Physics 2102 Physics 2102 Lecture: 03 TUE 26 JAN Lecture: 03 TUE 26 JAN

Electric Fields IIElectric Fields IIMichael Faraday

(1791-1867)

Physics 2102

Jonathan Dowling

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What Are We Going to What Are We Going to Learn?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−12F1q+ 21F

12r

Coulomb’s LawCoulomb’s Law

212

2112

||||||r

qqkF =

2

212

00

1085.8with 41

mNCk −×== επε

2

291099.8

CmN×k =

For Charges in aVacuum

Often, we write k as:

E-Field is E-Force Divided by E-E-Field is E-Force Divided by E-ChargeCharge

Definition of Electric Field:

€

r E =

r F q

€

|r F 12 |= k | q1 | | q2 |

r122

+q1 –q2

€

r F 12

P1 P2

€

|r E 12 |= k | q2 |

r122

–q2

€

r E 12

P1 P2

Units: F = [N] = [Newton] ; E = [N/C] = [Newton/Coulomb]

E-Force

on Charg

eE-Field at

Point

Force on a Charge in Electric Force on a Charge in Electric FieldField

Definition of Electric Field:

€

r E =

r F q

Force on Charge Due to Electric Field:

€

r F = q

r E

Force on a Charge in Electric Force on a Charge in Electric FieldField

E

E

Positive Charge Force in Same Direction as E-Field (Follows)

Negative Charge Force in Opposite Direction as E-Field (Opposes)

+ + + + + + + + + 

+ + + + + + + + + 

– – – – – – – – – – 

– – – – – – – – – – 

Electric Dipole in a Uniform Electric Dipole in a Uniform FieldField

• Net force on dipole = 0; center of mass stays where it is.

• Net TORQUE : INTO page. Dipole rotates to line up in direction of E.

• | | = 2(qE)(d/2)(sin ) = (qd)(E)sin = |p| E sin = |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 Electric Charges and FieldsFields

First: Given Electric Charges, We Calculate the Electric Field Using E=kqr/r3.

Example: the Electric Field Produced By a Single Charge, or by a Dipole:

Charge Produces E-Field

E-Field Then Produces Force on Another Charge

Second: Given an Electric Field, We Calculate the Forces on Other Charges Using F=qE

Examples: Forces on a Single Charge When Immersed in the Field of a Dipole, Torque on a Dipole When Immersed in an Uniform Electric Field.

Continuous Charge Continuous Charge DistributionDistribution

• Thus Far, We Have Only Dealt With Discrete, Point Charges.

• Imagine Instead That a Charge q Is Smeared Out Over A:– LINE

– AREA

– VOLUME

• How to Compute the Electric Field E? Calculus!!!

q

q

q

q

Charge DensityCharge Density

• Useful idea: charge density• Line of charge:

charge per unit length = • Sheet of charge:

charge per unit area = • Volume of charge:

charge per unit volume = r

= q/L

= q/A

r = q/V

Computing Electric FieldComputing Electric Field of Continuous Charge of Continuous Charge

DistributionDistribution• Approach: Divide the Continuous

Charge Distribution Into Infinitesimally Small Differential Elements

• Treat Each Element As a POINT Charge & Compute Its Electric Field

• Sum (Integrate) Over All Elements• Always Look for Symmetry to

Simplify Calculation!

dq

dq = dLdq = dSdq = r dV

Differential Form of Differential Form of Coulomb’s LawCoulomb’s Law

q2

€

r E 12

P1 P2

E-Field at

Point

dq2

€

dr E 12

P1

Differential dE-Field at Point

€

dr E 12 =

k dq2

r122

€

r E 12 =

k q2

r122

Field on Bisector of Charged RodField on Bisector of Charged Rod• Uniform line of charge

+q spread over length L

• What is the direction of the electric field at a point P on the perpendicular bisector?

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

located elements of length dq = dx

•x components of E cancel

q

L

a

P

o

y

x

dx

€

ds E

dx

€

dr E

€

↑E

Line of Charge: QuantitativeLine of Charge: Quantitative• Uniform line of charge,

length L, total charge q• Compute explicitly the

magnitude of E at point P on perpendicular bisector

• Showed earlier that the net field at P is in the y direction — let’s now compute this!

q

L

a

P

o

y

x

Line Of Charge: Field on bisectorLine Of Charge: Field on bisector Distance hypotenuse: r = a2 + x2( )

1/2

dE =k(d)r2

Lq=Charge per unit length:

2/322 )()(cos

xaadxkdEdEy +

== q

L

a

P

ox

dE

dx

r

cos =ar=

a(a2 + x2 )1/2

AdjacentOverHypotenuse

[C/m]

Line Of Charge: Field on Line Of Charge: Field on bisectorbisector

∫− +

=2/

2/2/322 )(

L

Ly xa

dxakE λ

Point Charge Limit: L << a

€

=2kλL

a 4a2 + L2

2/

2/222

L

Laxaxak

−⎥⎦⎤

⎢⎣⎡

+=

€

Ey = 2kλLa 4a2 + L2

≅2kλ

a

Line Charge Limit: L >> a

€

Ey = 2kλLa 4a2 + L2

≅kλLa2 = kq

a2

Integrate: Trig Substitution!

Coulomb’sLaw!

€

Nm2

C1m

Cm

⎡ ⎣ ⎢

⎤ ⎦ ⎥= N

m ⎡ ⎣ ⎢

⎤ ⎦ ⎥

Units Check!

Binomial Approximation from Taylor Series:x<<1

€

2kλLa 4a2 + L2

= kλLa2 1+ L

2a ⎛ ⎝ ⎜

⎞ ⎠ ⎟2 ⎡

⎣ ⎢

⎤

⎦ ⎥

−1/ 2

≅kλLa2 1− 1

2L2a ⎛ ⎝ ⎜

⎞ ⎠ ⎟2 ⎡

⎣ ⎢

⎤

⎦ ⎥≅ kλL

a2 ; (L << a)

2kλLa 4a2 + L2

= 2kλLaL

1+ 2aL ⎛ ⎝ ⎜

⎞ ⎠ ⎟2 ⎡

⎣ ⎢

⎤

⎦ ⎥

−1/ 2

≅2kλ

a1− 1

22aL ⎛ ⎝ ⎜

⎞ ⎠ ⎟2 ⎡

⎣ ⎢

⎤

⎦ ⎥≅ 2kλ

a ; (L >> a)

€

1± x( )n≅1± nx

Example — Arc of Charge: Example — Arc of Charge: QuantitativeQuantitative

• Figure shows a uniformly charged rod of charge -Q bent into a circular arc of radius R, centered at (0,0).

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

y

x450

x

y

dQ = Rdd coscos 2R

kdQdEdEx ==

∫∫ ==2/

0

2/

02 coscos)( ππ

θθλθθλ dRk

RRdkEx

RkEx=

RkEy=

RkE 2= = 2Q/(πR)

E–Q

E = Ex2 + Ey

2

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

• A uniformly charged circular disk (with positive charge)

• What is the direction of E at point 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 uniformly

charged rod of charge –Q bent into a circular arc of radius R, centered at (0,0).

• What is the direction of the electric 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 at any point in space is the force per unit charge they produce at that point. • We can draw field lines to visualize the electric field produced 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 align itself with the field. • Use CALCULUS to find E-field from a continuous charge distribution.