Ch 20: Electric Charge and Electric Fields

Electric Charge Experiments involving the rubbing of plastic, glass and wood rods with wool, silk, and other materials cause the material to inherit a property that allows for long-range attractive and repulsive forces. This property is called charge Examples:

• Unrolling plastic rap • Running a comb through hair • Rubbing rubber/plastic/glass rods with fur and silk • Walking on carpet with slippers

3 A charged object always attracts an uncharged macroscopic object

Pithballs charged by being touched by same object always repel

Pithballs can both be charged such that they are attracted to each other.

Results 4

Charge Model:

•  There are two kinds of electric charge: positive and negative (a 3rd type of charge that is attracted to both positive and negative has never been found).

•  Like charges repel, opposite charges attract.

•  Choice of what is a positive charge and a negative charge is arbitrary (glass rubbed with silk is defined as positive charging).

This charge model is consistent with experiments…

• Rubbing causes objects to be charged. • Charged object attracts neutral object • Sometimes two charged objects attract, sometimes they repel. • Objects charged the same way always repel.

Clicker question

Conductors, insulators, and dielectrics •  Materials in which charge is free to move are conductors. •  Materials in which charge isn’t free to move are

insulators.

A metallic sphere is initially neutral. A negatively charged rod is brought in contact with a connecting rod, which is also touching the metallic sphere. What type of rod should be used such that the metallic sphere is charged? 1.  a conducting rod 2.  a insulating rod 3.  Either kind of rod will charge the sphere.

Clicker question

A negatively charged glass rod is brought near a spherical conductor, which is initially neutral. The spherical conductor is initially touching another spherical conductor which is also initially neutral. While the glass rod is near the conductor, the two conductors are separated. The glass rod is then removed. What are the charged states of the two conductors?

1.  Left is positively charged, right is negatively charged.

2.  Right is positively charged, left is negatively charged.

3.  Left is positively charged, right is neutral.

Clicker question

Figure 21.8

•  How can a charged object attract a neutral, macroscopic object?

Coulomb’s law and the electric force •  Like charges repel, and opposite charges attract, with a

force that depends on •  The product of the two charges •  The inverse square of the distance between them

•  Mathematically, the electric force is described by Coulomb’s law:

12 1

2

1 2

2 29

Here is the force exertsˆ and is a unit vector

pointing toward is the ,

approximately 9.0 10 N m / C

.

.

F qon q r

from q qk

× ⋅Coulomb constant

r

�F12 =kq1q2r2

r̂

Clicker question

13

What is the angle between two strings attached to pithballs of mass m and charge Q?

The superposition principle •  The electric force obeys the superposition principle.

•  That means the force two charges exert on a third force is just the vector sum of the forces from the two charges, each treated without regard to the other charge.

•  The superposition principle makes it mathematically straightforward to calculate the electric forces exerted by distributions of electric charge.

•  The net electric force is the sum of the individual forces.

Consider the charge configuration shown below.

CT 25.11

+q

+Q +Q

h

s/2 s/2

What is the direction of the net force on the +q charge?

A

B

C

D E

Copyright Univ. of Colorado, Boulder

A charge q is to be placed at either A or B on the mid-line of two charges +q. Will the force on q be greater at point A or at point B? A) A B) B C) Can't tell without knowing magnitude of q.

Clicker question

F =2kqQy

(a2 + y2)3/2

The Electric Field, E •  The electric field at a point in space (due to a distribution

of charge) is the force per unit charge that a test-charge q’ placed at that point would experience:

�E =

�F

q�

The electric field exists in space (even if no charge exists there).

19

An electron is placed at the position marked by the dot. The force on the electron is

A. to the left. B. to the right. C. zero. D. There’s not enough information to tell.

Fields of point charges and charge distributions •  Based on Coulomb’s law, the

field of a point charge is radial, outward for a positive charge and inward for a negative charge.

�Ept charge =kq

r2r̂

•  The superposition principle states that, for a collection of charges, the total electric field at a point is simply the summation of the individual fields due to each charge

�E =�

�Ei =kqir2i

r̂i

The dipole: an important charge distribution •  An electric dipole consists of two point charges of equal

magnitude but opposite signs, held a short distance apart. •  The dipole is electrically

neutral, but the separation of its charges results in an electric field.

•  Many charge distributions, especially molecules, behave like electric dipoles.

•  The product of the charge and separation is the dipole moment: p = qd.

•  Far from the dipole, its electric field falls off as the inverse cube of the distance.

(E ∝ 1/r3)

phet: charges

Field lines •  Field lines show the direction of the electric field. The

density of field lines is proportional the the strength of the field.

•  Example:

Continuous charge distributions •  Charge is ultimately made of individual particles, but it’s

often convenient to consider it distributed continuously •  The electric field of a charge distribution follows by summing

—that is, integrating—the fields of individual charge elements dq, each treated as a point charge:

�E =

��dE =

�k r̂

r2dq

dq = λdl dq = σda dq = ρdV

1D 2D 3D

What is the vertical component, dEy, due to the little chunk of length dx? T 26.11e

x

y=H

+ + + + + + + + + + + + + + + + + + + + + ++ +

dE

dx dEy= A) B) C) D) E) |

•  The electric field of an infinite line of charge: •  The line carries charge density λ (units are C/m):

CT 26.11 An infinite line of charge with linear charge density λ is along the x-axis and extends to ± ∞.

from CU Boulder

kλdx/(x2 +H2) kλx dx/(x2 +H

2)3/2

kλH dx/(x2 +H2)3/2kλH dx/(x2 +H

2)kλx dx/(x2 +H

2)

2

direction radially outwardfor + charge;inward for – charge

kEyλ

=Ey =

� ∞

−∞

kλy dx

(x2 + y2)3/2

Figure 21.49

The electric field at point P is

1.  kQ/a2 2.  less than kQ/a2 but greater than zero 3.  zero 4.  Depends on the value of the charge at point P

•  The electric field on the axis of a charged ring:

Ex =

�k dq

r2cos θ =

k Q

r2x

r

Ex =kQx

(x2 + a2)3/2

•  Electric field due to disk of radius R and charge Q (a distance x from center):

E =

�dERing =

�k dQRing x

(x2 + r2)3/2

E = 2πσkx1

(x2 + r2)1/2|R0

E = 2πσk

�1− x√

x2 +R2

�

E =

� R

0

k 2πσr dr x

(x2 + r2)3/2