課程名稱︰電磁學上
課程性質︰系必修
課程教師︰陳正弦
開課學院：理學院
開課系所︰物理學系
考試日期（年月日）︰2009/01/13
考試時限（分鐘）：150
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
(NOTE: If the calculated quantity is a vector, you need to specify the
direction in addition to its magnitude.)
Part A: short questions(21 points)
(a) Find the volume and surface bound charge density distributions for a
→
uniformly polarized sphere with polarization Ｐ, assuming the polarization
direction is parallel to the z-axis.
(b) Find the volume and surface bound current density for a uniformly
→
magnetized sphere with magnetization Ｍ, assuming the magnetization
direction is parallel to the z-axis.
(c) Write down the continuity equation and describe its physical meaning.
→ → → →
(d) Among the four field vectors in electromagnetics, Ｅ, Ｄ, Ｂ, and Ｈ, name
the field vectors that we can create and control easily in our
laboratories.
(e) A phonograph record of radius R carries a uniform surface charge density σ
and it rotates at an angular frequency ω. Find its magnetic dipole moment.
(f) A uniformly charged sphere, of radius R and total charge Q, is center at
origin and spinning at a constant angular velocity ω about the z-axis.
Find the current density J at any point (r,θ,φ) within the sphere.
(g) Find the volume bound charge density ρb in a homogenous linear dielectric
characterized by an electric susceptibility χe with a free cjarge density
ρf embedded in the dielectric.
Part B: regular problems
1. (15 points) A metal sphere of radius a carrying a charge Q is surrounded,
out to radius b, by a linear dielectric material of permittivity ε. Find
→
(i) the electric field Ｅ in the region a < r < b and r > b, (ii) the
potential at the center, and (iii) the surface bound charge density σb at
the outer and the inner surface of the dielectric layer.
2. (15 points) Starting from the Biot-Savart law, calculate the magnetic field
Ｂ (i) at a distance s from a very long thin wire carrying a steady current
I, and (ii) at a distance z above the center of a circular loop of radius R
which carries a steady current I.
3. (15 points) An infinite long solenoid with n turns per unit length and
→
radius R carries a current I. Find (i) the magnetic field Ｂ and (ii) the
→
vector potential Ａ both inside and outside the solenoid.
4. (12 points) (i) A point charge Q is situated at a distance d above a thick
semi-infinite planar dielectric material of susceptibility χe. Calculate
the induced bound charge density σb at the surface of the dielectric. (ii)
→
Find the magnetic field Ｂ above and below an infinite surface on the xy
→ →
plane which carries a uniform surface current Ｋ = Kｘ.
5. (12 points) A long circular cylinder of radius R carries a magnetization
→ ︿
Ｍ = k*(s^2)*ψ, where k is a constant, s is the distance for the axis of
︿
the cylinder, and ψ is the usual azimuthal unit vector. Find (i) the
→ →
volume bound current density Jb and the surface bound current density Kb,
→
and (ii) the magnetic field Ｂ both inside and outside the cylinder.
6. (15 points) A coaxial cable consists of two very long cyclindrical metallic
tubes, separated by linear material of magnetic susceptibility χm. A
current I flows down the inner tube (with radius a) and returns along the
outer one (with radius b); in each case the current distribution is uniform
over the surface. In the region between the tubes, find (i) the
→ →
magnetization Ｍ, (ii) the volume bound current density Jb, and (iii) the
→
surface bound current density Kb at the inner and the outer surfaces of the
magnetic layer.
7. (20 points) (i) A sphere of homogeneous linear dielectric material
characterized by permittivity ε is placed in an otherwise uniform electric
→ →
field E0. Find the electric field Ｅ inside the sphere. (ii) By applying
the result obtained from (i) to a similar porblem in magnetostatics in
which a magnetic sphere characterized by permeability μ is placed in a
→ →
uniform magnetic field B0, find the magnetic field Ｂ inside the magnetic
material. (You can also do it by any method you know.)
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