E = V/d → Volt / Meter


Example of Potential of Point ChargePoint charge q (using V = 0 at r = ∞)

Example: Potential at surface of proton (r = 10-15 m)

( )( )9 196

15

9 10 1.6 101.44 10 1.44 MV

10kqVr

−

−

× ×= = = × =

kqVr

=


Energy Units: Electron Volts

1 eV = energy of charge e accelerated through 1 Volt

Let q = 4e and V = 2000 V

( )19

19

1eV 1.6 10 C 1V

1.6 10 J

−

−

= ×

= ×

i

( )19 15

4 2000 8000eV 8keV

8000 1.6 10 1.28 10 J

K

K − −

= × = =

= × × = ×


ConcepTest: Electric Energy A proton and an electron are each accelerated across a region of constant E field. Which has larger acceleration?

(a) proton (b) electron (c) both have equal acceleration(d) neither one accelerates

F = Eea = F/m = Ee/mme mp Electron is much lighter than proton


ConcepTest: Electric EnergyWhich has the biggest increase in KE?

(a) proton (b) electron (c) both have the same increase in KE(d) KE = 0 for both

K = Fd = EedVe > Vp


Equipotential SurfacesEquipotentials: Contours of constant potential

No work to move charge along contour: W = -qΔV = 0

E ⊥ equipotential surfaceIf E⎥⎥ ≠ 0, would need work to move charge along surfaceSee http://www.falstad.com/emstatic/


Equipotential: Constant E Field

Example: CapacitorConstant E


Equipotential: Point Charge

Equipotentials


Equipotential: Dipole


Topographic Map:Equal Altitude Contours

Contour: Line of constant gravitational potential


Calculating E From Electric Potential VElectric field in terms of potential

x y z

x y z

x y z

dU F ds F dx F dy F dz

dV E ds E dx E dy E dz

V V VE E Ex y z

= − ⋅ = − − −

= − ⋅ = − − −

∂ ∂ ∂= − = − = −

∂ ∂ ∂

Divide by q


Example: Electric Field of Point ChargeGet E by differentiating potential

2 2ˆ, ,kq x y z kqE r

r r rr r⎛ ⎞= ≡⎜ ⎟⎝ ⎠

( ) ( )1/ 2 3/ 2 32 2 2 2 2 2

, etc.

x

y z

kq kqx kqxEx rx y z x y z

E E

∂= − = =

∂ + + + +

( )1/ 22 2 2r x y z= + +

Coulomb’s law

kqVr

=

what you already kno · 2007-09-12 · what you already know Îcoulomb’s law Îelectric fields...

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