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課程名稱︰電磁學
課程性質︰必修
課程教師︰王立民
開課學院：理學院
開課系所︰物理系
考試日期（年月日）︰100. 10. 27
考試時限（分鐘）：120分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
notation:
a. The arrow of the vector is denoted by "bar"; e.g., the position vector
_
is denoted r. If the vector is an unit vector, "bar" will be replaced by
"hat", ^.
_
b. л denotes the separation vector from the source pointing to the observer.
^
1. (a) Find the components of the unit vector n perpendicular to the plane
shown in Fig.1.
(b) Also find the vector area of the plane shown in Fig.1.
_ _ _ _
(c) For the vector function v=r, calculate the integral ∮v╳dl,
where the integral is around the boundary line of the plane shown
in Fig.1.
_
(d) Check the Stoke's Theorem for v over the triangular surface with
corners at (1,0,0), (0,1,0), and (0,0,1) as shown in Fig.1.
Fig.1
The portion of plane x+y+z=1 in the first octant
_
2. Check the gradient theorem of ∫(▽T)dτ=∮Tda, for the scalar function
T=r^2, using the volume of a hemisphere with radius of R as shown in Fig.2.
Note. The gradient formula in spherical coordinate is given.
Fig.2
A hemisphere with radius R
3. A hollow spherical shell carries charge density ρ=kr^(-2) in the region
a≦r≦b.
(a) Find the total charge Q.
(b) Find the electrostatic energy W stored.
(c) Find the net force that southern hemispherical shell exerts on the
northern hemispherical shell, in terms of a, b and k.
(d) If the same charge Q redistributing on a conducting hollow spherical
shell with the same inner and outer radii of a and b. Again find the
net force that southern hemispherical shell exerts on the northern
hemispherical shell.
_ ^
[Hint: the formula of the force per unit area f=σ^2/2ε_0n]
4. A solid conical ice-cream cone carries a uniform volume charge density
ρ. The height of the cone is h, as is the radius of the top. Find the
potential at the vertex point a.
5. If the actual force of interaction between two point charges is found
to be
_ 1 q_1q_2 л ^
F= ------- --------(1+---) exp[-л/λ]л,
4πε_0 л λ
where λ is a huge-number constant. This makes a new scalar potential
admitted the electric field.
(a) Find the potential of a point charge with using infinity as the
reference point.
(b) Using the result in (a), find the potential at a distance z above
the center of a circular disk with uniform surface charge σ
as seen in Fig.3.
(c) Show that the slab looks like a point charge for z>>R, using the
approximation: exp[x]~1+x and (1+x)^n~1+nx for x<<1.
Fig.3
A circular disk of radius R and surface charge density σ. P:(0,0,z),
regarding the center of the disk as origin and the disk as x-y plane.
6. Consider two concentric spherical shells, of radii a and b. Suppose the
inner one carries a charge 2q, and the outer one a charge -q(both uniformly
distributed). Calculate the energy of this configuration in two different
ways:
(a) Use the energy density ε_0E^2/2, and
_ _
(b) the superposition principle: W_{tot}=W_1+W_2+ε∫E_1。E_2dτ.
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