Activity 2: Solution for electric potential due
to a ring
Find the electrostatic potential in all space due to a ring
with total charge
Q
and radius
R
V
(
vector
r
) =
1
4
πǫ
0
N
summationdisplay
i
=1
q
i

vector
r

vector
r
i

(1)
For a ring of charge this becomes
V
(
vector
r
) =
integraldisplay
ring
1
4
πǫ
0
λ
(
vector
r
′
)

dvector
r
′


vector
r

vector
r
′

(2)
where
vector
r
denotes the position in space at which the potential is measured and
vector
r
′
denotes the position of the charge.
In cylindrical coordinates,

dvector
r
′

=
R dφ
′
, where
R
is the radius of the ring.
Thus,
V
(
vector
r
) =
1
4
πǫ
0
2
π
integraldisplay
0
λ
(
vector
r
′
)
R dφ
′

vector
r

vector
r
′

(3)
Assuming constant linear charge density for a ring with charge Q and radius
R,
λ
(
vector
r
′
) =
Q
2
πR
Thus,
V
(
vector
r
) =
1
4
πǫ
0
Q
2
π
2
π
integraldisplay
0
dφ
′

vector
r

vector
r
′

(4)
Since
vector
r
and
vector
r
′
are not necessarily in the same direction, we cannot simply
leave

vector
r

vector
r
′

in curvilinear coordinates and integrate directly. One solution
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 none
 Power Series, Taylor Series, Charge, Electric Potential, Complex number, Polar coordinate system

Click to edit the document details