January 17, 07

Physics 2102Physics 2102Lecture 2Lecture 2

Electric FieldsElectric FieldsCharles-Augustin de Coulomb (1736-1806)

Physics 2102

Jonathan Dowling

Version: 1/17/07

What are we going to learn?What are we going to learn?A road mapA road map

• Electric charge Electric force on other electric charges Electric field, and electric potential

• Moving electric charges : current• Electronic circuit components: batteries, resistors, capacitors• Electric currents Magnetic field

Magnetic force on moving charges• Time-varying magnetic field Electric Field• More circuit components: inductors.• Electromagnetic waves light waves• Geometrical Optics (light rays).• Physical optics (light waves)

2q!

12F

1q+

21F

12r

CoulombCoulomb’’s laws law

2

12

2112

||||||

r

qqkF =

2

212

00

1085.8with 4

1

mN

Ck

!"== #

$#

2

2

91099.8

C

mN!k =

For charges in aVACUUM

Often, we write k as:

Electric FieldsElectric Fields

• Electric field E at some point inspace is defined as the forceexperienced by an imaginarypoint charge of +1 C, divided by1 C.

• Note that E is a VECTOR.• Since E is the force per unit

charge, it is measured in units ofN/C.

• We measure the electric fieldusing very small “test charges”,and dividing the measured forceby the magnitude of the charge.

2

||||R

qkE =

+1 C

R

Eq

Electric field of a point charge

SuperpositionSuperposition

• Question: How do we figure out the fielddue to several point charges?

• Answer: consider one charge at a time,calculate the field (a vector!) produced byeach charge, and then add all the vectors!(“superposition”)

• Useful to look out for SYMMETRY tosimplify calculations!

Example

• 4 charges are placed at thecorners of a square as shown.

• What is the direction of theelectric field at the center ofthe square?

(a) Field is ZERO!(b) Along +y(c) Along +x

-q

-2q

+2q

+q

y

x

Total electric field

Electric Field Lines• Field lines: useful way to

visualize electric field E• Field lines start at a

positive charge, end atnegative charge

• E at any point in space istangential to field line

• Field lines are closerwhere E is stronger

Example: a negative pointcharge — note sphericalsymmetry

Electric Field of a DipoleElectric Field of a Dipole• Electric dipole: two point

charges +q and –q separatedby a distance d

• Common arrangement inNature: molecules, antennae,…

• Note axial or cylindricalsymmetry

• Define “dipole moment”vector p: from –q to +q, withmagnitude qd

Cancer, Cisplatin and electric dipoles:http://chemcases.com/cisplat/cisplat01.htm

Electric Field ON axis of dipoleElectric Field ON axis of dipole

!++= EEE :ionSuperposit

2

2!"#

$%& '

=+

ax

kqE

2

2!"#

$%&

+

'='a

x

kqE

!!!!

"

#

$$$$

%

&

'()

*+,

+

-

'()

*+, -

=22

2

1

2

1

ax

ax

kqE 22

2

4

2

!!"

#$$%

&'

=

ax

xakq

Pa x

-q +qa x

-q +q

Electric Field ON axis of dipoleElectric Field ON axis of dipole

22

2

22

2

4

2

4

2

!!"

#$$%

&'

=

!!"

#$$%

&'

=

ax

kpx

ax

xakqE

What if x>> a? (i.e. very far away)

p = qa“dipole moment”-- VECTOR

- +

34

22

x

kp

x

kpxE =!

E~p/r3 is actually true for ANY point far from a dipole(not just on axis)

3r

pE

rr!

Electric Dipole in a Uniform FieldElectric Dipole in a Uniform Field• Net force on dipole = 0;

center of mass stays whereit is.

• Net TORQUE τ: INTOpage. Dipole rotates to lineup in direction of E.

• | τ | = 2(QE)(d/2)(sin θ)= (Qd)(E)sinθ

= |p| E sinθ = |p x E|• The dipole tends to “align”

itself with the field lines.• What happens if the field is

NOT UNIFORM??

Distance between charges = d

Electric charges and fieldsElectric charges and fieldsWe work with two different kinds of problems, easily confused:

• Given certain electric charges, we calculate the electric fieldproduced by those charges(using E=kqr/r3 for each charge) Example: the electric field produced by a single charge, or by a dipole:

• Given an electric field, we calculate the forces applied by thiselectric field on charges that come into the field, using F=qE

Examples: forces on a single charge when immersed in the field of a dipole,torque on a dipole when immersed inan uniform electric field.

Continuous Charge DistributionContinuous Charge Distribution

• Thus far, we have only dealtwith discrete, point charges.

• Imagine instead that a chargeQ is smeared out over a:– LINE– AREA– VOLUME

• How to compute the electricfield E??

Q

Q

Q

Q

Charge DensityCharge Density

• Useful idea: charge density• Line of charge:

charge per unit length = λ• Sheet of charge:

charge per unit area = σ• Volume of charge:

charge per unit volume = ρ

λ = Q/L

σ = Q/A

ρ = Q/V

Computing electric fieldComputing electric field of continuous charge distribution of continuous charge distribution

• Approach: divide the continuouscharge distribution intoinfinitesimally small elements

• Treat each element as a POINTcharge & compute its electricfield

• Sum (integrate) over all elements• Always look for symmetry to

simplify life!

Example: Field on Bisector of Charged RodExample: Field on Bisector of Charged Rod

• Uniform line of charge +Qspread over length L

• What is the direction of theelectric field at a point P onthe perpendicular bisector?

(a) Field is 0.(b) Along +y(c) Along +x

• Choose symmetricallylocated elements of length dx

• x components of E cancel

q

L

a

P

o

y

x

dx dx

Example --Line of Charge: QuantitativeExample --Line of Charge: Quantitative

• Uniform line of charge,length L, total charge Q

• Compute explicitly themagnitude of E at point Pon perpendicular bisector

• Showed earlier that the netfield at P is in the ydirection -- let’s nowcompute this!

Q

L

a

P

o

y

x

Line Of Charge: Field on bisectorLine Of Charge: Field on bisector Distance

22xad +=

2

)(

d

dqkdE =

L

q=!Charge per unit length

2/322 )(

)(cos

xa

adxkdEdEy

+

==!

"

Q

L

a

P

ox

dE

!

dx

d

2/122 )(cos

xa

a

+

=!

Line Of Charge: Field on bisectorLine Of Charge: Field on bisector

!" +

=

2/

2/

2/322 )(

L

L

yxa

dxakE #

What is E very far away from the line (L<<a)?What is E if the line is infinitely long (L >> a)?

224

2

Laa

Lk

+

=!

2/

2/

222

L

Laxa

xak

!"#

$%&

'

+

= (

a

k

La

LkEy

!! 22

2==

Example -- Arc of Charge: QuantitativeExample -- Arc of Charge: Quantitative• Figure shows a uniformly

charged rod of charge −Q bentinto a circular arc of radius R,centered at (0,0).

• Compute the direction &magnitude of E at the origin.

y

x450

x

y

θ

dQ = λRdθdθ!! coscos

2R

kdQdEdEx ==

!! ==

2/

0

2/

0

2cos

cos)(""

##$##$

dR

k

R

RdkEx

R

kEx

!=

R

kEy

!=

R

kE

!2=

λ = 2Q/(πR)

Example : Field on Axis of Charged DiskExample : Field on Axis of Charged Disk

• A uniformly charged circulardisk (with positive charge)

• What is the direction of E atpoint P on the axis?

(a) Field is 0(b) Along +z(c) Somewhere in the x-y plane

P

y

x

z

Example : Arc of ChargeExample : Arc of Charge

• Figure shows a uniformlycharged rod of charge -Qbent into a circular arc ofradius R, centered at (0,0).

• What is the direction of theelectric field at the origin?

(a) Field is 0.(b) Along +y(c) Along -y

y

x

• Choose symmetric elements• x components cancel

SummarySummary• The electric field produced by a system of charges atany point in space is the force per unit charge theyproduce at that point.• We can draw field lines to visualize the electric fieldproduced by electric charges.• Electric field of a point charge: E=kq/r2

• Electric field of a dipole: E~kp/r3

• An electric dipole in an electric field rotates to alignitself with the field.• Use CALCULUS to find E-field from a continuouscharge distribution.