April 24, 2023

Physics 2102 Physics 2102 Lecture: 04 WED 21 JAN Lecture: 04 WED 21 JAN

Electric Fields IIElectric Fields II

Physics 2102

Jonathan Dowling

Version: 04/24/23

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Michael Faraday (1791-1867)

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 =

= q/L

= q/A

= 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 = dV

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

q2

€

E 12

P1 P2

E-Field at

Point

dq2

€

dr E 12

P1

Differential dE-Field at Point

€

dr E 12 =

k dq2

r122

€

E 12 =

k q2

r122

Field on Bisector of Charged Field on Bisector of Charged RodRod

• 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:

€

d = a2 + x 2( )1/ 2

2)(

ddqkdE =

Lq=Charge per unit length:

2/322 )()(cos

xaadxkdEdEy +

== θq

L

a

P

ox

dE

θ

dx

d

2/122 )(cos

xaa

+=θ

AdjacentOverHypotenuse

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

θ

dQ = Rdθdθθθ coscos 2R

kdQdEdEx ==

∫∫ ==2/

0

2/

02 coscos)( ππ

θθλθθλ dRk

RRdkEx

RkEx=

RkEy=

RkE 2= = 2Q/(R)

E–Q

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 Example : Arc of ChargeCharge

• 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.