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Tipler: 21-7
Electromagnetism I
Electric Dipoles and their Interactions in
Electric Fields
An electric dipole consists of a positive charge separated from a negative charge of the same magnitude by a small distance. Which, if any, of the diagrams best represents the electric field lines around an electric dipole?
ET
EP
Learning Objectives
•To calculate E produced by an electric dipole
•To investigate the forces and torques an electric dipole experiences in an external uniform E-field
An electric dipole:
l
+q -q
Electrically neutral
A Reminder of Electric Dipoles
Define dipole moment as p = ql
The vector of p is drawn from the negative to the positive point charge
p is a vector.
p
Molecular example: H2O
p = 6.1 x 10-30 C m •The electric dipole of water makes it an excellent solvent
•Used in heating food in a microwave oven
Calculation of the E-field at an arbitrary (r, )
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
⎟⎠⎞
⎜⎝⎛ +
−
⎟⎠⎞
⎜⎝⎛ −
= 220
22
4
1
ar
q
ar
qE
πε
The E-field due to an Electric Dipole - Calculation
To simplify the calculation, we will only compute the field along the axis
r
E
-q +q
a
a/2
304
2
r
qaE
πε≈
For r >> a
This applies to any points along the line of the dipole,
and only for points along the line of the dipole.
Along this particular direction, the E field from the
positive charge is in opposite direction to that from
the negative charge.
To obtain E-field:
1) Coulomb’s Law, involving vector sums.
2) Gauss’s law, if the charge distribution
has a high degree of symmetry.
3) Get V first, then differentiate. No vector sums.
How to make a decision:
Choose the method that consumes the
least number of brain cells.
l
+q -q
•V due to the two charges at P
−+
−=r
q
r
qV
00P 44 πεπε
Assumption: r >> l
cos2
lrr ±≈±
r-r+
P
r
l/2
No nasty vectors here.
€
VP =q
4πε0
−lcosθ
r2 −l2
4cos2 θ
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
Hence
€
≈−pcosθ
4πε0r2
as r >> l
The sign of Vp depends on .
⎟⎠
⎞⎜⎝
⎛∂∂
+∂∂
−=
ˆV
rr̂
r
VE
1
In Plane Polar coordinates (lecture 4)
€
≈−pcosθ
4πε0r2PV
r
VEr ∂
∂−= P
∂∂
−=P1 V
rET€
=−2pcosθ
4πε0r3
€
=−psinθ
4πε0r3
l
+q l/2
r-r+
P
r
-q
Er points towards the centre of the dipole
ET points towards decreasing .
Er change signwhen is greater than 90 degrees.
€
=−2pcosθ
4πε0r3
€
=−psinθ
4πε0r3
€
Er =−2pcosθ
4πε0r3
€
ET =−psinθ
4πε0r3
Electric Dipoles in Uniform Electric Fields
(Tipler 671-672)
-q
+q
pl
E qE
-qE
No Net Force
But Torque - rotates the dipole clockwise
Two equal and opposite forces whose lines of action do not coincide constitute a couple. The two forces always
have a turning effect, called a torque
€
=r × F
A torque is defined as the moment of a force
Mathematically (STMR)
Torque (of a couple) (Tipler – pages 309-311)
The resultant torque is:
( ) FdxFdxF =×−+×
The magnitude of the torque of a couple is calculated from
Fd=i.e. torque = one force perpendicular distance between forces
-q
+q
l
qE
-qE
Torque
sinpE
sinqlEsinlqEdqE
=
=×=×=
d
The torque tends to align p and E
In vector form:
€
r = rp ×
r E = pE sinθ
The direction of
p
E
An electric dipole of moment p is placed in a uniform external electric field. The dipole moment vector is in the positive y direction. The external electric field vector is in the positive x direction. When the dipole is aligned as shown in the diagram, the net torque is in the
A)positive x direction. B)positive y direction. C)negative x direction. D)positive z direction. E)negative z direction.
Electric potential energy of a dipole
+q
-q
E
Electric potential at +q is V+, the potential energy is qV+
Electric potential at -q is V-, the potential energy is -qV-
The total potential energy of the dipole is:
€
U = qV+ − qV− = q(V+ −V−) = q(E .d) = qlcosθ • E
= pE cosθ
Thus the P.E. of an electric dipole in an E-field is:
E.pcospEU −=−=
Minimum at = 0, maximum at = π,
and zero at = π/2
The electric dipole is like an electric version of
a compass
P
E
The potential energy of a magnetic dipole is
€
U = −r μ •
r B
What is the torque on the dipole for
the above configuration?
€
= pE sinθ
Cooking instructions:
Molecules with dipole moment,
Molecules that are mobile
H2O= 6.1 x 10-30 C m
Electric Dipole Moment of
Some Gas Molecules
HCl 1.1
HBr 0.8
H2O 1.8
SO2 1.6
N2O 0.2
NH3 1.5
•An electric dipole in an electric field experiences a torque:
•The potential energy for an electric dipole in an electric field E depends on the orientation of the dipole moment p with respect to the field:
E.pU −=
Ep ∧=
p = 0.02 e.nm
E = 3 × 103 N/C
Calculate
(a) the magnitude of the torque
(b) The potential energy
Review and Summary
•An electric dipole is a pair of electric charges of equal magnitude q but opposite sign, separated by a distance l
•The electric dipole moment is defined to have magnitude p = ql
•We calculated the E of an electric dipole at any position in space by a method far easier than using Coulomb’s law and superposition
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