the electric dipole -q +q d an electric dipole consists of two equal and opposite charges (q and -q...
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The Electric Dipole
-q
+qd
An electric dipole consists of two equal and opposite charges (q and -q ) separated a distance d.
The Electric Dipole
-q
+qd
We define the Dipole Moment p magnitude = qd,
p direction = from -q to +q
p
The Electric Dipole
-q
+qE
q
d
Suppose the dipole is placed in a uniform electric field (i.e., E is the same everywhere in space).
Will the dipole move ??
The Electric Dipole
-q
+qE
q
d
What is the total force acting on the dipole?
The Electric Dipole
-q
+q
F-
F+E
q
d
What is the total force acting on the dipole?
The Electric Dipole
-q
+q
F-
F+E
q
d
What is the total force acting on the dipole? Zero, because the force on the two charges cancel: both have magnitude qE. The center of mass does not accelerate.
The Electric Dipole
-q
+q
F-
F+E
q
d
What is the total force acting on the dipole? Zero, because the force on the two charges cancel: both have magnitude qE. The center of mass does not accelerate.
But the charges start to move. Why?
-q
+q
F-
F+E
q
d
What is the total force acting on the dipole?
Zero, because the force on the two charges cancel: both have magnitude qE.
The center of mass does not accelerate.
But the charges start to move (rotate). Why?
There’s a torque because the forces aren’t colinear and aren’t acting exactly at the center of mass.
-q
+qF-
F+
q
d d sin q
The torque is: t = (magnitude of force) (moment arm)
t = (2qE)(d sin /2)=q qE dsin q
and the direction of t is (in this case) into the page
but we have defined : p = q d and the direction of p is from -q to +q
Then the torque can be written as: t = pxE t = p E sin q
-q
+q
q
dp
E
q
t = qE dsin q
Field Due to an Electric Dipole at a point x straight out from its midpoint
Electric dipole momentp = qd
E+
d
+q
-q
x
l
E-
E
qX
Y
Calculate E as a function of p, x,and d
E+
d
+q
-q
x
l
E-
E
qX
Y
You should be able to find E at different points around a dipole where symmetry simplifies the problem.
Torque on a Dipole in an Electric Field(another version of the derivation)
sinsin)(sin FdxdFFx
qdp
qEF
p E
sin|| pE
sinqEdF
A Dipole in an Electric Field
)cos(cos]cos[
sinsin
fi
if
pEpE
dpEdpEdUU
f
i
f
i
f
i
f
i
ddW
cospEU
0iU
EpU
90i
4. In which configuration, the potential energy of the dipole is the greatest?
E
a b c
d e
Example: Electric Field of a Dipole
])2
1()2
1[(4
)2/(4
1
)2/(4
1
4
1
4
1
222
0
20
20
20
20
z
d
z
d
z
q
dz
q
dz
q
r
q
r
qEEE
We can use the binomial expansion on these two terms
z
Lets find the E field here at a distance Z from the dipole center
For any power of n, the binomial (a + x) can be expanded two ways
or
This first expression is particularly useful when x is very much less than a so that the first few terms provide a good approximation of the value of the expression. There will always be n+1 terms
The Binomial Expansion (a +x)n
30
30
20
2
1
2
12
4
z
pE
qdp
z
qd
z
d
z
qE
z
d
z
d
z
d
z
d
z
d 2...)]
)!1(2
21(...)
)!1(2
21[(])
21()
21[( 22
E ~ 1/z3
E =>0 as d =>0
Valid for “far field”