Chapter 24 Electric Fields

第二十四章 電場

Action at a distance

1 212 212

21

ˆkq q

F rr

q1

q1

q2

q2

21r̂

9 2 29.0 10 /k Nm C

Another point of view -- Fields

1 212 212

21

ˆkq q

F rr

)( 212112 rEqF

rr

qr

r

kqrE ˆ

4

1ˆ)(

22

022

2

where E2 is the electric field generated by charge q2 and r is a position vector from the position of q2 to any point in space.

22120 /10854.8 NmC

Electric field lines

Electric field linesThe concept of electric field lines was originally due to Faraday who envisioned that the space around a charge is filled with “lines of force”.

The field lines are originated from positive charges and terminated at negative charges, and they are unbroken in vacuum.

The relationship between the electrical field and field lines

Electric field lines with arrows indicate the local field directions, and their density in a plane perpendicular to the local field represents the magnitude of the local field.

Suppose that there are N field lines originated from a positive charge q, then we have the density of field lines given by N / 4πr2 at a distance r from the charge.

Note that if N = q / ε0 then we recover E = q / 4πε0 r2.

Field lines repel each other

Field lines are smooth

The electric field due to a point charge

)(4

13

0i

i

ii rr

rr

qE

)(...)()()( 21 rErErErE n

Sample problem

What is the electric field at point P?

Sample problem

What is the electric field at point P?

The electric field due to an electric dipole

x

y

-a a

P

3 30

( )4

r r r rqE

r r r r

r

r

r

ˆr ax

ˆr ax

Electric dipole moment:

ˆ( ) 2p q r r qax

2p qa

The electric field due to an electric dipole

Along x-axis: ˆr xx

For 2 2 2

0

21

4 ( )

x pE

x a

For 2 2

2 2 20

1 ( )

4 ( )

a x pE

a a x

30

1 2

4

pE

x

:ax

:ax

:ax

The electric field due to an electric dipole

In y-z plane: ˆ ˆr yy zz

2 2 2 3/ 20

1

4 ( )

pE

y z a

30

1

4

pE

r

Note that 2

1E

r for net charge 0

3

1E

r for net charge 0

but net dipole moment 0

:ar

Electric field due to a line of charge

Electric field due to a line of charge

Electric field due to a line of charge

Electric field due to a line of charge

Electric field due to a charge disk

A point charge in an electric field

F qE

Millikan oil-drop experiment (1910-1913)

Cathode-ray tube

Deflecting charge particles

A dipole in an electric field

2

306.2 10H Op Cm

A dipole in an electric fieldThere is no net force on a dipole.

But there is a torque.

sin ( )( sin )

sin

qEa qE a

pE

p E

Potential energy of a dipole in a uniform electric field

cosU pE

ˆ ˆ( ) sinU

pE

Home work

Question ( 問題 ): 6, 8, 19

Exercise ( 練習題 ): 5, 10, 16

Problem ( 習題 ): 8, 18, 31, 32