Transcript Electric Field of a Dipole

Electric Field of a Dipole
Montwood High School
AP Physics C
R. Casao
• An electric dipole consists of two charges of
the same magnitude q, but opposite sign,
separated by a distance L.
• If the dipole is molecular, it could be an
induced dipole created by external charges
or a permanent dipole such as a water
molecule.
• We will calculate the electric field at a
location along the line of the dipole and at a
location along the perpendicular bisector of
the dipole.
• Consider a horizontal dipole with charge +q and –q
and separation s and determine the electric field E
at a location x along the x-axis, measured from the
center of the dipole.
• The net electric field E is horizontal to the right,
because at this location the field E+ of the closer +q
charge is larger than the field E- of the more distant
–q charge.
• The magnitude of the electric field is E1 = E+ - E-.
• Each charge is considered as a point charge:
El  E  E




k q
k q
1
1


El 


k

q


2
2
2
2

L
L
L
L 





x  
x  
x  
x  
2
2
2
2  
 





1

El  k  q 
2

  x  L 
2
 
2
L

x  
1
2



2
2
L
L


x  
x  
2
2


L 

x   
2 


2
L 

x  
2  

2
2
2


L
L




x  
x  
2
2




El  k  q 

2
2
2
2

L 
L 

  x  L    x  L 
x   x  
2 
2
2 
2  
 

2
2

L
L

 x     x   
2
2 



El  k  q 
2
2 

  x  L    x  L  
2 
2  
 
2  
2 
 2
L
L
L
L
   x2  2 x  

  x  2  x  
2
4  
2
4  

El  k  q  

2
2
L 
L



 x    x  


2
2






2  
2 
 2
L
L
2




x

L

x


x

L

x


 

4
4





El  k  q  

2
2
L 
L



 x    x  


2
2










2Lx


El  k  q 
2
2

L
L



  x     x   
2 
2  
 
2k qLx
El 
2
2
L 
L

 x    x  
2 
2

• If the point in question is far from the dipole (which
is usually the case when interacting with molecular
dipoles), L is small and is then divided by 2, so the
bottom of the equation simplifies to:
L
L
x-  x
x x
2
2
2k qLx 2k qLx
El 

2
2
4
x
x   x 
2k qL
El 
3
x
• To determine the electric field E2 at a location y on
the y-axis:
• The electric field equation for both +q and –q
is:
E
k q
L
y  
2
2
2
• These electric field vectors can be resolved in
to an x-component and a y-component.
• The vertical y-components of the two fields
E+ and E- are equal and opposite and
cancel.
• Both E+ and E- have an x-component that
contributes to the net electric field. The net
electric field E2 is the vector sum of the xcomponents of E+ and E-.
Ex
cos θ 
E
E x  E  cos θ
• The net electric field E2 is horizontal and
directed to the left.
• Using the distances to express the cos q in terms
of y and L:
L
2
cos θ 
1
2
 2 L 
y    


2




• The charges +q and –q can be considered as
point charges. The electric field contribution of
each one is given by:
L
2
Ex 
k q
L
y  
2
2
2

2
 2  L 2 
y   


2




1
2
• The net electric field is:
E 2  2  E x  2  E  cos θ
E2  2 
k q
L
y  
2
2

 2  L 2 
y   


2




2
E2 
L
2
k qL
 2  L 2 
y   


2




3
2
1
2
• If the point in question is far from the dipole (which
is usually the case when interacting with molecular
dipoles), L is small and is then divided by 2, so the
bottom of the equation simplifies to:
2
L
y     y2
2
k qL
k qL
k qL
E2 


3
3
3
 2  L 2  2
y 2 2  y 2 
y   




2




k qL k qL
E2 

3
y3
y 
2
 
• E2 is a vector and points to the left in the
direction of the negative y-axis, so add the
negative sign to E2.
• The electric field at both locations is
proportional to the product of q·L, called the
“dipole moment” and defined as: p = q·L.
– The dipole moment is a quantity that is
measurable for molecules such as HCl and H20
that are permanent dipoles.
– Nonpolar molecules have no permanent dipole
moment; however, in the presence of an external
electric field, they can acquire an induced dipole
moment as they become polarized.
• The dipole moment describes the
strength and orientation of the dipole.
• The dipole moment vector points from
the negative charge to the positive
charge.
• A uniform external electric field
exerts no net force on a dipole,
but it does exert a torque that
tends to rotate the dipole into
the direction of the field.
• The electric field will exert an equal an opposite
force on the two charges resulting in a couple (we
will pick one charge as the pivot and determine the
torque about the other charge).
• Using the negative charge as the pivot:
T  F  r
• Resolve F1 into components that are parallel and
perpendicular to the distance L.
F
sin θ 
F1
F  F1  sin θ
• Keep in mind, the angle given determines
whether you use sin or cos in the torque
equation.
• The electric force F1 on the positive charge is
equal to q·E.
T  F  r  q  E  sin θ  L
• q·L = dipole moment p
T  q  E  sin θ  L  p  E  sin θ
• The torque is the cross product of the dipole
moment and the electric field and the torque
will rotate the dipole so that the dipole
moment is aligned with the electric field.
  
T  pxE
• Maximum torque occurs
when the dipole moment
is perpendicular to the
electric field.
• The closer the angle
between the dipole
moment and the electric
field is to 90°, the
greater the torque. As
the angle q decreases,
so does the torque.
• When the angle
between the
dipole moment
and the electric
field is 0°, the
torque is 0 N·m.
• In this
picture, the
dipole
moment has
been rotated
in the
direction of
the electric
field.
Torque & Microwave Cooking
• In liquid water where molecules are relatively
free to move around, the electric field produced
by each molecular dipole affects the
surrounding dipoles.
• As a result, the molecules bond together in
groups of 2 or 3 because the negative oxygen
end of one dipole and a positive hydrogen end
of another dipole attract each other.
• Each time a group is formed, electric potential
energy is transferred to the random thermal
motion of the group and the surrounding
molecules.
Torque & Microwave Cooking
• Each time collisions between the molecules
breaks up a group, the transfer of energy is
reversed.
• The temperature of the water (which is
associated with the average thermal motion)
does not change because, on average, the net
transfer of energy is zero.
• When a microwave oven is turned on, the
microwaves produce in the oven an electric
field that rapidly oscillates back and forth in
direction.
Torque & Microwave Cooking
• If there is water in the oven, the oscillating field
exerts oscillating torques on the water
molecules, continually rotating them back and
forth to align their dipole moments with the
electric field direction.
• Molecules that are bonded as a pair can twist
around their common bond to stay aligned, but
molecules that are bonded in a group of 3 must
break at least 1 of their 2 bonds.
Torque & Microwave Cooking
• The energy to break the
bonds comes from the
electric field produced by
the microwaves.
• Then molecules that
have broken away from
groups can form new
groups, transferring the
energy they just gained
into thermal energy.
Torque & Microwave Cooking
• Thermal energy is added to the water when
the groups form but is not removed when the
groups break apart, and the temperature of
the water increases.
• Foods that contain water can be cooked in
microwave ovens because of the heating of
that water.
• If the water molecule were not an electric
dipole, this would not happen and microwave
ovens would be useless.
• MIT Visualzations:
• http://web.mit.edu/8.02t/www/802TEAL3D/
visualizations/electrostatics/index.htm
• The Electric Field of a Dipole
• Creation of an Electric Dipole
• Torque on an Electric Dipole in a Constant
Field