L02 Coulomb Post

7/30/2019 L02 Coulomb Post

1/19

LECTURE 2 slide 1

Lecture 2

Coulombs Law, Charge Density

Sections: 2.1, 2.2, 2.3, 2.4, 2.5

Homework: D2.1; 2.1, 2.3, 2.5, 2.7, 2.9, 2.11, 2.13


2/19

LECTURE 2 slide 2

The electric charge is a fundamental property of matter. It ismeasured in coulombs (C). The electric current unit ampere (A) waschosen as a basic unit in SI. Thus, coulomb is a secondary unit

derived from

C=AsdQ

i

dt

=

i is the electric current in amperes (A)

Q is the electric charge in coulombs (C)t is time

Electric Charge 1


3/19

LECTURE 2 slide 3

Atoms contain charged particles: electrons and protons. These particles react in opposite way to the influence of external electricfields. Therefore, they have opposite charges. It was agreed that

the protons would have the positive charge, and the electronswould have the negative charge. The charge of an electron isequal in magnitude to the charge of a proton and is

19

1.602 10 , CeQ e

=

In this course, we are concerned with macroscopic charges, i.e.,charge distributions much larger than the dimensions of thelargest atomic nucleus ( 10 -15 m).

Electric Charge

The electron charge is the smallest indivisible amount of charge.


4/19

LECTURE 2 slide 4

Charge occupies a finite volume. However, a volumetric chargecan be always considered made of smaller charges, so small thatthey tend to a point.

Point charge is a charge whose volume can be consideredinfinitesimal (a point) in comparison with the distance from itscenter to the observation point. This definition implies that it is

an infinitesimal sphere of homogeneous charge distribution(charge density v is constant).

Point Charge as an Approximation 1

v vQ

Q vv

= =

far away


5/19

LECTURE 2 slide 5

( ) 1v v vavv

Qdv

v v = =

( ) ( )i i P P

i ii i

Q v

Q Q v

= = =

Average Charge Density

From close up, the details in thecharge distribution matter. We thenconsider the total charge Q acollection of charges:

( , , )v dv

Q x y z dx dy dz

=

( ) P Q

close up

The homogeneous charge distribution is an approximation wherean averaged charge density is assumed


6/19

LECTURE 2 slide 6

The point charge features spherical symmetry, which impliesthe spherical symmetry of its field.

Equipotential lines and forcelines of a point charge

Field of Point Charge

From far away, the details in the charge distribution do not matter.We then consider the total charge Q only.


7/19

LECTURE 2 slide 7

N

1 22 12 12

12

1, N

4k

Q Q

R = = F a F

The inverse square law is a universal property of fields in 3-D.

Charges of the same sign repel each other, and charges of theopposite signs attract each other.

Coulombs Law (1785)


8/19

LECTURE 2 slide 8

TRUE OR FALSE?

Two charges Q1 and Q2 are 8 cm apart. The forceacting on Q2 is 122 9 10 , N. y

= F a

Q1: If we set the charges 24 cm apart, the force will become12

2 3 10 , N. y = F a

Q2: If we replace Q1 with Q3 = Q1/3, the force will become12

2 3 10 , N. y

= F a


9/19

LECTURE 2 slide 9

where c is the speed of light.

The constant of proportionality k depends on the systemof units used. In SI, 1/(4 )k =

2 2

2 2 2

m Nm Vm[ ] = = CC A sk

=

By experiment (in air/vacuum), if the force is measured innewtons , the distance in meters , and the charge in ampere-

seconds (coulombs):99.0 10k

Theoretically, this constant in the SI system must be exactly7 2

10k c

=

Coulombs Law 2


10/19

LECTURE 2 slide 10

The constant1

4 k

=

is called dielectric permittivity , which in vacuum is

9

0 2 7 9

1 1 10

364 10 4 9 10c

= =

A more precise value is

120 8.856 10 , F/m=C/(V m)

=

Dielectric Permittivity in Coulombs Law


11/19

LECTURE 2 slide 11

It is usually specified relative to that of vacuum via therelative dielectric permittivity (dielectric constant) r :

0r =

Air: 1.0006r =

Water: 80r =

Urban (dry) ground: 3r

Rural (moist) ground: 14r

Dielectric Permittivity in Coulombs Law 2

The dielectric permittivity of materials is usually different fromthat of vacuum due to polarization.


12/19

LECTURE 2 slide 12

The electric field vector is the force exerted on a unit charge.

0lim , N/C=V/mq q

= FE

Here, q is a test (probe) charge , which means that it is smallenough not to disturb the measured original field of the source.

The electric field of a positive point charge located at theorigin of a spherical coordinate system is

2

1, V/m4 r

Qr =E a

, Nq =F E

Electric Field (Intensity) Vector


13/19

LECTURE 2 slide 13

Volume charge density is a differential (local) description of the charge distribution

3

0lim , C/m , Cv vv v

Q dQQ dvv dv

= = =

Surface charge is another approximation, which is very usefulwhen the physical 3-D charges are spread in wide thin sheets

whose thickness is negligible in comparison with their lengthand width. It is assumed that the charge density variationswith the height are negligible.

Charge Density 1


14/19

LECTURE 2 slide 14

/ 2

/ 2

v s

h

v v sv S h S

h

Q dv dh ds ds

=

= = =

sS

Q ds = 20

lim , C/m s s

Q dQ s ds

= =

h

+++++

+ + ++

+

++ +++ds

Surface charge distribution is described by surface chargedensity s

Charge Density 2


15/19

LECTURE 2 slide 15

The surface charges imply the symmetry of the field withrespect to their plane.

e q u

i p o

t e n

t i a l

l i n e s force lines

Charge Density 3


16/19

LECTURE 2 slide 16

Linear charges are a useful approximation for charges whosevolume has two of its dimensions negligibly small comparedto the other dimension (the length). The variations of thecharge distribution in the cross-section are negligible.

v l

v vV l s

s

Q dv ds dl

=

= =

l l

Q dl =

The linear charge distribution is described by the linear charge density . The field has cylindrical symmetry.l

N

, C/ml v

dv

dQ dQ s s

dl s dl = = =

l

s

Charge Density 4


17/19

LECTURE 2 slide 17

How much charge Q is there on a charged surface

of area A = 1 mm 2 if s = 4 C/m 2?

Q =

How long is a uniformly charged string whose total charge isQ = 1 C and its linear charge density is l = 10

6 C/m?

L =


18/19

LECTURE 2 slide 18

The infinitesimal (differential) charge elements

point charge surface charge linear charge

, andv s l dQ dv dQ ds dQ dl = = =

are often used to evaluate the field of complicated chargeconfigurations through the principle of superposition(integration over the volume of the charge distribution).

The fields (potential and force) of the point, surface and linear charges are known and serve as building blocks in solutions of complicated problems.

The field of the point charge is of fundamental importance.All other solutions follow from it.

Differential Charge dQ


19/19

LECTURE 2 slide 19

You have learned about

volume charge distribution, its density and its relation to the totalcharge Q

surface and line charge distributions, their densities and their

relation to the total charge Qthe symmetry of the field of a point charge, a planar charge, and aline charge

the E -vector as the electric-field normalized force

Coulomb's law for the force between two point charges

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L02 Coulomb Post