Midterm 1, Alternative Solution to Problem 20
Instructors: Korytov, Takano
PHY 2049, Spring 2014
February 10, 2014
(20) The electric field produced by the charged sphere is
E=
1 Q
,
4πε 0 r 2
which leads to the energy density
1
1
Q2
u = !0E 2 =
2
4
2
2 ( 4" ) ! 0 r
in the space surrounding the sphere. Integrate this to find the total
energy stored in the electric field in the space:
#
1
Q2 &
U = " u ! dV = " %
4" r 2 dr =
2
4 (
r '
R $ 2 ( 4! ) ! 0
)
(
)
Q2
dr
Q2 # 1&
!
=
%* (
8!" 0 "R r 2
8!" 0 $ r '
)
)
R
Q2
=
.
8!" 0 R
Here we use concentric shells of radius r and thickness dr as elements
of volume increments dV. Such shells have volume dV=4πr2⋅dr.