EE381

Solutions of HW5 & Practice Problems

Solution (d)

Solution:

Solution

Notes: (i) R in Spherical coordinates is always positive.

(ii) There is a singularity at R = 0. The integral from z

= -4 m to +2 m needs to be carried out in 2 separate

integrals.

Since the field is in the radial direction described by Spherical

coordinates (spherical symmetry), the potential energy change

from z = -4 to 0 is exactly the same as that from z = 4 to 0.

To make it less confusing, the following integral is performed to

avoid using negative z:

V5.41818

ˆ18

ˆˆ18

ˆ

2

0

0

4

2

0 2

0

4 2

=

−−

−−=

⋅+⋅−= ∫∫

RR

dRR

dRR

VAB

zzzz

This is the same as moving from z = 4 to 2m:

V5.418

ˆ18

ˆ

2

4

2

4 2=

−−=⋅−= ∫

RdR

RV

ABzz