Electromagnetic Fields and Waves 1, Final
Lecturer : Sanghoek Kim
1:30-2:45pm, June 16, 2017
Name Sanghoek Kim, Sanghoek Kim
1
Problem 1 (15 pts)
(Columb’s force) The Thomson model of a hydrogen atom consists a sphere of positive charge
and an electron as a point charge within the sphere. The electron has a negative charge of
−e and the total positive charge equals the electronic charge Q = e. Denoting the radius of
the sphere as R, the charge density is uniform inside the sphere as ρ0 = e/(4πR3 /3). When
an electron is at a distance r from the center of the sphere of positive charge, show that the
electron is being attracted inside the sphere of positive charge (r < R) with a force
F =
e2 r
4π0 R3
Problem 2 (15 pts)
(Columb’s force) In Thomson model of Problem 1, if an electron is outside the sphere of
positive charge, i.e., r > R, what is the attracting force as a function of distance r?
2
Problem 3 (15 pts)
(Gauss’s law) Assume we have an electric dipole at the origin along the ẑ-direction. In class,
we learned that the electric field due to the dipole can be obtained by
E=
Qd 2
cos
θr̂
+
sin
θ
θ̂
,
4π0 r3
where Q is the charge of dipole and d is the distance between the positive and the negative charge in the dipole. Considering a large sphere with radius R surrounding the dipole,
evaluate
Z
E · dS
S
over the surface of the sphere S.
3
Problem 4 (30 pts)
(Gauss’s law and electric potential)A positive point charge Q is at the center of a spherical
dielectric shell of an inner radius Ri and an outer radius Ro . The dielectric constant of the
shell is r = 5.
Dielectric
shell
(a) Determine and plot D as a function of the radial distance r.
(b) Determine and plot E as a function of the radial distance r.
(c) Determine and plot V as a function of the radial distance r.
4
Problem 5 (15 pts)
(Capacitance) We have two concentric conducting shell of spheres. The radius of inner and
outer sphere are a and b, respectively. The dielectric between the two conductors has a
permittivity of . Derive that the capacitance between two spheres is
C=
4π
.
1/a − 1/b
5