1. Consider a dipole in an external electric field £ = Eoŷ where Eo is a constant. For the four cases of orientation of the dipole (p in the magnitude of the dipole moment): (i) F= på (ii) - på (iii) F (iv) = -Pâů
1. Consider a dipole in an external electric field £ = Eoŷ where Eo is a constant. For the four cases of orientation of the dipole (p in the magnitude of the dipole moment): (i) F= på (ii) - på (iii) F (iv) = -Pâů
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Transcribed Image Text:→ the system is unstable.
The P.E. of an electric dipole in an
E-field
Example:
A water molecule has an electric dipole moment of 6.3 × 1030 Cm. A sample contains 102 water molecules,
with all the dipole moments aligned parallel to the external electric field of magnitude 3 × 10° NC'. How
much work is required to rotate all the water molecules from 0 = 0° to 90°?
Solution:
When the water molecules are aligned in the direction of the electric field, it has minimum potential energy.
The work done to rotate the dipole from 0 = 0° to 90° is equal to the potential energy difference between
these two configurations.
W= AU =U (90°)-U (0°)
we write U= - pE cose, Next we calculate the work done to rotate one water
molecule from e = 0° to 90°.
For one water molecule
w --pE cos 90° + pE cos0° - pE
w=6.3×10-30 x 3 ×105 = 18 .9×10-25 J
For 1022 water molecules, the total work done is
Wtot - 18. 9 x10 25 x102 -18.9x10-3 J
H.W. :
Consider a dipole in an external electric field E = Eoý where Eo is
a constant. For the four cases of orientation of the dipole (p is the
magnitude of the dipole moment):
1
(i) f= på
(ii) = Pg
(iii) F= -pât
(iv) p=-pâý
a. Find the potential energy of the dipole in this external field. Order
your answers from highest to lowest potential energy.
b. If the dipole were free to rotate in this external electric field, which
orientation would be the equilibrium alignment? (as a hint, consider
the analogy of a particle in the gravitational field of the earth;
its potential energy increases with increasing altitudle above the
surface. If left free, in what direction of change of potential energy
does it go?)
2. A point dipole p = Poy is located at the origin of the coordinates
and po = 1.00 x 10-15 Cm. At a distance of 0.01 meters from the
dipole, find the electric field at each of the following locations and, on
a diagram, sketch in the direction of the field at each point:
a. along the positive x-axis.
b. along the positive y-axis.
c. along the negative x-axis.
d. along the negative y-axis.
e. at a point in the first quadrant in the z-y plane, making an angle
of 45° with both the positive-z and positive-y axes.
3. An electric dipole of dipole moment = pê, with p = 5.0 x 10-1 Cm,
is placed in a uniform electric field E = Eoê with Eo= 4.0 x 10 N/C.
a. What torque is exerted on the dipole by the field?
b. What is the potential energy of the dipole in this field?
c. If you (with tweezers?) rotated this dipole so that it aligned per-
pendicular to the field, = pê, how mnuch work would you (not the
field!) have to do?
4. Given a finite-sized dipole with a charge of -2.5 x 10 C at one end-
point and a charge of +2.5 x 10 C at the other end-point, and - die
tance between the two points of 0.0010m, and with the dipol
at 40.0° to a uniform external electric field, find:
aa. the dipole moment.
b. with external electrie field strength equal to 7.5 N/Cê, ind the
potential energy of the dipole in this field.
e. the torque on the dipole due to the field.
d.
the work done by the field on the dipole in rotating it from 40.0°
with the field to itas equilibrium orientation.
Brief Answers:
. (i) 0; (ii) -Eop; (iii) 0; (iv) +Eop. Order of deereasing emergy: (iv),
(i) and (iii), (ii).
b. The lowest energy alignment: (i) E- PO.
2. a. E = -99 N/C.
b. E
e. E- -90 N/c.
- 18ON/C.
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