課程名稱︰電磁學上
課程性質︰物理系大二必帶
課程教師︰石明豐
開課學院：理學院
開課系所︰物理學系
考試日期（年月日）︰108年11月6日
考試時限（分鐘）：175分
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
註：為方便表示數學符號，部分符號以LaTeX碼輸入。
1.Prove the divergence theorem for any vector field \vec{A}(u,v,w)=A_u\vec{u}
+A_v\vec{v}+A_w\vec{w} in a general coordinate (u,v,w), in which \hat{u}⊥
\hat{v}⊥\hat{w} and d\vec{l}=fdu\hat{u}+gdv\hat{v}+wdw\hat{w}. (5pts)
2.The integral \vec{a}≡∫_S d\vec{a} is the vector area of the surface S. (5
pts)
(a) Find the vector area of a hemispherical bowl of radius R.
(b) Show that \vec{a}≡0 for any closed surface.
(c) Show that \vec{a} is the same for all surfaces sharing the same boundary.
(d) Show that \vec{a}=1/2∫\vec{r}×d\vec{l} where the integral is around the
boundary line.
(e) Show that ∮(\vec{c}\cdot\vec{r})d\vec{l}=\vec{a}×\vec{c}.
3.An uncharged spherical conductor centered at the origin has a cavity of some
shape. Somewhere inside the cavity is a charge q. Show that the field outsi-
de the spherical conductor is \vec{E}=1/(4πε0)q/r^2\hat{r}. (5pts)
4.In a vacuum diode, electrons are "boiled" off a hot cathode, at potential
zero and position x=0, and accelerated across a gap to the anode, which is
held at positive potential V_0. The cloud of moving electrons within the gap
(called space charge) quickly builds up to the point where it reduces the
field at the surface of the cathode to zero. From then on, a steady current I
flows between the plates.
Suppose the plates are large relative to the separation (A>>d) so that edge
effects can be neglected. Then V, ρ, and v (the speed of the electrons) are
functions of x alone. (10pts)
(a) Write Poisson's equation for the region between the plates. (1pt)
(b) Assuming the electrons start from rest at the cathode, what is their
speed at point x, where the potential is V(x)? (1pt)
(c) In the steady state, I is independent of x. What, then, is the relation
between ρ and v. (2pts)
(d) Use (a)~(c) to obtain a differential equation for V, by eliminating ρ
and v. (2pts)
(e) Solve this equation for V as a function of x, V_0, and d. Plot V(x), and
compare it to the potential without space-charge. Also, find ρ and v as
functions of x. (2pts)
(f) Show that I=KV_0^{3/2} and find K. (2pts)
5.Prove that the field is uniquely dterminded when the charge density ρ is
given and either V or the normal derivative \frac{\partial V}{\partial n} is
specified on each boundary surface. Do not assume that boundaries are conduc-
tors or that V is constant over any given surface.
6.Solve Laplace's equation by separation of variables in cylindrical coordinat-
es for a general solution, assuming there is no dependence on z. (5pts)