Lecture  4  1.  Review  superposi.on  principle  and  dipole  field  pa5ern  

+ -E

E-

E+

E+ + E- = EE-

E+

E- + E+

+ -Field  Pa5ern  of  Charge:  

Method  of  Parallelogram:  

+ -

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Clicker  4-­‐1  

Lecture 5 Dipole field pattern and dipole interactions

Lec5-1 Sketch a dipole field pattern

Consider a dipole centered at the origin. A parallelogram of E fieldvectors due to the dipole charges has been constructed at point A.

x

y

~E+

~E�

~Enet

A

B

Please sketch in your notebook the parallelogram formed by the fieldvectors at B. Wait. At the appropriate time discuss the parallelogramconstruction with your neighbour. Complete the parallelogramconstruction and label the three field vectors. Enter your clicker answer.

Choice Have you completed your parallelogram construction at B?1 Yes2 No

Clicker  401:  Dipole  field  pa5ern  and  dipole  interac.on  

Field  at  a  point  P  

xP

-q +q

+s

2�s

2

s

r

EP = E+ + E�

=kq

(r � s2 )

2� kq

(r + s2 )

2

=kq

r2

1

(1� ✏)2� 1

(1 + ✏)2

�

where: ✏ =s

2r

Digression  on  small  epsilon  expansion  

General  Formula:    

1

1� ✏= 1 + ✏Where  a  is  nega.ve:  

Product:  

So  the  general  formula  above  works  for  both  posi.ve  and  nega.ve!  

(1� ✏)a ⇡ 1� a✏

(1� ✏)a = 1� a✏+O(✏2) ⇡ 1� a✏

(1� ✏)2 = 1� 2✏+ ✏2 = 1� 2✏+O(✏2) ⇡ 1� 2✏

(1� ✏)3 = 1� 3✏+ 3✏3 + ✏3 = 1� 3✏+O(✏2) ⇡ 1� 3✏

(1� ✏)(1 + ✏) ⇡ (1� ✏2) ⇡ 1

Digression  on  small  epsilon  expansion  con4nued...  Examples:  

1

(1� ✏)2=

1

1� (2✏)⇡ 1 + 2✏

1

(1 + ✏)2=

1

1 + 2✏⇡ 1� 2✏

At  P  (from  slide  2):  

EP =kq

r

2

1

(1� ✏)2� 1

(1 + ✏)2

�

[ ] = (1 + 2✏)� (1� 2✏) = 4✏ = f

⇣s

2x

⌘=

2s

x

✏ =(s/2)

x

p = qsDipole:  

xP

-q +q

+s

2�s

2s

r

E =kq

x

2· 2sx

=2kqs

x

3=

2kp

x

3

More  on  dipoles...  

P1

E+

E�

~Edipole

P1 = ~E+P1 +

~E�P1

dir of EP1 : (�x̂)

~E = 2E+cos↵

= 2⇥ kq

r2· s/2

r=

kqs

r3

Dipole  in  Uniform  E:  

~Fdipole

= ~F+q

+ ~F�q

= q ~E � q ~E = 0

Uniform  dipole  in  uniform  E-­‐field  con.nued...  

F+

F� ↵

⌧ = Fd = Eqd = Eqs sin↵

~⌧ = ⌧ n̂

= (Ep sin↵)n̂

= ~p⇥ ~E Direc.on:  CW  

Is  electric  field  real?  Source  creates  field:  field  spreads  out  with  the  speed  of  light.  

 Time  taken  to  spread  out  1  foot:  

�t ⇡ 0.3

3⇥ 108= 10�9 sec = ns

e+ + e� ! ��Dipole  field   r

e+e� system

2kp

r3

At  t=0,  reac.on  occurs  à  dipole  system  dissolves  At  P:  Electric  field  due  to  the  dipole  con.nues  to  be  present  for  t=0  à  t1  =  r/c