Download Example: The Electric Dipole

10/26/2004
Example The Electric Dipole.doc
1/7
Example: The Electric
Dipole
Consider two point charges (Q1 and Q2), each with equal
magnitude but opposite sign, i.e.:
Q1 = Q
and Q2 = − Q
so Q1 = − Q2
Say these two charges are located on the z-axis, and separated
by a distance d .
z
The location of charge Q1 = -Q
is therefore specified by
′ d
position vector r1 = ˆaz
2
z =d/2
Q1
r1′
z =0
r2′
Q2
z =-d/2
The location of charge Q2 = -Q
is therefore specified by
−d
ˆaz
position vector r2′ =
2
The Electric Dipole
Jim Stiles
The Univ. of Kansas
Dept. of EECS
10/26/2004
Example The Electric Dipole.doc
2/7
We call this charge configuration an electric dipole. Note the
total charge in a dipole is zero (i.e.,Q1 + Q2 = Q − Q = 0 ). But,
since the charges are located at different positions, the
electric field that is created is not zero !
Q: Just what is the electric field created by an
electric dipole?
A: One approach is to use Coulomb’s Law, and
add the resulting electric vector fields from each
charge together.
However, let’s try a different approach. Let’s find
the electric potential field resulting from an
electric dipole. We can then take the gradient to
find the electric field !
Note that this should be relatively straightforward! We
already know the electric potential resulting from a single point
charge—the electric potential resulting from two point charges
is simply the summation of each:
V ( r ) = V1 ( r ) +V2 ( r )
where the electric potential V1 ( r ) , created by charge Q1, is:
V1 ( r ) =
Jim Stiles
Q1
4πε 0 r − r1
=
Q
4πε 0 r − d 2 ˆaz
The Univ. of Kansas
Dept. of EECS
10/26/2004
Example The Electric Dipole.doc
3/7
and electric potential V2 ( r ) , created by charge Q2 , is:
V2 ( r ) =
Q2
4πε 0 r − r2
=
−Q
4πε 0 r + d 2 ˆaz
Therefore the total electric potential field is:
V (r ) =
Q
4πε 0 r − d 2 ˆaz
−
Q
4πε 0 r + d 2 ˆaz
⎛
Q ⎜
1
1
=
−
4πε 0 ⎜⎜ r − d ˆaz
r + d ˆaz
2
2
⎝
⎞
⎟
⎟⎟
⎠
If the point denoted by r is a significant distance away from
the electric dipole (i.e., r >> d ), we can use the following
approximations:
1
1 d cosθ 1 d cosθ
≈
+
= +
r
2r
r
2r2
r − d 2 ˆaz
1
1 d cosθ 1 d cosθ
≈
−
= −
d
r
2
r
r
2r2
r + 2 ˆaz
where r and θ are the spherical coordinate variables of the
point denoted by r .
Jim Stiles
The Univ. of Kansas
Dept. of EECS
10/26/2004
Example The Electric Dipole.doc
4/7
Therefore, we find:
⎛
Q ⎜
1
1
−
V (r ) =
4πε 0 ⎜⎜ r − d ˆaz
r + d 2 ˆaz
2
⎝
⎞
⎟
⎟⎟
⎠
⎛ ⎛ 1 d cosθ ⎞ ⎛ 1 d cosθ ⎞ ⎞
⎜⎜ +
⎟−⎜ −
⎟⎟
2
2
r
2r
r
2r
⎝
⎠
⎝
⎠⎠
⎝
Q d cosθ
=
4πε 0
r2
=
Q
4πε 0
Note the result. The electric potential field produced by an
electric dipole, when centered at the origin and aligned with
the z-axis is:
V (r ) =
Qd cosθ
4πε 0 r 2
Q: But the original question was, what is the electric
field produced by an electric dipole?
A: Easily determined! Just take the gradient of the
electric potential function, and multiply by -1.
Jim Stiles
The Univ. of Kansas
Dept. of EECS
10/26/2004
Example The Electric Dipole.doc
5/7
E ( r ) = −∇V ( r )
⎛ Qd cosθ ⎞
= −∇ ⎜
2 ⎟
4
r
πε
0
⎝
⎠
−Qd ⎡
d ⎛ 1 ⎞
1 d ( cosθ ) ⎤
ˆ
ˆaθ ⎥
=
+
a
cos
θ
⎜ 2⎟ r
3
⎢
dr ⎝ r ⎠
r
dθ
4πε 0 ⎣
⎦
=
− Q d ⎡ ⎛ -2 cosθ
⎢
4πε 0 ⎣ ⎜⎝ r 3
sinθ ⎤
⎞
ˆ
ˆaθ ⎥
−
a
r
⎟
3
r
⎠
⎦
The static electric field produced by an electric dipole, when
centered at the origin and aligned with the z-axis is:
E (r ) =
Qd 1
⎡ 2cosθ ˆar + sinθ ˆaθ ⎤⎦
3 ⎣
4πε 0 r
Yikes! Contrast this with the electric field of a single point
charge. The electric dipole produces an electric field that:
1) Is proportional to r-3 (as opposed to r-2).
2) Has vector components in both the ˆar and âθ directions
(as opposed to just ˆar ).
In other words, the electric field does not point away from
the electric dipole!
Jim Stiles
The Univ. of Kansas
Dept. of EECS
10/26/2004
Example The Electric Dipole.doc
6/7
The electric potential produced by an electric dipole looks like:
4
2
y
0
−2
−4
−4
−2
0
x
2
4
4
2
y
0
0.4
0.2
0 volts
− 0.2
− 0.4
−2
−4
−4
−2
0
x
2
4
Jim Stiles
The Univ. of Kansas
Dept. of EECS
10/26/2004
Example The Electric Dipole.doc
7/7
And the electric field produced by the electric dipole is:
4
y
2
0
−2
−4
−4
−2
0
x
2
4
−4
−2
0
x
2
4
4
y
2
0
−2
−4
Jim Stiles
The Univ. of Kansas
Dept. of EECS