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Physics 202, Lecture 2

Today’s Topics

Announcements

Electric Fields More on the Electric Force (Coulomb’s Law) The Electric Field Motion of Charged Particles in an Electric Field

Announcements Homework Assignment #1:

WebAssign (corresponds to Ch. 23 #5,10,12,15,19,26,42)

Due Friday, Sept 14 at 10PMFuture assignments will be due Mondays at 10 PM.

IMPORTANT: Exam policy revisionIf you have an evening class conflict, let us know

right away (now!) and we will accommodate it. Recall exam dates: 10/1, 10/29, 11/26, 5:30-7 PM.

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Coulomb’s Law (Point Charges)

!F21= !ke

q1q2

r2r̂

Force on q2 by q1:

Force on q1 by q2:

r

!F12= ke

q1q2

r2r̂

Vector direction of F: more explicitly

!r1

!r2

q1

q2

!r2!!r1

!F12= ke

q1q2

|!r2!!r1|2

(!r2!!r1)

|!r2!!r1|= ke

q1q2(!r2!!r1)

|!r2!!r1|3

in the above language: r =|!r2!!r1|

r̂ =

!r2!!r1

|!r2!!r1|

Force on q2 by q1:

Coulomb’s Law: Multiple charges

Many point charges: linear superposition

!F1=

!Fi1

i

! = keq1qi

ri12r̂i1

i

!

!r1

!r2

q1

q2

!r1!!r2

!F1= ke

q1qi

|!r1!!ri |

2

i

"(!r1!!ri )

|!r1!!ri |

= keq1qi (!r1!!ri )

|!r1!!ri |

3

i

"

q3

!r3

Force on charge 1:

More explicitly:

Important: vector sum!

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Coulomb’s Law: DistributionsContinuous distributions of charge: divide into

!qii

" # dq$

Line segment (1 dimension): dq = !dx

!qi

Surface (2 dimensions): dq = !dA

Volume (3 dimensions): dq = !dV

more about this shortly!

Interaction of point charge with continuous distribution:

!F1= keq1

dq

r2r̂!

One of four fundamental forces: Strong >Electromagnetic > weak >> gravity Magnitude: Proportional to 1/r2 : r doubled ¼ F Direction: repulsive for like signs, attractive for opposite signs

A conservative force (work independent of path):

A potential energy can be defined. (Topic of Ch. 25)

Properties of the Electric Force

r

qqkU e 21

=

)()(2121if

i

e

f

er

rUU

r

qqk

r

qqkrdFW

f

i

!!!=+!=•= "!!

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The Electric Field

Original (Coulomb’s) view of electric force q applies electric force on q0 (action at a distance)

Force on charge always proportionalto strength of that charge!

“Modern” view of electric force: q is the source of an electric field E

The electric field E applies a force on q0

!F = ke

q0q

r2r̂

!F = q

0keq

r2r̂ = q

0

!E

E

q: source chargeq0: “test charge”E independent of q0!

q0

qF

Aside: Vector and Scalar Fields

What is a field?•A field represents some physical quantity

(e.g., temperature, wind speed, force)•It can be a scalar field (e.g., temperature field)•It can be a vector field (e.g., electric field)•It can be a “tensor” field (e.g., spacetime curvature)

The field concept is extremely useful in physics,both conceptually and for practical purposes

Using the concept of the electric field,one can map the electric field anywhere in spaceproduced by any arbitrary charge distribution

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A Scalar Field

7782

83685566

8375 8090 91

757180

72

84

73

82

8892

7788

887364

These isolated temperatures sample the scalar field(you only learn the temperature at the point you choose,

but T is defined everywhere (x, y)

A Vector Field

7782

83685566

8375 809091

7571

80

72

84

73

57

8892

77

5688

7364

That would require a vector field(you learn both wind speed and direction)

Perhaps more interesting to know which way the wind is blowing…

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The Electric Field, Recap Charge q sources an electric field E

(charge at origin) More generally,

Multiple sources:

Continuous distributions:

!!r

!r

q

!E(!r ) = ke

q

r2r̂

!r

q

P

P

!E(!r ) = ke

q

|!r !!"r |2

(!r !!"r )

|!r !!"r |= ke

q(!r !!"r )

|!r !!"r |3

!r !!"r

!E(!r ) = ke

qi

ri2r̂i

i

!

!E(!r ) = ke

qi (!r !!"ri )

|!r !!"ri |3

i

#

!E(!r ) = ke

dq(!r !!"r )

|!r !!"r |3#

!E(!r ) = ke

dq

r2r̂!

(more general)

Visualizing the Electric Field

+

+ chg

field lines

+ charge

vector map map

+

Consider the electric field of a positive point charge at the origin:

Magnitude: length of arrows Magnitude: local density of lines

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Rules for Field Lines

Field lines can start or terminate only on charges, never empty space.

Property 1.

Property 2.Field lines of point charge go off to infinity.(true for any localized distribution with nonzero net charge)

Property 3.Field lines originate on positive charges, terminate on negative charges.

Property 4.

No two field lines ever cross, even when multiple charges present.

Example: Point-Like Charges

+q -q

!E(!r ) = ke

q

r2r̂

!E(!r ) = ke

q

r2r̂

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Example: Two Charged Particles

+q and +q +q and -q

Two Charged Particles

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Example: Uniformly Charged Sphere A uniformly charged sphere has a radius R and total

charge Q, find the electric field outside the sphere. See notes (to be posted on web)

Doable but complicated with the conventional method! Please preview Ch. 24 “Gauss Law” before Tuesday!

R

x

y

z

E(x,y,z)=?